Eleonore Coppens

Didier Desmaële

Sébastien Timothée Naret

Sophie Feuillastre

Grégory Pieters

Catherine Cailleau

Jean-Louis Paul

Bastien Prost

Audrey Solgadi

Sébastien Garcia-Argote

Jean-Philippe Michel

Magali Noiray

Patrick Couvreur

Simona Mura
email: simona.mura@universite-paris-saclay.fr

Gemcitabine lipid prodrug nanoparticles: switching the lipid moiety and changing the fate in the bloodstream

Keywords: Lipoproteins, drug delivery, nanoparticles, lipid prodrugs, gemcitabine

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Introduction

The ability of lipoproteins (LPs) to transport drugs has been described as early as 1981 [START_REF] Counsell | Lipoproteins as potential site-specific delivery systems for diagnostic and therapeutic agents[END_REF][START_REF] Firestone | Selective delivery of cytotoxic compounds to cells by the LDL pathway[END_REF][START_REF] Gal | Low-density lipoprotein as a potential vehicle for chemotherapeutic agents and radionucleotides in the management of gynecologic neoplasms[END_REF][START_REF] Masquelier | Low-density lipoprotein as a carrier of antitumoral drugs: in vivo fate of drug-human low-density lipoprotein complexes in mice[END_REF].

Unlike most synthetic nanocarriers, LPs provide biocompatibility, biodegradability and nonimmunogenicity [START_REF] Thaxton | Lipoproteins and lipoprotein mimetics for imaging and drug delivery[END_REF]. In addition, their structure makes them suitable for drug delivery thanks to: (i) a small size ranging from 5 nm (high-density lipoproteins (HDL))

to 100 nm (very low-density lipoproteins (VLDL)) preventing rapid clearance by the cells of the reticuloendothelial system, (ii) a hydrophobic core that could transport hydrophobic drugs, and (iii) a more hydrophilic outer envelope for amphiphilic molecules loading [START_REF] Davis | Chapter 17 -Structure, assembly and secretion of lipoproteins[END_REF].

On the other hand, lipoproteins are attractive drug carriers to treat, at least theoretically, any pathological situation that implies their tissue accumulation, in particular neoplastic diseases. Indeed, due to higher cholesterol and cholesteryl esters requirements compared to healthy tissues, most tumors overexpress the low-density lipoprotein (LDL) receptor (LDLR) [START_REF] Firestone | Low-density lipoprotein as a vehicle for targeting antitumor compounds to cancer cells[END_REF][START_REF] Vitols | Uptake of low-density lipoprotein by malignant cells--possible therapeutic applications[END_REF] and/or the HDL scavenger receptor type B1 (SR-B1) [START_REF] Mooberry | Targeting the SR-B1 Receptor as a Gateway for Cancer Therapy and Imaging[END_REF]. Accordingly, along with the fact that VLDL and chylomicrons suffer from a low plasma half-life [START_REF] Thaxton | Lipoproteins and lipoprotein mimetics for imaging and drug delivery[END_REF], LDL and HDL have been the predominant lipoproteins investigated for selective drug delivery to tumors [START_REF] Mahmoudian | Natural low-and high-density lipoproteins as mighty bio-nanocarriers for anticancer drug delivery[END_REF].

Applications in oncology have been largely documented in the past years, but similar strategies have been pursued to target other LP-accumulating tissues such as atherosclerotic plaques (Chen et al., 2020a;[START_REF] Tabas | Subendothelial Lipoprotein Retention as the Initiating Process in Atherosclerosis[END_REF] or organs such as the brain (Chen et al., 2020b;[START_REF] Dehouck | A new function for the LDL receptor: transcytosis of LDL across the blood-brain barrier[END_REF].

In this context, the in vitro loading of isolated lipoproteins with various drug molecules has been proposed. It includes conjugation of the drug to the apolipoprotein moieties, its insertion in the phospholipid outer layers or its encapsulation into the hydrophobic core of the LP (Zhu and Xia, 2017). Lipoproteins can be obtained from fresh donor plasma through a variety of separation techniques according to their size, density or apolipoprotein content (e.g., ultracentrifugation, gel permeation, size exclusion chromatography, affinity chromatography, immunocapture assays) [START_REF] Busbee | Separation and detection of lipoproteins in human serum by use of size-exclusion liquid chromatography: a preliminary report[END_REF][START_REF] Havel | The distribution and chemical composition of ultracentrifugally separated lipoproteins in human serum[END_REF][START_REF] Tadey | Chromatographic techniques for the isolation and purification of lipoproteins[END_REF][START_REF] Wasan | A comparison of step-gradient and sequential density ultracentrifugation and the use of lipoprotein deficient plasma controls in determining the plasma lipoprotein distribution of lipid-associated nystatin and cyclosporine[END_REF][START_REF] Watanabe | Proteomic profiling following immunoaffinity capture of highdensity lipoprotein: Association of acute-phase proteins and complement factors with proinflammatory high-density lipoprotein in rheumatoid arthritis[END_REF]. However, the need to isolate LP from fresh plasma is a major burden for their use in routine clinical practice. All these methods are indeed low yielding and time consuming, not to mention variations between plasma donors making batch reproducibility difficult. The inherent risk they pose in terms of blood contamination and LP denaturation during the disruptive extraction and loading processes represents another concern with this approach [START_REF] Busatto | Lipoprotein-based drug delivery[END_REF].

Formulation of lipoprotein-like carriers has emerged as an alternative strategy. It relies on the in vitro assembly of commercially available lipids in proportions close to that found in natural lipoproteins, therefore bypassing the LP-harvesting step [START_REF] Corbin | Mimicking nature's nanocarrier: synthetic low-density lipoprotein-like nanoparticles for cancer-drug delivery[END_REF][START_REF] Huang | Learning from biology: synthetic lipoproteins for drug delivery[END_REF][START_REF] Pittman | Synthetic high density lipoprotein particles. Application to studies of the apoprotein specificity for selective uptake of cholesterol esters[END_REF]. Going beyond simple mimicking, surface-functionalization of these artificial LPs with various ligands has even made it possible to reroute them from their natural pathways and promote their accumulation in target tissues overexpressing other specific receptors [START_REF] Chen | Ligand Conjugated Low-Density Lipoprotein Nanoparticles for Enhanced Optical Cancer Imaging in Vivo[END_REF][START_REF] Corbin | Near-infrared fluorescent imaging of metastatic ovarian cancer using folate receptor-targeted high-density lipoprotein nanocarriers[END_REF][START_REF] Han | Lipoprotein-Inspired Nanocarrier Composed of Folic Acid-Modified Protein and Lipids: Preparation and Evaluation of Tumor-Targeting Effect[END_REF][START_REF] Meng | A novel LDL-mimic nanocarrier for the targeted delivery of curcumin into the brain to treat Alzheimer's disease[END_REF][START_REF] Zhang | HDL-mimicking peptide-lipid nanoparticles with improved tumor targeting[END_REF]Zheng et al., 2005). Although promising, this approach remains quite laborious since it requires the opportune formulation of a variety of lipids (i.e., phospholipids, cholesterol, cholesterol esters, triglycerides) and apolipoproteins. It also suffers from the fact that these apolipoproteins have to be extracted from human serum, and are difficult to handle because of their poor water-solubility and large size [START_REF] Corbin | Mimicking nature's nanocarrier: synthetic low-density lipoprotein-like nanoparticles for cancer-drug delivery[END_REF].

Overall, the complex production and storage of both the plasma-derived and the artificial lipoproteins, along with the aforementioned concerns, have hindered any industrial transposition and clinical application of LP-based drug delivery systems to date.

It is therefore evident that being able to trigger a drug-lipoprotein interaction directly in the blood circulation would be an easier way to exploit the natural transport capacity of lipoproteins.

We have recently discovered that the conjugation of gemcitabine, a hydrophilic anticancer drug, to the lipophilic squalene chain resulted in a preferential accumulation of the prodrug into lipoproteins, while the free drug did not show any affinity for the latter (Sobot et al., 2017a). Following this interaction, the LPs acted as indirect carriers transporting the squalenegemcitabine (SQGem) prodrug toward high LP-accumulating cancer cells, leading to a higher therapeutic efficacy compared to free gemcitabine (Sobot et al., 2017b).

As lipoproteins are the natural transporters of cholesterol in the blood circulation, the question remains whether the conjugation of gemcitabine (as a model drug) to cholesterol could provide an even better interaction with endogenous lipoproteins.

Therefore, in the present study, we have explored the fate of nanoparticles (NPs) made of the cholesterol-gemcitabine (CholGem) conjugate both in vitro after incubation with human blood and in vivo after intravenous administration to rodents. Further information was then obtained by directly analyzing the interactions of these NPs with isolated blood components (i.e., LDL, HDL, albumin). The conjugation to the cholesterol resulted in the formation of a prodrug and the activity of the gemcitabine is recovered only after its release from the conjugate following the in vivo enzymatic cleavage of the carbamate bond [START_REF] Ghosh | Organic Carbamates in Drug Design and Medicinal Chemistry[END_REF][START_REF] Rautio | Prodrugs: design and clinical applications[END_REF]. However, the assessment of a potential therapeutic activity of the CholGem NPs was not within the scope of this work, which was voluntarily focused on the evaluation of interactions with the components of biological media.

Altogether, these results contribute to shed light on what could drive the in vivo fate of NPs made of lipid prodrugs.

Materials and methods

Materials

Regular analytical grade and deuterated solvents were supplied by Carlo Erba (France) and Eurisotop (France), respectively. Tetrahydrofuran (THF) was distilled from sodium/benzophenone ketyl. Pyridine, triethylamine and dimethylformamide (DMF) were distilled from calcium hydride under a nitrogen atmosphere. All reactions involving air-or water-sensitive compounds were routinely conducted in glassware which was flame-dried under a positive pressure of nitrogen. Gemcitabine (Gem) base was purchased from Carbosynth Ltd (UK). Squalene (SQ), cholesteryl chloroformate, ethyl chloroformate, pyridine, triethylamine, sodium hydrogen carbonate, hydrogen chloride, sodium chloride and sodium bromide (NaBr) were purchased from Sigma-Aldrich (France). Chemicals obtained from commercial suppliers were used without further purification. Deionized and filtered MilliQ water was produced using a water purification system (Millipore, France).

Chemical characterization

Infrared spectra (IR) were obtained as solid or neat liquid on a Fourier transform Bruker Vector 22 spectrometer. Only significant absorptions are listed. Optical rotations were measured on a PerkinElmer 241 polarimeter at 589 nm. The 1 H and 13 C NMR spectra were recorded on a Bruker Avance 300 spectrometer (300 and 75 MHz for 1 H and 13 C, respectively). Recognition of methyl, methylene, methine, and quaternary carbon nuclei in 13 C NMR spectra rests on the J-modulated spin-echo sequence. Mass spectra (MS) and high resolution mass spectra (HRMS) were recorded using electrospray ionization (ESI) on an LTQ Velos Pro mass spectrometer (ThermoScientific, Germany). Analytical thin-layer chromatography was performed on Merck silica gel 60F 254 glass pre-coated plates (0.25 mm layer). Column chromatography was performed on Merck silica gel 60 (230-400 mesh ASTM). Semi-preparative high-performance liquid chromatography (HPLC) was performed using a Phenyl 5 µm XBridge Prep column 10 x 150 mm (PN 186003273, batch 121331521, Waters) with a cell Berthold YG50-S5P. Counting of radiolabeled prodrugs after synthesis was performed on a PerkinElmer Tricarb 2910 TR liquid scintillation analyzer equipped with automatic external standardization. Samples were prepared with the Ultima Gold TM scintillation cocktail in 10 mL super polyethylene vials (PerkinElmer).

NP preparation and characterization

NPs were prepared by the nanoprecipitation solvent evaporation technique. Briefly, conjugates were dissolved at 2 mg.mL -1 in EtOH. The resulting organic solution (500 µL) was added dropwise to a stirred solution of MilliQ water (1 mL). The organic solvent was then removed under reduced pressure using a rotary evaporator to reach a 1 mg.mL -1 aqueous dispersion of NPs. 3 H-SQGem and 3 H-CholGem NPs were prepared in a similar manner. An appropriate volume of the radiolabeled conjugate was mixed with the cold one prior to the nanoprecipitation step in water, in order to obtain a NP dispersion with a final activity of 40 µCi.mL -1 and 50 µCi.mL -1 for in vitro and in vivo studies, respectively. Dextrose (5% m/v) was added to the final formulations to ensure isotonic conditions. The NP mean diameter was measured by dynamic light scattering with a Nano ZS (173° scattering angle, Malvern Panalytical, France) at 25 °C and monitored over a two-week period at room temperature. The zeta potential of NPs was measured at 25 °C after dilution with 1 mM NaCl solution by applying the Smoluchowski equation and using the same apparatus. All measurements were performed at least in triplicate.

Cryogenic transmission electron microscopy (cryo-TEM)

The morphology of SQGem and CholGem NPs was examined by cryo-TEM. Briefly, 5 µL of NP dispersion (1 mg.mL -1 ) was deposited onto a 200-mesh holey carbon copper grid (TedPella Inc., USA) and flash-frozen in liquid ethane cooled at liquid nitrogen temperature.

Cryo-TEM images were acquired on a JEOL 2200FS energy-filtered (20 eV) field emission gun electron microscope (JEOL, USA) operating at 200 kV using a Gatan ssCCD 2048 × 2048 pixels (Gatan Inc., USA).

Atomic force microscopy (AFM)

AFM experiments were performed in liquid conditions on mica substrates with surface area of 1.5 x 1.5 cm 2 . Because of its plate-like structure, composed of an octahedral alumina sheet sandwiched by two tetrahedral silicate sheets, the cleavage of muscovite mica surface reveals a molecular smooth negatively charged surface. For experiments using pre-treated mica, the latter was immersed in a fresh 0.1 g.L -1 poly-L-lysine aqueous solution for 30 min before rinsing with water. 1 mL of NP dispersion (10 µg.mL -1 in MilliQ water) was deposited during 2 h onto the mica surface. AFM experiments were performed using the Nanowizard 3 Ultra Speed (JPK Instruments, Germany) installed on an air-buffered table coupled to a dynamic anti-vibration device and enclosed in an acoustic box. Imaging of the surface morphology was performed in water in Hyper Drive mode (JPK Instruments), with gold-coated silicon cantilever of 24  0.5 N.m -1 spring constant and 140  2 kHz resonance frequency. The pyramid-shaped tips had a radius of curvature less than 20 nm. A free amplitude oscillation of 10 nm with 0.67 nm setpoint in amplitude and phase setpoints ranging between 0.9° and 1.2° was chosen, allowing the best resolution of the imaged surface. Images were taken at scan rate of 1 or 0.5 Hz for large scale pictures. Image processing (flatten, plane fit, edge, and hole detection) was performed with the JPK Data Processing software (JPK Instruments). At least three different areas of each sample were scanned, and representative images are presented.

Surface Plasmon Resonance (SPR)

Interaction analyses were performed on a Biacore T100 instrument using CM5 Series S sensor chips (Cytiva, France). The activation of the carboxymethylated dextran matrix was performed with a 1/1 ratio mixture of 0.4 M N-ethyl-N'-(3-diethylaminopropyl)-carbodiimide (EDC)/0.1 M N-hydroxysuccinimide (NHS) (GE Healthcare, France) for 7 min. Solutions of either human serum albumin (HSA) (A3782, Sigma-Aldrich) or rat serum albumin (RSA) (A6414, Sigma-Aldrich) at 20 µg.mL -1 in 10 mM acetate buffer (pH 4.5) were injected over the activated surface at 5 µL.min -1 during 7 min. The average immobilization level was of 660 resonance units (RU) for HSA and 620 RU for RSA. A solution of ethanolamine hydrochloride (1 M, pH 8.5) was injected during 7 min to block the unreacted sites of the sensing surfaces. For reference, an additional blank flow channel was prepared in accordance with the same process but without injecting the albumin over the surface. Binding capacities of the functional surfaces were tested with 3 min injections at 10 µL.min -1 of SQGem or CholGem NPs (concentration range: 17.5-560 µM in MilliQ water). An additional washing step with EtOH was added to prevent lipids from sticking to the fluidic system. After each cycle, the sensing surfaces were regenerated by injection of 70% EtOH (30 s, 30 µL.min -1 ), followed by a 10 mM NaOH solution (60 s, 30 µL.min -1 ). All experiments were carried out in duplicate at 37 °C in Dulbecco-Phosphate Buffered Saline (PBS) running buffer.

Plasma lipid and protein profile determination

Fresh citrated blood of healthy volunteers was obtained from the Etablissement Français du Sang (EFS, Paris, France, Agreement n°17/EFS/031). Lipid composition (total cholesterol, triglycerides, LDL-C, HDL-C, ApoB, ApoA, Lip(a)) and albuminemia were assessed at the Georges Pompidou European Hospital (France) using standard laboratory procedures. Molar concentrations of LDL and HDL in the plasma of each blood bag were determined as previously reported (Sobot et al., 2017a) taking into account the molecular weight of LDL and HDL (2.3 x 10 6 and 1.8 x 10 5 Da respectively), and the percentage of apolipoprotein (Apo) present in each of them (21% of ApoB and 50% of ApoA in LDL and HDL, respectively) [START_REF] Ohnishi | Frontal analysis of drug-plasma lipoprotein binding using capillary electrophoresis[END_REF]. All blood samples had normal lipid and albumin concentrations.

Transmission electron microscopy (TEM)

A physical mixture of NP aqueous dispersion (final concentration 0.74 mM) and LPs in phosphate buffer (final concentration 0.1 µM) was incubated for 5 min at 37 °C in a thermoshaker (400 rpm) immediately before grid preparation. Five microliters of the mixture was then deposited on Formvar/carbon 400 mesh copper grid. After 5 min at room temperature, the remaining drop was blotted with a filter paper. Then, a drop of 0.22 µm-filtered 2% uranyl acetate solution was added for 30 s for negative staining and the excess of volume was removed with a blotting filter paper. The grids were then observed with a JEOL 1400 120 kV electron microscope operating at 80 kV at a nominal magnification of 5,000-40,000. Digital images were directly recorded on a CCD postcolumn high-resolution (11 MegaPixel) high-speed camera (SC1000 Orius, Gatan Inc.) using the Digital Micrograph image acquisition and processing software (Gatan Inc.). Images of isolated NPs (0.74 mM) and LP dispersion (0.1 µM) were acquired following the same protocol. All samples were prepared at a constant water/phosphate buffer ratio (57/1 v/v).

Incubation of radiolabeled NPs with human blood and LP separation

3.9 mL of whole human blood was incubated with 100 µL of 3 H-SQGem or 3 H-CholGem NPs (final volumetric activity in blood: 1 µCi.mL -1 , final NP concentration in blood: 39 µM and 37 µM for SQGem and CholGem, respectively) for 5 min at 37 °C. Following a first centrifugation step to remove the red blood cells (3,000 g, 15 min, 15 °C), plasma components were separated by NaBr gradient density ultracentrifugation [START_REF] Cassidy | Plasma lipoprotein distribution of liposomal nystatin is influenced by protein content of high-density lipoproteins[END_REF]Sobot et al., 2017a). All NaBr solutions and plasma were cooled at 4 °C for 1 h prior to ultracentrifuge tube preparation. 1 mL of 1.006 g.mL -1 NaBr solution was placed at the bottom of a 4 mL Ultraclear ultracentrifuge tube (344062, Beckman Coulter), and was then underlaid with 1 mL of increasing density NaBr solutions (1.063 and 1.21 g.mL -1 ). Finally, 1 mL of plasma, whose density was adjusted to 1.25 g.mL -1 with NaBr, was carefully added at the bottom of the tube. The tubes were placed on a SW 60Ti swinging bucket rotor (Beckman Coulter) and ultracentrifuged (164,326 g, 18 h, 15 °C) using an Optima™ LE-80K Ultracentrifuge (Beckman Coulter). At the end of the centrifugation cycle, twenty fractions of 200 µL each were collected from the top to the bottom of the tube using single use 1 mL insulin syringes (324891, BD). The density of each fraction was then measured, and the radioactivity counted.

Radioactivity measurements

100 µL of each fraction was mixed to 10 mL of Ultima Gold TM scintillation cocktail (6013329, PerkinElmer) in scintillation vials. Samples were vortexed vigorously for 30 s and kept aside for 1 h before radioactivity measurement using a β-scintillation counter (LS6500, Beckman Coulter).

Dialysis of separated fractions from human plasma

Prior to electrophoresis and cholesterol and albumin dosages, the collected plasma fractions were dialyzed against a Tris-buffered saline solution (50 mM Tris, 150 mM NaCl, pH 7.4) for 2 h using 3.5 kDa MWCO Slide-A-Lyzer™ MINI dialysis devices (10321114, Fisher Scientific). Buffer was changed every 30 min. The dilution factor was determined by measuring the volume of each fraction before and after the dialysis step.

Electrophoresis

The quality of the LP separation was assessed by electrophoresis using the HYDRAGEL LIPO + Lp(a) K20 separation kit (3007, SEBIA). Following the manufacturer instructions, 10 µL of each dialyzed fraction was loaded onto the agarose gels and allowed to migrate at a constant voltage of 50 V for 90 min. Then, gels were dried at 80 °C for at least 45 min, stained with Sudan Black, bleached and washed with the supplied reagents. The LP identity within each fraction was confirmed by the migration distance of the electrophoretic bands from the loading point.

Cholesterol and albumin dosage

Cholesterol and albumin content in each plasma fraction was quantified using the cholesterol assay kit (ab65390, Abcam) and the human albumin ELISA kit (ab227933, Abcam), according to the provider instructions.

Animals

Male Sprague Dawley rats (4-week-old) were bought from Janvier Labs (France). All animals were housed in appropriate animal care facilities during the experimental period and were handled according to the principles of laboratory animal care and legislation in force in France. Experimental approval was obtained from the Ethical Committee C2EA-26 (Authorization number #3805.02).

In vivo 3 H-SQGem and 3 H-CholGem NPs administration to rats and LP separation

Radiolabeled 3 H-SQGem and 3 H-CholGem NPs (1.0 mg.kg -1 eq. Gem, 20 µCi per rat) were administered to healthy male Sprague Dawley rats by intravenous injection in the tail vein.

After 5 min, blood was collected by cardiac puncture into EDTA K 3 Vacuette tubes (VK054STK, Labelians). Rats were kept under anesthesia with a mixture of ketamine (100 mg.kg -1 ) and xylazine (10 mg.kg -1 ) during the whole process. The separation protocol was performed in a similar manner as in the in vitro assay on human blood with small adjustments to best fit the murine model. Following a first centrifugation of the blood (1,700 g, 15 min, 15 °C), plasma was separated by sodium bromide gradient density ultracentrifugation using NaBr solutions of 1.006, 1.054 and 1.210 g.mL -1 densities which have been chosen to match those of the different LP populations in rodents. Afterwards, separation was performed as described above (see 2.9.).

Results and discussion

NP formulation and physicochemical characterization

As for the SQGem prodrug (Fig. 1a), the cholesterol moiety has also been conjugated to the amino group of gemcitabine (Fig. 1c), allowing protection of the drug from the metabolization by plasma deaminases [START_REF] Couvreur | Squalenoyl Nanomedicines as Potential Therapeutics[END_REF]. Radiolabeled conjugates were synthesized in a similar manner, using a 3 H-Gem radiolabeled in the position C-5 of the cytosine moiety (Fig. 1b,d). The nanoprecipitation of CholGem prodrugs resulted in their self-assembly into nanoparticles with a mean diameter of 97 nm, a narrow size distribution and a negative surface charge of -21 mV, similar to SQGem NPs, herein used as a well-characterized reference. However, CholGem NPs showed almost a three-fold longer stability at room temperature, comparatively to SQGem NPs which displayed visible aggregates from day 7 (Fig. 2c). Only a slight decrease in size occurred when the radiolabeled 3 H-CholGem prodrug was introduced in the formulation (Table 1). 15

Both SQGem and CholGem NPs displayed spherical structures (Fig. 2a,b) but cryo-TEM images revealed a different supramolecular organization, the CholGem NPs assembling as vesicle-like structures (Fig. 2b). The latter showed a highly contrasted envelope likely corresponding to a bilayer made of amphiphilic CholGem prodrugs and stabilized by hydrophobic interactions between the cholesterol moieties. Values represents mean ± standard deviation (n=3).

In vitro distribution of 3 H-SQGem and 3 H-CholGem in human plasma

The distribution of SQGem and CholGem prodrugs within the plasma fractions has been first investigated in vitro following incubation with whole human blood at 37 °C for 5 min. After sodium bromide gradient density ultracentrifugation, twenty plasma fractions were isolated (numbered from top to bottom of the tube). According to their electrophoretic profile (Fig. S9b), their density (Fig. S12a) and their cholesterol content (Fig. 3), these fractions were attributed to the different lipoprotein subclasses: (i) VLDL: fraction 1, (ii) LDL: fractions 2-5, (iii) HDL: fractions 7-13 and (iv) to the lipoprotein deficient fraction (LPDF: fractions 14-19) in which albumin accumulated (Fig. S9a and Fig. S11). A small pellet was present in fraction 20 which likely corresponded to lipoprotein residues and cellular debris.

Prodrug content in these fractions was determined by radioactivity quantification which was then normalized to the radioactivity in plasma. SQGem NPs have been herein used as a reference and, in agreement with the previous data (Sobot et al., 2017a), a main accumulation in the cholesterol-rich lipoprotein fractions was detected. On the contrary, CholGem was only detected in the lipoprotein-deficient fractions at the bottom of the tube and not in the LP fractions (Fig. 3). Of note, the higher variability of the CholGem results among replicates compared to the more reproducible distribution profile of SQGem NPs, suggested the absence of any association of CholGem with plasma components and a rather nonspecific partition at the bottom of the ultracentrifugation tubes.

In vivo distribution of 3 H-SQGem and 3 H-CholGem in rat plasma

We additionally investigated whether a similar plasma distribution of CholGem NPs also occurred in vivo upon intravenous administration of NPs to rats. In vivo studies were carried out with the radiolabeled NPs for better detection sensitivity. 5 min after the injection of The in vivo blood distribution profile of the two types of NPs closely resembled to that observed in vitro. Now again, the SQGem was recovered in the cholesterol-rich LP 1-13 fractions, while the level of CholGem in the different fractions was opposite and inversely proportional to their content in endogenous cholesterol (Fig. 4).

Of note, electrophoretic mobility (Fig. S10b) and density profile (Fig. S12b) revealed a similar LP distribution among the fractions compared to the one found in human plasma (i.e., VLDL: fractions 1-2, LDL: fractions 1-5, HDL: fractions 3-13 and LPDF: fractions 14-20).

Yet, the distribution of cholesterol between the LP classes in vivo (Fig. 4, Fig. S11) differed from those observed in vitro (Fig. 3, Fig. S11), reflecting the different lipoprotein metabolism in humans and rodents and confirming the role of HDL as main cholesterol carrier in the latter [START_REF] Camus | Distribution and characterization of the serum lipoproteins and apoproteins in the mouse, Mus musculus[END_REF][START_REF] Innerarity | Disparities in the interaction of rat and human lipoproteins with cultured rat fibroblasts and smooth muscle cells. Requirements for homology for receptor binding activity[END_REF][START_REF] Oschry | Rat plasma lipoproteins: re-evaluation of a lipoprotein system in an animal devoid of cholesteryl ester transfer activity[END_REF].

As also observed for human samples, the gradient ultracentrifugation of rat plasma led to albumin accumulation in the LP-deficient 16-20 fractions (Fig. S10a,S11). In vitro, in human blood, the CholGem profile partially overlapped the albumin distribution (Fig. S9a), while in vivo in rats the two traces were clearly separated (Fig. S10a). In these LP-deficient fractions, the cumulated amount of the SQGem conjugate reached around 18% of the total radioactivity in human plasma (Fig. S9a) and was twice as high in rodent plasma (39%) (Fig. S10a).

Whether such different trend was related to a different affinity toward the human or rat serum albumin deserved to be further investigated.

Interaction of SQGem and CholGem NPs with serum albumin

The affinity of SQGem and CholGem NPs for both human serum albumin (HSA) and rat serum albumin (RSA) was thus assessed by SPR. HSA and RSA were fixed onto separate flow channels of a CM5 chip, and SQGem and CholGem NPs were injected at different concentrations (Fig. 5).

For both HSA and RSA (Fig. 5a,c), SQGem NPs injections resulted in a concentration-dependent response which supported SQGem affinity for albumin, in agreement with previous in silico studies [START_REF] Yesylevskyy | Low-Density Lipoproteins and Human Serum Albumin as Carriers of Squalenoylated Drugs: Insights from Molecular Simulations[END_REF]. Signal amplitude was slightly higher for the couple SQGem NPs/RSA (Fig. 5c) compared to the SQGem NPs/HSA (Fig. 5a), which may explain the different quantities of SQGem detected in the albumin-rich fractions when NPs entered into contact with human (Fig. S9a) or rat blood (Fig. S10a). On the contrary, the interaction of CholGem NPs with rat or human albumin was almost inexistent (Fig. 5b,d). Indeed, injection of these NPs at the highest concentrations (i.e., 280 µM and 560 µM) in the channel containing HSA resulted in a signal almost ten times lower in amplitude than the one measured for the SQGem NPs at the same concentration (Fig. 5b). Similarly, when CholGem NPs were introduced in the RSA channel at concentrations above 140 µM, a concentration dependent trend appeared, but again the signal intensity was very weak, (i.e., 6-times lower than the one observed with SQGem NPs), which did not validate a significant interaction, too (Fig. 5d).

Taken together with the in vivo and in vitro plasma distribution profiles (Fig. 3, 4), these results confirmed the capacity of SQGem NPs to interact with albumin in addition to lipoproteins. On the contrary, the presence of CholGem in some albumin-containing plasma fractions could not be correlated with any interaction between the two components.

Analysis of NP morphology by AFM

AFM analysis of SQGem and CholGem NPs has been carried out to ascertain whether differences in their respective behavior toward plasma components could (or not) be attributed to structural differences between them.

Both nanoparticles, which displayed a negative surface charge (Table 1), did not adsorb on an untreated negatively-charged mica surface (Fig. S13) but rapidly adhered to a lysine-coated positively-charged surface (Fig. 6) therefore revealing an attractive electrostatic-driven adsorption. The deposition of SQGem NPs onto the latter quickly led to the formation of 4 nm-height patches which spread out unevenly. This effect was likely due to their physisorption onto the surface, followed by a disruption of their supramolecular structure (Fig. 6a,b). On the contrary, heights from 20 to 40 nm were measured for the CholGem NPs suggesting that these 100 nm NPs flattened upon adsorption onto the surface but did not break (Fig. 6c,d). These results further suggested a difference in NP supramolecular organization:

the squalene prodrug assembled in easily deformable NPs while the cholesterol derivatives led to stiffer NPs which prevented their rupture on the positively-charged mica surface.

TEM observations of NP/ lipoprotein mixtures

The likely more stable and rigid structure of the CholGem NPs has been confirmed

qualitatively by TEM observations in the presence of isolated lipoproteins (Fig. 7). Upon physical mixture with LDL (Fig. 7e) or HDL (Fig. 7g) a merge of SQGem NPs and lipoproteins was observed, probably triggering the previously described disassembly of these nanoparticles resulting from the insertion of SQGem as single molecular entities into the lipoprotein core [START_REF] Cayre | In Vivo FRET Imaging to Predict the Risk Associated with Hepatic Accumulation of Squalene-Based Prodrug Nanoparticles[END_REF]Sobot et al., 2017a). On the contrary, CholGem NPs coexisted with either LDL (Fig. 7f) or HDL (Fig. 7h) without any apparent interaction as suggested by the lack of modification of their well-defined bilayer. These NPs tended to stick together and to form large clusters independently of the presence of lipoproteins.

Further imaging of the same mixtures after a 19 h incubation period revealed the greater deformation of SQGem NPs in presence of LPs in comparison to CholGem in the same conditions (Fig. S14). Altogether, AFM and TEM observations pointed toward a more fragile nature of the SQGem NPs that could explain their easier disassembly in biological conditions and their interaction with lipoproteins with subsequent transfer of SQGem molecules into the LP core. On the contrary, the CholGem NPs were more stable and maintained their structural integrity over time, limiting interactions with blood components (both LPs and albumin).

Conclusion

Aiming to exploit endogenous lipoproteins for selective drug delivery to neoplastic tissues, we have herein investigated whether the drug conjugation to a LP component, like cholesterol, could have been a sufficient condition. Using gemcitabine as a model drug, we have conjugated this anticancer compound to cholesterol, because it is one of the main lipids transported by LPs. The resulting amphiphilic conjugate was capable to self-assemble as nanoparticles. But, unexpectedly, no interaction with any lipoprotein fraction could be detected neither in vitro after incubation with human blood nor in vivo after intravenous administration to rats. Comparison with SQGem NPs, which clearly interacted with LPs, highlighted that other key parameter than the prodrug lipophilicity and the natural affinity of the lipid moiety for endogenous lipoproteins were involved in the prodrug interaction with LPs.

And, it was discovered that the structure of the NPs formed by the gemcitabine prodrug supramolecular assembly played a key role in the sequence of events occurring in the blood, the first biological medium encountered after NPs intravenous administration. Electron microscopy and atomic force microscopy experiments suggested, in comparison to CholGem, a much higher deformability of SQGem NPs and their modification over time in the presence of plasma components.

It remains an open question whether the physical constants of the lipid moieties can affect the properties of the prodrug nanoparticles, namely their rigidity, and thus the behavior of the NPs in the biological media. Cholesterol has a crystal structure and a higher melting point (148.2 °C) compared to squalene, an oil with a melting point of -4.5 °C [START_REF] Haynes | CRC Handbook of Chemistry and Physics. 97[END_REF]. Since conjugation of squalene and cholesterol to the gemcitabine resulted in prodrugs whose physical form closely resembled that of the parent lipid (an oily product (SQGem) and a solid powder (CholGem)) these two prodrugs are expected to have very distinct melting points. It can be speculated that the higher the melting point, the stronger the intermolecular interactions and thus the molecular packing, as previously observed for binary crystals [START_REF] Kilinkissa | Melting point-solubility-structure correlations in chiral and racemic model cocrystals[END_REF].

Although further studies are needed to validate such a hypothesis, a trend might exist since in our study we showed that the lipid moiety with the lowest melting point (i.e., squalene) led to nanoparticles with lower colloidal stability in the biological medium.

On the contrary, the lack of interaction with the plasma fractions suggests that the cholesterol moiety triggered the formation of more closely packed nanoparticles and conferred greater resistance to destabilization. This is in agreement with previous studies that demonstrated a crucial role of cholesterol in decreasing permeability of unsaturated phospholipid vesicles, increasing the packing density of phospholipid molecules in liposomes, and increasing the colloidal stability of these vesicles in the presence of blood components [START_REF] Allen | A study of phospholipid interactions between high-density lipoproteins and small unilamellar vesicles[END_REF][START_REF] Demel | The function of sterols in membranes[END_REF][START_REF] Kirby | Effect of the cholesterol content of small unilamellar liposomes on their stability in vivo and in vitro[END_REF][START_REF] Papahadjopoulos | Phase transitions in phospholipid vesicles Fluorescence polarization and permeability measurements concerning the effect of temperature and cholesterol[END_REF][START_REF] Patel | Inhibitory effect of cholesterol on the uptake of liposomes by liver and spleen[END_REF].

As the interaction of CholGem or SQGem prodrugs with LPs is likely to happen at the molecular scale, the nanoparticle destabilization upon injection might be a crucial prerequisite for the interaction to occur. In that respect, it is obvious that colloidal stable NPs such as CholGem are less prone to interact with endogenous lipoproteins.

Overall, these results point out that the interplay between the physicochemical properties of the prodrugs and the resulting supramolecular assembly as nanoparticles may strongly influence the fate of the conjugates in the bloodstream, further affecting pharmacological and toxicological properties.

  Fig. 1. Chemical structures of (a) squalene-gemcitabine (SQGem) and (c) cholesterol-gemcitabine (CholGem) prodrugs and their tritiated equivalent (b) 3 H-SQGem and (d) 3 H-CholGem (*, site of 3 H incorporation).

Fig. 2 .

 2 Fig. 2. Cryo-TEM images of (a) SQGem NPs (1 mg.mL -1 ) and (b) CholGem NPs (1 mg.mL -1 ), scale bar: 100 nm; (c) evolution of NP mean diameter and polydispersity index

Fig. 3 .

 3 Fig. 3. Distribution of 3 H-SQGem, 3 H-CholGem and endogenous cholesterol among the

Fig. 4 .

 4 Fig. 4. Distribution of 3 H-SQGem, 3 H-CholGem and endogenous cholesterol among the

Fig. 5 .

 5 Fig. 5. Reference-subtracted SPR sensorgrams (resonance units (RU) versus time) obtained

Fig. 6 .

 6 Fig. 6. AFM images obtained in liquid medium of (a, b) SQGem NPs and (c, d) CholGem

Fig. 7 .

 7 Fig. 7. TEM images of (a) SQGem NPs, (b) CholGem NPs, (c) LDL, (d) HDL and physical

Table 1

 1 SQGem and CholGem nanoparticle characterization.

	Conjugate	Mean diameter a (nm)	Polydispersity index a (PdI)	Zeta Potential a (mV)	Drug loading b (%)
	SQGem	82 ± 7	0.11 ± 0.03	-20 ± 7	40.7
	CholGem	97 ± 15	0.09 ± 0.02	-21 ± 7	38.9
	3 H-SQGem	86 ± 5	0.15 ± 0.02	-	-
	3 H-CholGem	77 ± 10	0.10 ± 0.04	-	-

a Measured by dynamic light scattering (values ± standard deviation, n ≥ 15). b Calculated as molecular weight (gemcitabine)/molecular weight (conjugate).

H-SQGem or 3 H-CholGem NPs, the blood was collected by cardiac puncture and the plasma fraction separation was carried out as described above.

Acknowledgments

This work was supported by the French Ministry of Research, the CNRS, the Fondation pour la Recherche Médicale (Application FDT202001010776) and the Ligue contre le Cancer (Subvention Recherche HB2019-22 and NH2020-32). It has benefited from the Imagerie-Gif core facility supported by l'Agence Nationale de la Recherche (ANR-11-EQPX-0029/Morphoscope, ANR-10-INBS-04/FranceBioImaging; ANR-11-IDEX-0003-02/ Saclay Plant Sciences), the Electron Microscopy Facility of the Multimodal Imaging Centre at Institut Curie, Orsay and the INTERMOL molecular interactions platform of the IPSIT/US31/UMS3679. We thank Miss Claire Boulogne and Miss Cynthia Gillet (Imagerie-Gif) for their assistance during the TEM experiments, Miss Assia Hessani (Institut Galien Paris-Saclay, France) for her help with radiolabeled materials, and Dr Sylvain Trépout (Institut Curie, France) for the cryo-TEM images. G.P. and S.F. thank ANR for funding T.N. (Carb2zyme, ANR-17-CE11-0014).

Appendix A. Supplementary information

Supplementary information to this article can be found online at xxx. NMR spectra; radio-HPLC chromatograms and elution conditions; absorbance calibration curves used to determine radiolabeled products concentration; albumin distribution among human and rat plasma fractions; electrophoresis of human and rat plasma fractions; AFM images of SQGem and CholGem NPs on bare mica surface; TEM images of isolated NPs, LPs, and their physical mixtures after 19 h incubation at 37 °C. Zheng, G., Chen, J., Li, H., Glickson, J.D., 2005. Rerouting lipoprotein nanoparticles to selected alternate receptors for the targeted delivery of cancer diagnostic and therapeutic agents. Proc. Natl. Acad. Sci. U.S. A. 102, 17757. Zhu, C., Xia, Y., 2017. Biomimetics: reconstitution of low-density lipoprotein for targeted drug delivery and related theranostic applications. Chem. Soc. Rev. 46,[7668][7669][7670][7671][7672][7673][7674][7675][7676][7677][7678][7679][7680][7681][7682]


Integer Ring Sieve for Constructing Compact QC-LDPC Codes with Girths 8, 10, and 12 Alireza Tasdighi and Emmanuel Boutillon Senior Member, IEEE Université Bretagne-Sud, Lab-STICC, UMR 6285, CNRS -Lorient, France ar.tasdighi@gmail.com, emmanuel.boutillon@univ-ubs.fr Abstract-This paper proposes a new method of constructing compact fully-connected Quasi-Cyclic Low Density Parity Check (QC-LDPC) codes with girth g = 8, 10, and 12. The originality of the proposed method is to impose constraints on the exponent matrix P to reduce the search space drastically. For a targeted lifting degree of N , the first step of the method is to sieve the integer ring ZN to make a particular sub-group with specific properties to construct the second column of P (the first column being filled with zeros). The remaining columns of P are determined recursively as multiples of the second column by adapting the sequentially multiplied column (SMC) method whereby a controlled greedy search is applied at each step. The codes constructed with the proposed semi-algebraic method show lengths that can be significantly shorter than their best counterparts in the literature.

Index Terms-QC-LDPC Code Construction, Girth, Multiplicative Group, Cyclic Subgroup, Greedy Search Method.

I. INTRODUCTION

It has been more than two decades since the rediscovery of low-density parity-check (LDPC) codes as a class of modern channel coding [START_REF] Mackay | Good error-correcting codes based on very sparse matrices[END_REF]. Quasi-cyclic (QC) LDPC codes, a special class of LDPC codes, enable efficient parallel hardware implementation and have been adopted in many communication standards. Examples include the WIFI standard [START_REF]IEEE Draft Standard for Air Interface for Broadband Wireless Access Systems[END_REF], digital video broadcasting (DVB) standard [START_REF] Etsi | Digital Video Broadcasting (DVB)[END_REF], CCSDS standards [START_REF] Ccsds | Short Blocklength LDPC codes for TC synchronization and channel coding[END_REF], and more recently, the 5G standard [START_REF]3GPP, The Mobile Broadband Standard[END_REF]. The Performance of an LDPC iterative decoder is significantly impacted by the minimum size of a cycle of the Tanner graph (the so-called girth). Maximizing the girth for a given code size and code rate, or reciprocally, finding the minimum code size for a given code rate and girth size, has proved challenging for the past two decades [START_REF] Fossorier | Quasi cyclic low-density parity-check codes from circulant permutation matrices[END_REF]- [START_REF] Naseri | Construction of Girth-8 QC-LDPC Codes Free of Small Trapping Sets[END_REF]. This paper is focused on the construction of a QC-LDPC matrix constructed from a fully connected exponent matrix. QC-LDPC codes can be used on their own for applications requiring a very low Frame Error Rate (e.g. memory storage). They can also be combined with a masking technique to build good irregular QC-LDPC codes [START_REF] Xu | Construction of regular and irregular ldpc codes: Geometry decomposition and masking[END_REF]- [START_REF] Han | Construction of Irregular QC-LDPC Codes via Masking with ACE Optimization[END_REF]. In addition, it has recently been shown that by using some spreading techniques, a class of SC-QC-LDPC convolutional codes with very low syndrome memory could be constructed based on QC-LDPC codes constructed from a fully-connected exponent matrix [START_REF] Battaglioni | Compact QC-LDPC Block and SC-LDPC Convolutional Codes for Low-Latency Communications[END_REF], [START_REF] Jimenez Felstrom | Time-varying periodic convolutional codes with low-density parity-check matrix[END_REF]- [START_REF] Battaglioni | Girth Analysis and Design of Periodically Time-Varying SC-LDPC Codes[END_REF]. Specifically, [START_REF] Battaglioni | Design and Analysis of Time-Invariant SC-LDPC Convolutional Codes With Small Constraint Length[END_REF] asserts that given a fixed girth and degree distribution, the smaller the lifting degree of QC-LDPC codes, the smaller the size of the syndrome memory of SC-QC-LDPCC codes and this results in better performance of such code under windowed decoding. This study focuses solely on constructing short length QC-LDPC codes with girth g = 8, 10, and 12. However, we keep in mind that SC-QC-LDPCC codes are potential candidates for beyond 5G applications, and good QC-LDPC codes are the basis of good SC-QC-LDPC codes.

QC-LDPC codes can be divided into two major classes: 1) random-like codes constructed by means of a computer search under efficient algorithms, and 2) structured codes constructed using algebraic tools [START_REF] Li | LDPC code designs, constructions, and unification[END_REF]. Individually, these methods all have drawbacks. Search-based methods (even heuristic or exhaustive ones) require high search complexity but may find codes with shorter length than the ones obtained with algebraic methods. In contrast, algebraic methods determine the code explicitly (e.g. array code [START_REF] Li | LDPC code designs, constructions, and unification[END_REF] of girth 6) but to date, algebraic methods are known to construct small-girth codes only, not high-girth codes. Defining algebraic properties that are perfectly matched with high-girth conditions resulting in the explicit construction of short-length codes is one of the main shortcomings of algebraic methods. In this paper, we try to combine the two methods in order to construct large-girth QC-LDPC codes with a short length in considerably lower search complexity. We take the search-based sequentially multiplied column (SMC) construction method [START_REF] Tadayon | Efficient Search of Compact QC-LDPC and SC-LDPC Convolutional Codes With Large Girth[END_REF] as our search algorithm and modify it by introducing an algebraic property for the second column of the exponent matrix of the code. The second column with the asserted algebraic property is found by an Integer Ring Sieve (IRS) method in a way that eventually leads to reducing the search space. As a result, a semialgebraic fast search-based method of constructing high-girth QC-LDPC codes is proposed, and many constructed codes of girth g = 8, 10, and 12 with different rates and degrees are reported. Most of the constructed codes have lengths shorter than (by up to 35%), or equal to, their counterparts in the literature. The paper also proposes matrices with sizes not yet reported in the literature.

The rest of the paper is organized as follows: Section II presents earlier results regarding SMC construction-based QC-LDPC codes. Section III presents the proposed IRS construction method. Numerical results are provided in Section IV.

II. PRELIMINARIES

In this section, we review the construction of Quasi-Cyclic LDPC codes. Next we discuss the conditions that result in QC-LDPC codes with good topological properties.

A. QC-LDPC block codes

Let us consider a fully-connected QC-LDPC block code in which the parity-check matrix is an m × n array of N × N circulant permutation matrices (CPMs),

I(p ij ), 0 ≤ i ≤ m-1, 0 ≤ j ≤ n-1,
where N is the lifting degree of the code. I(p ij ) is obtained from the identity matrix through a cyclic shift of its rows by p ij positions, with 0 ≤ p ij ≤ N -1. The code length is L = nN , the column degree of the parity-check matrix is presented by m and the row degree of the paritycheck matrix is presented by n1 . The resulting rate of the code is r ≥ 1-m/n. The m×n matrix P having the integer values p ij as its entries is referred to as the exponent matrix of the code.

Definition 2.1:

A cycle C 2k of length 2k in the exponent matrix P is defined as an ordered set of 2k positions {(m s , n s )} s=0,1,...,2k-1 in the matrix P satisfying the following three conditions: 1) m s = m s+1 ; n s = n s+1 when s is even, 2) m s = m s+1 ; n s = n s+1 when s is odd, 3) additions in the indexes are done modulo 2k, i.e. when s = 2k-1, n s+1 = n 2k = n 0 . For example, the length-6 cycle C 6 = {(0, 0), (0, 1), (1, 1), (1, 2), (2, 2), (2, 0)} is represented in Fig. 1.c. Definition 2.2: For a given exponent matrix P, we define θ P as a function that associates to a cycle C 2k the integer θ P (C 2k ) as

θ P (C 2k ) = 2k-1 s=0 (-1) s p msns . (1)
According to [START_REF] Fossorier | Quasi cyclic low-density parity-check codes from circulant permutation matrices[END_REF], a necessary and sufficient condition for the existence of a cycle of length 2k in the Tanner graph of the QC-LDPC block code is the existence of a cycle C 2k in P satisfying θ P (C 2k ) = 0 mod N . Thus, a necessary and sufficient condition to ensure a QC-LDPC code of girth g is that all cycles C 2k , 2k < g of the exponent matrix P satisfy

θ P (C 2k ) = 0 mod N. (2)
We define an inevitable cycle of length 2k to be a cycle C i 2k so that θ P (C i 2k ) = 0 regardless of what the values of p ij s are.

In [START_REF] Fossorier | Quasi cyclic low-density parity-check codes from circulant permutation matrices[END_REF] it is shown that fully-connected CPM-based QC-LDPC codes always contain inevitable cycles of length 12, and thus their girth cannot be larger than 12.

B. Code design using sequentially multiplied columns

Searching for an exponent matrix P m×n of a given girth g = 2k and a lifting degree N is a complex task. First, the raw number of exponent matrices is exponential in m and n (N mn exactly); second, the number of cycles C 2k of length 2k = g -2 (and thus, the number of equations (2) to satisfy), increases in the order of O(m k n k ). In fact, a cycle of length 2k can involve up to k columns and k rows. Solutions with reduced complexity were proposed in [START_REF] Gholami | An explicit method to generate some QC-LDPC codes with girth 8[END_REF] and [START_REF]Symmetrical Constructions for Regular Girth-8 QC-LDPC Codes[END_REF], but the corresponding design methods result in girth g = 8. For constructing short codes with higher girths (i.e. g = 10, 12), many methods have been developed. To the best of the authors' knowledge, the results in [START_REF] Tadayon | Efficient Search of Compact QC-LDPC and SC-LDPC Convolutional Codes With Large Girth[END_REF] for QC-LDPC codes with girth g = 10, 12 found by applying the SMC construction technique are the shortest ones in the literature. Let us recall the basic assumptions of the design method proposed in [START_REF] Tadayon | Efficient Search of Compact QC-LDPC and SC-LDPC Convolutional Codes With Large Girth[END_REF]. The design of the parity-check matrix of a QC-LDPC block code with lifting degree N starts from an exponent matrix with the following form (SMC assumption):

P m×n = 0 P 1 γ 2 ⊗ P 1 γ 3 ⊗ P 1 . . . γ n-1 ⊗ P 1 , (3)
with m, n, ∈ N, m < n, and 0 and P 1 being m × 1 column vectors with entries in {0, 1, • • • , N -1}. The vector 0 is filled with all zero entries, while the entries of the vector P 1 are chosen as follows: the first entry is zero, the second entry is one, and the other entries are chosen in {2, 3, • • • , N -1}. Then, the subsequent vectors have the form γ j ⊗ P 1 (j = 2, 3, • • • , n -1), where ⊗ denotes multiplication modulo N of each term of P 1 with γ j ∈ {2, 3, • • • , N -1}, γ j < γ j+1 .

To achieve this result, the authors of [START_REF] Tadayon | Efficient Search of Compact QC-LDPC and SC-LDPC Convolutional Codes With Large Girth[END_REF] established a recursive greedy search algorithm (see algorithm 1 in [START_REF] Tadayon | Efficient Search of Compact QC-LDPC and SC-LDPC Convolutional Codes With Large Girth[END_REF]) to determine, for a large enough lifting factor N , the exponent matrix P SM C m×n that satisfies (2) for all cycles of girth lower than g. This search algorithm is supposed to find n-2 (resp., m-2) non-zero and distinct elements to be placed in the second row (resp., column) of P SM C m×n . These elements vary from 1 to N -1, and, in the worst case, the overall possibilities are equal to N -1 n-2 N -1 m-2 . For high-rate and high-girth codes, the lifting degree is much bigger than m and n (i.e. m, n N ), and the whole search space is of order O (N -1) m+n-4 .

It has to be noted that for a desired girth g, each realization of the matrix P SM C m×n requires checking all the constraints of type (2) with k < g/2.

In the next section, an improved construction method is proposed.

III. INTEGER RING SIEVE TO FIND PERMISSIBLE ELEMENTS FOR THE VECTOR P 1

This section is divided into four parts. In Part A, we propose the definition of strictly equivalent relations between cycles of a fully connected exponent matrix P based on [START_REF]IEEE Draft Standard for Air Interface for Broadband Wireless Access Systems[END_REF]. Next, we give a theorem for counting the number of strictly equivalent classes of length 2k (k = 2, 3, 4, 5) in P. In Part B, we show that a careful selection of the second column of matrix P m×n (i.e. P 1 ) can create a new type of equivalence relation called "Integer-Ring (IR) equivalence". When combining strict equivalence and IR-equivalence together, the number of equivalent classes is divided by a factor close to 3 for m = 3 and close to m -1 for m ≥ 4. In Part C, we show results regarding the existence of integer rings with a property defined in Part B. Our greedy search algorithm is explained in Part D with a pseudocode. Complexity analyses to highlight the important role of our sieving method in reducing the search space are also provided in this final part.

A. Equivalent relations between the set of cycles Definition 3.1 (strictly equivalent cycle): Two cycles C and C of an exponent matrix P are said to be strictly equivalent if and only if θ(C) = 0 ⇔ θ P (C ) = 0 regardless of the value of P. This relation of equivalence defines equivalent classes. By convention, if two cycles of different lengths are equivalent, only the cycle with the smallest length will be considered. The matrices T C4 , T C6 , T C8 and T C10 are respectively

T C4 = 0 0 0 1 , T C6 = 0 0 0 0 0 0 0 0 6 T C8 =    0 0 0 0 0 1 3 3 0 3 18 36 0 3 36 72    , T C10 =     0 0 0 0 0 0 0 0 0 0 0 0 60 180 180 0 0 180 900 1440 0 0 180 1440 1440     .
It has to be noted that T C 2k is symmetrical (i.e. T C 2k = T C 2k T ) as the number of non-equivalent cycles involved in a i × j matrix is equal to the number of such cycles involved in a matrix of size j × i. Note that the matrices T C4 to T C10 have been obtained using a computer program.

In an exponent matrix P m×n of size (m × n), the total number #C m,n 2k of non-equivalent cycles of girth 2k can be computed as the number of distinct sub-matrices (i, j) compatible with a cycle of length 2k (i.e.

2 ≤ i ≤ k and 2 ≤ j ≤ k) multiplied by T C 2k (i, j), this results in #C m,n 2k = min{k,m} i=2 min{k,n} j=2 T C 2k (i, j) m i n j , (4)
where n r represent the binomial coefficient

n! r!(n-r)! . Table I gives the values of #C m,n 2k for k = 2, 3, 4, 5, 2 ≤ m ≤ 5 and 2 ≤ n ≤ 10.
The total number of strictly equivalent cycles of length lower than g of an m × n exponent matrix P m×n is thus given as

#P m×n (g) = g/2-1 k=2 #C m,n 2k . (5)
For example, if g = 12, m = 3 and n = 10, then satisfying that a given exponent matrix P 3,10 of size (3, 10) generates a girth 12 QC-LDPC code with a given lifting degree N implies checking a total of #P 3,10 (12) = 135+720+12960+90360 = 104175 equations. To reduce the number of non-equivalent classes, some constraint can be added to the exponent matrix.

B. Integer Ring-based construction of P 1

Let us recall the usual notations used in integer ring theory [START_REF] Riesel | Prime numbers and computer methods for factorization[END_REF]. The greatest common divisor of a and b is noted gcd (a, b). The ring of integers modulo N is denoted by

Z N . The multiplicative group of integers modulo N (set of values of Z N coprime with N ) is denoted by Z × N . An element a ∈ Z × N generates a finite cyclic subgroup a = {1, a, a 2 , a 3 , • • • , a O N (a)-1 }, with O N (a) the smallest non- null integer satisfying a O N (a) = 1 mod N . The value O N (a) is called the order of a in Z × N .
In this part, we try to pick the non-zero elements of P 1 (see ( 3)) from a specific cyclic subgroup of Z × N . Depending on the value of m, we propose to allocate some or all of the elements in this subgroup to p j1 (1 ≤ j ≤ m -1). The main reason behind such allocation is to add to the strictly equivalent cycles a new type of equivalent cycles that are called "Integer Ring equivalent" (IR-equivalent) cycles. Reducing the number of equivalent cycles (i.e. the number of equations that should be satisfied) has several positive side effects. First, it directly accelerates the research algorithm since the number of constraints to be checked is reduced. Second, it reduces the search space: because of the a priori selection of the P 1 column coefficients, only the determination of the elements γ j (j = 2, 3, • • • , n-1) is required. Third, and above all, it increases the likelihood of finding a solution. The question is open whether, for a given girth and exponent matrix size, the smallest lifting factor can be obtained only with IRS exponent matrices. Note that, by serendipity, the authors forgot to kill an unconstrained search based on the method proposed in [START_REF] Derrien | Additive, Structural, and Multiplicative Transformations for the Construction of Quasi-Cyclic LDPC Matrices[END_REF] to generate an (m, n) = (3, 7) exponent matrix with a lifting degree N = 133. After five months of silent computation, a solution was found, which involved an exponent matrix with the IRS structure (see Appendix I).

ߠ ۾ ܥ ସ :  െ  ଵ  ଵଵ െ  ଵ ≅  ଵ െ   ଵ െ  ଵଵ   ଵ  ଵ  ଵଵ b)   ଵ  ଶ  ଵ  ଵଵ  ଵଶ  ଶ  ଶଵ  ଶଶ ߠ ۾ ܥ :  െ  ଵ  ଵଵ െ  ଵଶ  ଶଶ െ  ଶ ≅  ଵ െ   ଶ െ  ଶଶ  ଵଶ െ  ଵଵ c)   ଵ  ଵ  ଵଵ ߠ ۾ ܥ ଼ : 2ሺ െ  ଵ  ଵଵ െ  ଵ ሻ e) a) ۾ ൈ ൌ   ଵ ⋮  ሺିଵሻ  ଵ  ଵଵ ⋮  ିଵ ଵ ⋯ ⋯ ⋱ ⋯  ሺିଵሻ  ଵሺିଵሻ ⋮  ሺିଵሻሺିଵሻ   ଵ  ଵ  ଵଵ  ଶ  ଵଶ  ଷ  ଵଷ d) ߠ ۾ ܥ ଼ :  െ  ଵ  ଵଵ െ  ଵଶ  ଶ െ  ଷ  ଵଷ െ  ଵ ≅  െ  ଷ  ଵଷ െ  ଵଶ  ଶ െ  ଵ  ଵଵ െ  ଵ   ଵ  ଵ  ଵଵ  ଶ  ଶଵ ߠ ۾ ܥ ଼ :  െ  ଵ  ଵଵ െ  ଵ  ଶ െ  ଶଵ  ଵଵ െ  ଵ f)   ଵ  ଶ  ଵ  ଵଵ  ଶଵ  ଶ  ଵଶ  ଶଶ  ଷ  ଵଷ  ଶଷ  ସ  ଵସ  ଶସ ߠ ۾ ܥ ଵ :  ଵ െ  ଷ  ଶଷ െ  ଶସ  ଵସ െ  ଵଶ  ଶଶ െ  ଶ  ଵ െ  ଵଵ g)
(k = 2, 3, 4, 5) THAT ARE INVOLVED IN MATRIX P m×n , WHEN, 2 ≤ m ≤ 5 AND 2 ≤ n ≤ 10. - m = 2 m = 3 m = 4 m = 5 -#C 2,n 4 #C 2,n 6 #C 2,n 8 #C 2,n 10 #C 3,n 4 #C 3,n 6 #C 3,n 8 #C 3,n 10 #C 4,n 4 #C 4,n 6 #C 4,n 8 #C 4,n 10 #C 5,n 4 #C 5,n 6 #C 5,n 8 #C
Definition 3.4: A type-I IRS QC-LDPC matrix is a matrix obtained from the full SMC exponent matrix P a,I of size m × n with vector column P 1 defined as P 1 = (0, 1, a, a 2 , . . . , a m-1 ) T , with a an element of Z × N of order O N (a) = m and N representing the lifting degree. Definition 3.5: A type-II IRS QC-LDPC matrix is a matrix obtained from the full SMC exponent matrix P a,II of size (3, n) with vector column P 1 defined as P 1 = (0, 1, a) T , with a an element of Z × N satisfying a(1 -a) = 1 mod N and N representing the lifting degree.

The element (i, j) of the type-I IRS exponent matrix P a,I is defined as

P a,I (i, j) = u(i)a i-1 γ j . (6)
with u(i) = 0 if i = 0, 1 otherwise. Note that the elements of a type-II IRS exponent matrix P a,II have also the same form as [START_REF] Fossorier | Quasi cyclic low-density parity-check codes from circulant permutation matrices[END_REF].

Let π be a permutation on the number of rows (0, 1, . . . , m -1) of an exponent matrix P m×n . The permutation π is defined in the sequel by the vector (π(0), π(1), . . . , π(m -1)). From the permutation π, π(C 2k ) is defined as the function that is associated to a cycle C 2k = {(m s , n s )} s=0,1,...,2k-1 of P m×n the cycle of same length π(C 2k ) of P m×n defined as

π(C 2k ) = {(π(m s ), n s )} s=0,1,...,2k-1 .
Note that, in the general case, cycles C 2k and π(C 2k ) are not strictly equivalent. Definition 3.6: Let π I,1 be the permutation over the ordered vector (0, 1, . . . , m -1) defined as π I,1 = (0, 2, 3, . . . , m -1, 1) and π I,l the rotation obtained by applying the permutation π I,1 l times. Thus, for l = 1, 2, . . . , m -1, we have π I,l (0) = 0 and π I,l (i) = ((i + l) mod m) + ((i + l) ÷ m). Note that (i + l) ÷ m, the Euclidean division of (i + l) by m takes the value 0 when (i + l) < m, and 1 when m ≤ i + l < 2m. Theorem 3.7: Let P a,I m×n be a type-I IRS exponent matrix. Then, for all cycles C 2k of length 2k of P a,I m×n , θ P a,I (C 2k ) = 0 if and only if θ P a,I (π I,l (C 2k )) = 0, l = 1, 2, . . . m -1.

Proof: Let C 2k be a cycle of length 2k of P a,I m×n . According to (1) and ( 6),

θ P a,I (C 2k ) = 2k-1 s=0 (-1) s u(m s )a ms-1 γ ns . (7)
Since O N (a) = m in Z × N , then a m = 1 and a and N are coprime. Since a and N are co-prime, θ P a,I (C 2k ) = 0 mod N if and only if a l θ a,I

P (C 2k ) = 0 mod N for l = 1, 2, . . . , m-1. The equation a l θ P a,I (C 2k ) = 0 mod N gives 2k-1 s=0 (-1) s u(m s )a ms-1+l γ ns = 0 mod N. (8)
Since a m = 1, u(m s )a ms+l-1 = u(π I,l (m s ))a π I,l (ms)-1 , thus (8) is equivalent to θ P a,I (C 2k ) = 0.

Example 3.8: Suppose that P 4×9 is the exponent matrix in Fig. 2 (a), and C 2k (k = 4, 5) are the cycles with the paths depicted in Fig. 2.a. According to theorem 3.7, there are two other IR-equivalent cycles to C 2k , π I,1 (C 2k ) given in Fig. 2.b and π I,2 (C 2k ) given in Fig. 2.c. Thus, when a type-I IRS exponent matrix is used, the IRequivalence between cycles reduces almost by a factor of m-1 the number of equivalence classes. The reduction is not exactly m-1 because in some rare specific cases, π I,l (C 8 ) can also be strictly equivalent to C 8 . For example, for (m, n) = (5, 2), the length-8 cycle C 8 = {(0, 0), (0, 1), (2, 1), (2, 0), (0, 0), (0, 1), (4, 1), (4, 0)} gives with the permutation π I,2 = (0, 3, 4, 1, 2) the cycle π I,2 (C 8 ) = {(0, 0), (0, 1), (4, 1), (4, 0), (0, 0), (0, 1), (2, 1), (2, 0)}, which is strictly equivalent to C 8 (just a 4 position shift). Definition 3.10: Let us define π II,1 and π II,2 the permutations π II,1 = (1, 2, 0) and π II,2 = π II,1 • π II,1 = (2, 0, 1). Theorem 3.11: Let P a,II be a type-II IRS exponent matrix. Then, for all cycle C 2k of length 2k of P a,II , for l = 1, 2, θ P a,II (C 2k ) = 0 if and only if θ P a,II (π II,l (C 2k )) = 0.

Proof: Let C 2k be a cycle of length 2k of P a,II . Since O N (a) = 6 mod N , then according to the definition of a type-II exponent matrix, a(1-a) = 1 mod N and a 2 = a-1. Thus, θ P a,II (C 2k ) = 0 mod N if and only if aθ P a,II (C 2k ) = 0 mod N and if and only if a 2 θ P a,II (C 2k ) = 0 mod N . According to the definition of a cycle, n s = n s+1 when s odd. This property is equivalent to n s = n s-1 when s even, thus it is possible to write θ P a,II (C 2k ) = 0 mod N as k-1 s=0 p m2sn2s -p m2s-1n2s = 0 mod N.

Thus, using ( 6), θ P a,II (C 2k ) = 0 mod N can be expressed as

k-1 s=0 (u(m 2s )a m2s-1 -u(m 2s-1 )a m2s-1-1 ))γ n2s = 0 mod N. (10)
By taking λ(m 2s , m 2s-1 ) = u(m 2s )a m2s-1u(m 2s-1 )a m2s-1-1 , (10) gives

Since m 2s = m 2s-1 and m = 3, the only six possible couples (m 2s , m 2s-1 ) are (0, 1), (0, 2), (1, 2), (1, 0), (2, 0) and (2, 1). Table II shows that aλ(m 2s , m 2s-1 ) = -λ(π II,1 (m 2s ), π II,1 (m 2s-1 )) for all possible couples (m 2s , m 2s-1 ). Finally, since π II,2 = π II,1 • π II,1 , θ P a,II (π II,2 (C 2k )) = (-a) 2 θ P a,II (C 2k ) and thus:

θ P a,II (π II,2 (C 2k )) = 0 ⇐⇒ θ P a,II (C 2k ) = 0.
Example 3.12: Let P 3×9 be the exponent matrix of Fig. 3. Fig. 3.a shows examples of cycles of length 4, 6, and 8. Fig. 3.b and 3.c show the corresponding IR-equivalent cycles using π II,1 and π II,2 , respectively. Thus, when a type-I or type-II IRS-SMS exponent matrix is used, a new type of equivalence between cycles appears. This equivalence relation will be called IR-equivalence, to distinguish it from the strictly equivalent class, but its effect is identical: it reduces the number of equivalence classes by a factor close to m -1 for type-I IRS matrix and close to 3 for type-II IRS matrix. This reduction of the number of equivalence classes translates directly into a reduction of the

۾ ൌ 0 0 0 0 0 1 ܽ ܽ ଶ 0 ߛ ଶ ܽߛ ଶ ܽ ଶ ߛ ଶ 0 ߛ ଷ ܽߛ ଷ ܽ ଶ ߛ ଷ 0 ߛ ସ ܽߛ ସ ܽ ଶ ߛ ସ 0 ߛ ହ ܽߛ ହ ܽ ଶ ߛ ହ 0 ߛ ܽߛ ܽ ଶ ߛ 0 ߛ ܽߛ ܽ ଶ ߛ 0 ߛ ଼ ܽߛ ଼ ܽ ଶ ߛ ଼ ସൈଽ 0 0 0 0 0 1 ܽ ܽ ଶ 0 ߛ ଶ ܽߛ ଶ ܽ ଶ ߛ ଶ 0 ߛ ଷ ܽߛ ଷ ܽ ଶ ߛ ଷ 0 ߛ ସ ܽߛ ସ ܽ ଶ ߛ ସ 0 ߛ ହ ܽߛ ହ ܽ ଶ ߛ ହ 0 ߛ ܽߛ ܽ ଶ ߛ 0 ߛ ܽߛ ܽ ଶ ߛ 0 ߛ ଼ ܽߛ ଼ ܽ ଶ ߛ ଼ ସൈଽ c) a) 0 0 0 0 0 1 ܽ ܽ ଶ 0 ߛ ଶ ܽߛ ଶ ܽ ଶ ߛ ଶ 0 ߛ ଷ ܽߛ ଷ ܽ ଶ ߛ ଷ 0 ߛ ସ ܽߛ ସ ܽ ଶ ߛ ସ 0 ߛ ହ ܽߛ ହ ܽ ଶ ߛ ହ 0 ߛ ܽߛ ܽ ଶ ߛ 0 ߛ ܽߛ ܽ ଶ ߛ 0 ߛ ଼ ܽߛ ଼ ܽ ଶ ߛ ଼ ସൈଽ ܥ ଼ ܥ ଵ ߠ ۾ ሺܥ ଼ ሻ: 0 െ ߛ ଶ ܽ ଶ ߛ ଶ െ ܽ ଶ ߛ ଷ ߛ ଷ െ 1 ܽ ଶ െ 0 ߠ ۾ ሺܥ ଵ ሻ: 0 െ 0 ߛ ହ െ ߛ ܽߛ െ ܽߛ ܽ ଶ ߛ െ ܽ ଶ ߛ ଼ ܽߛ ଼ െ ܽߛ ସ ߨ ூ,ଵ ሺܥ ଼ ሻ ߨ ூ,ଵ ሺܥ ଵ ሻ ߨ ூ,ଶ ሺܥ ଼ ሻ ߨ ூ,ଶ ሺܥ ଵ ሻ b)
(m 2s , m 2s-1 ) λ(m 2s , m 2s-1 ) aλ(m 2s , m 2s-1 ) π II,1 ((m 2s , m 2s-1 )) λ(π II,1 (m 2s , m 2s-1 )) (0, 1) -1 -a (2,0) a (0, 2) -a -a 2 = 1 -a (2,1) a -1 (1, 2) 1 -a a -a 2 = 1 (0,1) -1 (1, 0) 1 a (0,2) -a (2, 0) a -a 2 = -1 + a (1,2) 1 -a (2, 1) a -1 a -a 2 = -1 (1,0) 1 ۾ ൌ 0 0 0 0 1 ܽ 0 ߛ ଶ ܽߛ ଶ 0 ߛ ଷ ܽߛ ଷ 0 ߛ ସ ܽߛ ସ 0 ߛ ହ ܽߛ ହ 0 ߛ ܽߛ 0 ߛ ܽߛ 0 ߛ ଼ ܽߛ ଼ ଷൈଽ 0 0 0 0 1 ܽ 0 ߛ ଶ ܽߛ ଶ 0 ߛ ଷ ܽߛ ଷ 0 ߛ ସ ܽߛ ସ 0 ߛ ହ ܽߛ ହ 0 ߛ ܽߛ 0 ߛ ܽߛ 0 ߛ ଼ ܽߛ ଼ ଷൈଽ ߠ ۾ ሺܥ ସ ሻ: 0 െ 0 ߛ ଶ െ 1 ߠ ۾ ሺܥ ሻ: 0 െ 0 ߛ ସ െ ߛ ହ ሺܽߛ ହ െ ܽߛ ଷ ሻ ߠ ۾ ሺܥ ଼ ሻ: 0 െ 0 ߛ ଼ െ ߛ 0 െ 0 ሺܽߛ ଼ െ ܽߛ ሻ 0 0 0 0 1 ܽ 0 ߛ ଶ ܽߛ ଶ 0 ߛ ଷ ܽߛ ଷ 0 ߛ ସ ܽߛ ସ 0 ߛ ହ ܽߛ ହ 0 ߛ ܽߛ 0 ߛ ܽߛ 0 ߛ ଼ ܽߛ ଼ ଷൈଽ ߨ ூூ,ଵ ሺܥ ସ ሻ ߨ ூூ,ଵ ሺܥ ሻ ߨ ூூ,ଵ ሺܥ ଼ ሻ ߨ ூூ,ଶ ሺܥ ସ ሻ ߨ ூூ,ଶ ሺܥ ሻ ߨ ூூ,ଶ ሺܥ ଼ ሻ a) b) c) ܥ ସ ܥ ܥ ଼ Fig. 3.
Example of cycles of length 4, 6, and 8 and IR-equivalent cycles number of constraints on the exponent matrix. Note that when m = 3, it is more efficient to search type-II IRS exponent matrices than type-I IRS exponent matrices. In fact, with a type-II IRS exponent matrix, the number of equations is reduced by a factor of 3 whereas is reduced by only a factor m -1 = 2 for a type-I IRS exponent matrix.

C. IRS technique as an a priori step of greedy search algorithm

Theorems 3.7 and 3.11 show that there might exist a proper cyclic subgroup a of Z × N from which we can pick non-zero components of P 1 as P 1 = 0, 1, a, a 2 , • • • a m-2 T . When a exists, P 1 is directly defined, and the number of non-equivalent cycles is reduced by at least a factor close to 3. The search algorithm will then take the sub-matrix [ 0 P 1 ] as a base and try to find proper values of γ j s so that P m×n meets the girth condition. Nevertheless, for a given N value, there is not always an a value satisfying the condition of theorems 3.7 and/or 3.11. Moreover, even if an a value exists, the resulting two column type-I or type-II exponent matrix [ 0 P 1 ] is always of girth 8 but not necessarily of girth 10 or 12. The question is thus which portion of N s remains after the sieved process for a given girth g and a given column weight m. Proposition 3.13: For N > 3, all type-II IRS QC-LDPC matrices with an exponent matrix P a,II 3×2 of size 3 × 2 are of girth 12.

Proof: First, the girth is lower than, or equal to, 12 according to [START_REF] Tanner | A recursive approach to low complexity codes[END_REF]. According to (4), there are in P a,II 3×2 three cycles of length 4, no cycle of length 6, six cycles of length 8 and no cycle of length 10. Owing to Theorem 3.11, the number of cycles to be checked is reduced to one cycle of length 4 and two cycles of length 8. Those cycles can be respectively the cycles C 4 , C b 8 and C c 8 depicted in parts (b), (e), and (f) of Fig. 1, respectively. Thus, For type-I IRS QC-LDPC code, the girth of the QC-LDPC code generated from exponent matrix P a,I m×2 , m > 3 is not always girth 12. For example, P a=73,I

θ P (C 4 ) = 1 mod N θ P (C b 8 ) = 2 mod N θ P (C c 8 ) = 2 -a mod N If N > 3, θ P (C 4

4×2

, N = 216 gives a QC-LDPC matrix of girth 8 only, while P a=5,I 4×2 , N = 215 gives a QC-LDPC matrix of girth 12. Nevertheless, finding a proper N and, accordingly, the existence of a suitable cyclic subgroup that results in P 1 is not time-consuming. Given a fixed m, it will take few milliseconds for MATLAB software to check if Z × N is a proper candidate or not.

Proposition 3.14: Let N ≥ 6, a, b be two different elements of Z × N of order m -1 with a = b = S. Owing to row/column permutations, the Tanner graph constructed from type-I exponent matrix P a,I m×2 is equivalent to the Tanner graph constructed from type-I exponent matrix P b,I m×2 .

Proof: Since a = b = S, the set of elements of the second column of P a,I m×2 , i.e. the set {0, 1, a, a 2 , . . . , a m-2 } is equal to the set of element of the second column P b,I m×2 , i.e. the set {0, 1, b, b 2 , . . . , b m-1 }. Thus, an appropriate permutation of rows allows the transformation of P a,I m×2 into P b,I m×2 (the coefficients of the first column are all zero, and thus not affected by any permutation). Since a necessary and sufficient condition for non-identity element z = x y ( x = S, y ∈ N) to be a generator of S is gcd (y, O(S)) = 1, it is easy to see that a and b = a 5 are the only generators of S. Since, gcd(a 5 , N ) = 1, according to [START_REF] Tasdighi | Efficient Search of Girth-Optimal QC-LDPC Codes[END_REF], a 5 P a,II 3×2 and P a,II 3×2 are equivalent exponent matrices due to row/column permutations. The second column of a 5 P a,II 3×2 is 0, a 5 , 1

T = [0, b, 1] T . Swapping the second and the third rows of a 5 P a,II 3×2 gives P b,II 3×2 , thus, the two exponent matrices are equivalent.

In short, the search algorithm needs to test one generator per each permissible cyclic subgroup S to find the exponent matrix P m×n of code with girth g (g = 8, 10, 12).

The final point is "there might be more than one permissible cyclic subgroup of Z × N that meet the conditions to construct type-I or type-II exponent matrices, but not all of them would necessarily result in the matrix P m×n with girth g (g = 8, 10, 12)". For example, Z × N =301 has two permissible cyclic subgroups S 1 = 80 and S 2 = 136 of order 6 where their generators satisfy the property a(1 -a) = 1 mod 301. The search algorithm in Section IV can find an exponent matrix P a=80,II 3×10 with girth 10 but not with P a=136,II 3×10 . Fig. 4 shows a 2D representation of integers between 1 and 10 4 sieved by type-I property for m = 4. The black pixels correspond to values of N with no admissible a value, and the red pixels represent N values with at least one admissible a value but no girth 12 type-I [ 0, P 1 ] exponent matrix. Finally, the white pixels indicate N values with at least one girth 12 type-I [ 0, P 1 ] exponent matrix. It may be noted that, on average, 60% of integers between 1 and 10 4 can give a girth 12 type-I [ 0, P 1 ] exponent matrix. We observe that the white pixels are uniformly spread between 1 and 10 4 . For N values corresponding to a white pixel, the number of distinct order 3 subgroups of Z × N varies from 2 to 13. The same uniform dispersion is observed also for type-II QC-LDPC codes and type-I QC-LDPC codes with m = 5 and m = 6. The proportion of values able to give a girth 12 two columns IRS exponent matrix are 51.9 %, 24.2%, and 13.4% for type-I m = 5, type-I m = 6 and type-II m = 3, respectively.

D. Controlled greedy search algorithm

In this section, we present a modified controlled greedy search algorithm that uses the SMC technique [START_REF] Tadayon | Efficient Search of Compact QC-LDPC and SC-LDPC Convolutional Codes With Large Girth[END_REF]. Let Γ k = {0, 1, γ 2 , γ 3 , . . . , γ k-1 } be a set of size k of elements of Z N . The property ρ g (Γ k , P 1 , N ) is true if and only if the exponent matrix matrix with a girth greater than or equal to g when expanded by a factor of N . We call Φ g (Γ k , P 1 , N ) the ordered set of coefficients of Z N so that a vector Γ k+1 of size k + 1 constructed by the concatenation of Γ k and any coefficient of Φ g (Γ k , P 1 , N ) also gives an exponent matrix of girth g. In a more formal way

0 P 1 γ 2 ⊗ P 1 γ 3 ⊗ P 1 . . . γ k-1 ⊗ P 1 gives a
β ∈ Φ g (Γ k , P 1 , N ) ⇐⇒ ρ g (Γ k ∪ {β}, P 1 , N ) is true. (12)
The search for a solution of degree (m, n) for a given lifting degree N is conducted in two steps. The first step involves the enumeration of a single element per class of the a values satisfying the condition of Theorem 3.7. This step is described in Algorithm 1 part I for m > 3. To do so, the set of values A is initialized as A = {2, 3, . . . , N -1}. The values of A are extracted one by one. The second step starts each time an extracted value a fulfills the condition of Theorem 3. Extract an element a of A.

4:

A ← A \ {a} 5:

if O N (a) = m -1 then 6:

P 1 ← 0, 1, a, a 2 , . . . , a m-2 T

7:

A ← A \ {a k } k=2,3,...,m-2

8:

S ← Φ g (Γ

s(i) ← |S ∩ Φ g (Γ ∪ S(i), P 1 , N )| (note: s is a vector).
S k ← S ∩ Φ g (Γ k , P 1 , N ) 23: if |Γ k | + |S k | ≥ n then 24: Γ n ← search (Γ k , S k , N, n, P 1 , G) 25: else 26:
Return ∅ It can be done in a limited time (less than a few days) only for low values of n. For a large n, the first values of G are set to 1 or 2 to reduce the search space to a reasonable size. Note that |X| represents the cardinal of the set X. We observe that when taking random coefficients in the exponent matrix P, θ P (C) (see [START_REF] Mackay | Good error-correcting codes based on very sparse matrices[END_REF]) takes random values between 0 and N -1. Thus, the probability that θ P (C) = 0 mod N is (1 -1/N ). Assuming that all the cycles are independent (which, of course, is not the case) then, under this groundless hypothesis, the expected number E 0 (m, n, g) of girth g m × n exponent matrices will be given by the size of the space multiplied by the probability that all the cycles satisfy (2), i.e.

E 0 (m, n, g) = N m•n (1 -1/N ) #P m×n (g) . (13)
Using the IRS method, the search space is greatly reduced, but, the number of Fossorier's equation that should be fulfilled is reduced by a factor close to max(3, m -1). The expectation of finding a solution is thus

E 1 (m, n, g) = N n-2 (1 -1/N ) #P m×n (g) max(3,m-1) (14)
For g = 10, m = 3, n = 10, and N = 301, the ratio E 0 /E 1 is of order 5 × 10 -46 , which shows that finding a girth g code is 10 46 times likely to happen if one uses IRS instead of a normal search, on average. Although the argument is not solid, it helps explain why most of the best IRS matrices found have a smaller size than the already published fully-connected exponent matrices.

Although the IRS construction method reduces the search space by a factor N m-2 compared to the SMC technique alone, the search space is still in O(N n-2 ). However, the numerical results presented in the next section demonstrate the efficiency of the proposed IRS construction method.

IV. NUMERICAL RESULTS

Table III shows, for several sets of parameters (m, n, g), the minimum lifting degree N min found to allow constructing a type-I and type-II IRS QC-LDPC matrix (the description of the found IRS matrices is shown in Appendix I). In Table III, the best state-of-the-art minimum N value is also provided with the associated reference.

A first conclusion of this study is that the state-of-theart lower bound on the lifting degree required to obtain a girth-10 expanded matrix [START_REF] Amirzade | Lower Bounds on the Lifting Degree of QC-LDPC Codes by Difference Matrices[END_REF] is a little over-estimated when m > 3 and n > 3. In fact, we found a counter example for (m, n) = (4, 7). The lower bound is supposed to be L(m, n) = 2 m 2 n 2 + 1, which gives, L(4, 7) = 253 whereas we found a girth-10 4 × 7 expanded matrix with a lifting degree of N = 247 (see Appendix I). A corrected lower bound derived from the Double Difference matrix defined in [START_REF] Amirzade | Lower Bounds on the Lifting Degree of QC-LDPC Codes by Difference Matrices[END_REF] is derived in Appendix II. The corrected lower bound L c (m, n) is given as

L c (m, n) = 2 m 2 n 2 -2 m -2 2 n -2 2 + 1. ( 15
)
Note that L c (4, 7) = 237, which is compatible with the found value N = 247.

A second conclusion is that we were always able to find an equal or better (sometimes, significantly better) solution than that of the state of the art for girth 10 and girth 12 matrices. For girth 8, no matrices of variable node degree m < 6 better than the state-of-the-art have been found. Hence, the result are not reported. For m = 6, except for n ∈ {7, 8, 9, 10}, better solutions have been found. Finally, future search simulation may help us find smaller lifting degree. Updated values will be reported online in [START_REF] Boutillon | Low Density Parity Check code[END_REF]. EXPONENTS a,b,c,d,e,f,g,h REFER TO [START_REF] Bocharova | Searching for Voltage Graph-Based LDPC Tailbiting Codes With Large Girth[END_REF], [START_REF] Tasdighi | Efficient Search of Girth-Optimal QC-LDPC Codes[END_REF], [START_REF] Battaglioni | Compact QC-LDPC Block and SC-LDPC Convolutional Codes for Low-Latency Communications[END_REF], [START_REF] Derrien | Additive, Structural, and Multiplicative Transformations for the Construction of Quasi-Cyclic LDPC Matrices[END_REF], [START_REF] Xu | On the Girth of Tanner (3,13) Quasi-Cyclic LDPC Codes[END_REF], [START_REF] Xu | Tanner $(J,L)$ Quasi-Cyclic LDPC Codes: Girth Analysis and Derived Codes[END_REF], [START_REF] O'sullivan | Algebraic construction of sparse matrices with large girth[END_REF], [START_REF]Symmetrical Constructions for Regular Girth-8 QC-LDPC Codes[END_REF] RESPECTIVELY.

P a,II 3×4 P a,II 3×5 P a,II 3×6 P a,II 3×7 P a,II 3×8 P a,II 3×9 P a,II 3×10 P a,II 3×11 P a,II 3×12 P a,II 3×13 P a,II 3×14 P a,II 3×15 P a,II 3×16 P a,II 3×18 P a,II 3×19 P a,II 3×20 P a,II 3×21 P a,II 3×22 P a,II 3×23 P a,II 3×24 P a,II 3×25 P a,II 3×26 P a,II 3×27 P a,II 3×28 P a,II 3×29 P a,II 3,30 P a,II

V. CONCLUSION

A new construction method of fully-connected QC-LDPC codes of girth g = 8, 10, and 12 has been presented. This method is called Integer Ring Sieve. For a lifting factor of size N , the IRS construction requires that the second column of the exponent matrix takes its values in an integer ring of Z × N with a specific property. The constraint on the second column has several beneficial effects: it reduces the search space, it reduces the number of constraints to be checked, and ultimately, for most of the exponent matrix size (m, n) with a girth objective of 8, 10, or 12, it gives a smaller lifting factor than the ones already reported in the literature. The table of best-found IRS QC-LDPC codes is also given in the paper. Finally, the IRS method allows us to find a counter-example showing that the lower bound proposed in [START_REF] Amirzade | Lower Bounds on the Lifting Degree of QC-LDPC Codes by Difference Matrices[END_REF] is a little overestimated. A corrected lower bound has been given.

ACKNOWLEDGMENT

This project has been partly funded by the Brittany Region in the frame of the call "Campagne d'Activité Postdoctorale 2017". The authors would like to thank the anonymous reviewers and the associated editor, M. Andrew Thangaraj, for their insightful comments that helped to improve the paper.

Fig. 1

 1 Fig.1gives several examples of strictly equivalent cycles in an exponent matrix P.

Fig. 1 .

 1 Fig. 1. Sample paths for cycles of length from 4 to 10 involved in an exponent matrix P: a) exponent matrix of size m × n. b) paths of length 4 of strictly equivalent cycles. c) paths of length 6 of strictly equivalent cycles. d) paths of length 8 of strictly equivalent cycles. e) path of a length 8 cycle non-strictly equivalent to (d). f) another path of a length 8 cycle non-strictly equivalent to the paths (d) and (e). g) path of a length 10 cycle.

Lemma 3 . 9 :

 39 If a ∈ Z × N , satisfies a(1 -a) = 1 mod N , then a 3 = -1 mod N . Moreover, for N > 3, O N (a) = 6. Proof: a(1 -a) = 1 mod N implies a 2 -a + 1 = 0 mod N , thus (1 + a)(a 2 -a + 1) = 0 mod N , and finally, (a 3 + 1) = 0 mod N . Since a 3 = -1 mod N , we can deduce that O N (a) = 3 and a 6 = 1. The case a 2 = 1 mod N gives a = 2 and N = 3. Thus, if N > 3, we have O N (a) = 6.

  2s , m 2s-1 )γ n2s = 0 mod N.

Fig. 2 .

 2 Fig. 2. Samples of cycles with different length in P 4×9 : a) primary underlined paths for cycles C 8 and C 10 . b) isomorphic paths to the paths in part (a) derived from transformation π I,1 (C). c) isomorphic paths to the paths in part derived from transformation π I,2 (C)

  ) and θ P (C a 8 ) are not equal to zero modulo N . The third cycle requires more attention. In fact, θ P (C b 8 ) = 0 mod N implies a = 2. When a = 2, a(1 -a) = 1 mod N gives -3 = 0 mod N and thus N = 3. Since N > 3, θ P (C b 8 ) = 0 mod N.

Proposition 3 . 15 :

 315 Let N ≥ 6, a, b be two different elements of Z × N that satisfy a(1 -a) = 1 mod N , b * (1 -b) = 1 mod N and a = b = S. The Tanner graph of the constructed matrix P a,II m×2 with the second column [0, 1, a] is equivalent, due to a row permutation, to the Tanner graph of the matrix P b,II m×2 with the second column [0, 1, b] T . Proof: The integers a and b verify a (1 -a) = 1 = b (1 -b), moreover, O N (a) = O N (b) = 6 (Lemma 3.9).

Fig. 4 .

 4 Fig. 4. 2D representation on a 100 × 100 grid of the first 10 4 integers. Integers N having no subgroup of Z × N of order 4 are represented by black pixels. If no type-I exponent matrix of size (4,2) exists, the integer is represented by a red pixel; it is white otherwise. The first top left pixel corresponds to N = 1, and its left neighbor corresponds to N = 2. The last pixel of the first line corresponds to N = 100. The last bottom right pixel corresponds to N = 10 4 .

Algorithm 1

 1 7. The function search is launched to try to find a solution Γ n . In case of success, the algorithm successfully stops. Otherwise, the elements of a are suppressed from the search space A. The process continues until no values remain in A, and the search is unsuccessful. Note that for m = 3, the condition O N (a) = m -1 of line 5 should be replaced by the condition (a(1 -a) = 1 mod N ), and line 7 should be replaced by the instruction A = A \ {a k } k=1,2,3,4,5 . The search function is described in Algorithm 1, part II. It is a recursive function that tries to increase recursively the size of Γ until it reaches a size of n. The arguments of the search function are Γ, S, N , n, P 1 , and a vector G of size n that controls the processing effort. Let us describe the processing during the first call of the function in line 9. The arguments of this first call are Γ 1 = {0} and S (defined in line 8), as the set of values compatible with Γ 1 (see (12)). Lines 14 and 15 set up the greedy search. For i = 1, 2, . . . , |S|, the number s(i) of triplets Γ 3 = {0, S(i), µ}, µ ∈ S satisfying the condition ρ(Γ 3 , P 1 , N ) is computed (note that s(i) < |S|). The s(i) are thus sorted in decreasing order (line 16), and the first G(|Γ|) = G(1) elements of S (line 17) associated with the highest values of vector s are tested. For each tested value, a vector Γ k of size 2 is generated (line 18). The tested value is suppressed from the set S (line 19), and then the subset S k of S of values compatible with Γ k is created (line 20). If the size of S k plus the size of Γ k is greater than or equal to n, or, if it is still possible to generate a Γ vector of length n, then the search function is called again with a Γ set of size 2. The process is recursively reiterated until a length n Γ vector is found or until no new possibility remains to be explored. The complexity of the search is controlled by a vector G of size n. The k th value G(k) of G indicates that only the most "promising" G(k) branches will be explored inside each depth k recursive call of the search function. Note that when all the values of G are equal to N the search algorithm is exhaustive. Controlled greedy search algorithm for m ≥ 3 Input: Parameters n, m, N of the code, targeted girth g, vector G of size n to control the greedy search effort. Output: Eventually, a set of coefficients Γ n of size n if success, an empty set otherwise. Part I: primary step 1: A ← {2, 3, . . . , N -1}, Γ n = ∅, Γ 1 = {0} 2: while A = ∅ and Γ n = ∅ do 3:

16 :I

 16 ← sort index (s) (note: s(I(1)) ≥ s(I(2)) ≥ . . . ≥ s(I(|S|))).

17

 17

  :

for j = 1

 1 to min (|S|, G(|Γ|)) do 18: if |Γ n | = n then Return Γ n 19:

else 20 :

 20 Γ k ← Γ ∪ {S(I(j))} 21:S ← S \ {S(I(j))} 22:

  37 a,b 61 a,b 91 b 139 c,d 181 c 241 c 313 c 397 c 523 c b 151 c 271 c 457 c 691 c 991 c 1447 c 2161 c 4730 a 5851 e

TABLE I NUMBER

 I OF STRICTLY EQUIVALENT CLASSES OF LENGTH 2k CYCLES

TABLE II COMPUTATION

 II OF aλ(m 2s , m 2s-1 ) FOR ALL POSSIBLE COUPLES.

  1 , P 1 , N ) Γ n ← search (Γ, S, N, n, P 1 , G) 11: Γ n ← Γ 12: if |Γ n | = n then Return Γ n 13: else

	9:	Γ n ← search (Γ 1 , S, N, n, P 1 , G)
		Part II: search function
	10: 14:	for i = 1 to |S| do
	15:

TABLE III MINIMUM

 III FOUND LIFTING DEGREE OF IRS EXPONENT MATRICES FOR GIRTH g = 8, 10 AND 12. N g min IS THE SMALLEST FOUND LIFTING DEGREE OF THE EXPONENT MATRIX WITH GIRTH g. THE LIFTING DEGREE OF THE SHORTEST EXISTING CODES IS GIVEN WITH AN EXPONENT THAT INDICATES THE CORRESPONDING REFERENCE.

  241 c 307 c 409 c 577 c 787 c 1039 c 1381 c 95 h 125 h 150 h 182 h 218 h 254 h 296 h 337 h 380 h 429 h

															3,31
	N g=10 min	1453 1723 2089 2197 2689 3049 3331 3577 -----5659 f -4801 f	--	--	--	--	--	--
		P a,I 4×4 P a,I 4×5 P a,I 4×6 P a,I 4×7 P a,I 4×8 P a,I 4×9 P a,I 4×10 P a,I 4×11 P a,I 4×12 P a,I 4×13 P a,I 4×14 P a,I 4×15 P a,I 4×16 P a,I 4×17
	N g=10 min	73 -	133 139 c -199 247 403 541 703 883 1123 1429 1933 2389 2881 3397 ----
	N g=12 min	254 -	571 1087 2203 4489 8966 607 c 1201 c 2371 c 6607 g 12071 g	--	--	--	--	--	--	--	--
		P a,I 5×4 P a,I 5×5 P a,I 5×6 P a,I 5×7 P a,I 5×8 P a,I 5×9 P a,I 5×10 P a,I 5×11 P a,I 5×12 P a,I 5×13 P a,I 5×14 P a,I 5×15 P a,I 5×16 P a,I 5×17
	N g=10 min	175 -	205 -	511 -	763 1067 1417 1903 2431 3445 4849 5933 1471 f -1621 f -2861 f -5981 f -	--	--	-
		P a,I 6×4 P a,I 6×5 P a,I 6×6 P a,I 6×7 P a,I 6×8 P a,I 6×9 P a,I 6×10 P a,I 6×11 P a,I 6×12 P a,I 6×13 P a,I 6×14 P a,I 6×15 P a,I 6×16 P a,I 6×17
	N g=8 min	41 -	61 -	101 -	101 70 h	121	151	181	181	181	241	281	331	341	401
	N g=10 min	251 P a,I 6×18 P a,I 421 6×19 P a,I 571 6×20 P a,I 971 1331 1891 2621 3421 4261 5611 7171 6×21 P a,I 6×22 P a,I 6×23 P a,I 6×24 P a,I 6×25 P a,I 6×26 P a,I 6×27 P a,I 6×28 P a,I -6×29 P a,I -6,30 P a,I -6,31
	N g=8 min	451 478 g 530 g 584 g 521 571	601 -	661 2113 f 967 f 751	781 -	881 1951 f 2029 f 941	941 1051 1111 1111 -3529 f 4003 f -	--

In the case that the QC-LDPC code is not fully-connected, m and n are often noted by dv and dc in the literature, respectively.

APPENDIX I

In this appendix, an abstract list of the parameters of our constructed codes is provided. The parameters of each code are embedded in braces as {N, P a,T m×n , [0, 1, γ 2 , γ 3 , • • • , γ n-1 ]}, with T indicating the type of IR exponent matrix, i.e. P a,I or P a,II .

APPENDIX II

In this appendix, we give first a toy counter-example showing that condition (ii) of Theorem 1 of [START_REF] Amirzade | Lower Bounds on the Lifting Degree of QC-LDPC Codes by Difference Matrices[END_REF] is not a necessary condition to prevent girth-8 cycles, and thus, that the derived lower bound is over-estimated. Next, we proposed an updated accurate lower bound.

Let us consider the following exponent matrix P 4×4 with an expansion factor N , where ∞ denotes the zero matrix.

The corresponding 4 2 × 4 2 = 6 × 6 double difference DD matrix defined in [START_REF] Amirzade | Lower Bounds on the Lifting Degree of QC-LDPC Codes by Difference Matrices[END_REF] is given as

In [START_REF] Amirzade | Lower Bounds on the Lifting Degree of QC-LDPC Codes by Difference Matrices[END_REF], the lower bound is derived from the claim that all the finite values of the L(m, n) = 2 m 2 n 2 elements of DD should be distinct. In the toy counter-example, the double difference matrix DD of P 4×4 contains 4 finite values, which gives a lower bound equals to 4 + 1 = 5. Nevertheless, setting a = 1, b = 1 and N = 3 gives a matrix of girth 12, which is in contradiction with the fact that the lower bound is equal to N = 5. The trick is that two elements of the matrix DD corresponding to two length-4 cycles that share neither lines nor columns can be equal without generating a length-8 cycle in the expanded matrix.

From this observation, it is possible to update the lower bound for a fully connected exponent matrix P m×n . Let DD be the double difference matrix associated to P m×n [START_REF] Amirzade | Lower Bounds on the Lifting Degree of QC-LDPC Codes by Difference Matrices[END_REF]. The couple DD(1, 1) is equal to DD(1, 1) = (θ P (C 0 4 ), -θ P (C 0 4 ) mod N ) with C 0 4 defined as the cycle that goes through the first two lines and the first two rows of P m×n , i.e., θ P (C 0 4 ) = p 1,1 -p 1,2 +p 2,2 -p 2,1 mod N (see Fig.