Xavier Montagutelli
email: xavier.montagutelli@pasteur.fr

Matthieu Prot

Grégory Jouvion

Laurine Levillayer

Laurine Conquet

Edouard Reyes-Gomez

Flora Donati

Melanie Albert

Sylvie Van Der Werf

Jean Jaubert

Etienne Simon-Lorière
email: etienne.simon-loriere@pasteur.fr

A mouse-adapted SARS-CoV-2 strain replicating in standard laboratory mice

Keywords:

SARS-CoV-2 has infected almost 200 million humans and caused over 4 million deaths worldwide. Evaluating countermeasures and improving our understanding of COVID-19 pathophysiology require access to animal models that replicate the hallmarks of human disease. Mouse infection with SARS-CoV-2 is limited by poor affinity between the virus spike protein and its cellular receptor ACE2. We have developed by serial passages the MACo3 virus strain which efficiently replicates in the lungs of standard mouse strains and induces age-dependent lung lesions. Compared to other mouse-adapted strains and severe mouse models, infection with MACo3 results in mild to moderate disease and will be useful to investigate the role of host genetics and other factors modulating COVID-19 severity.

Main text:

Since its emergence in 2019, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has infected almost 200 million humans and caused over 4 million deaths worldwide, resulting also in economic crisis. To help developing and assessing preventive and therapeutic interventions, animal models have been produced in several species. Unlike other species such as non-human primate, hamsters, ferrets, minks and cats, mice show low permissiveness to SARS-CoV-2 replication due to poor binding of the virus spike protein on the rodent version of the cellular receptor angiotensin-converting enzyme 2 (ACE2) [START_REF] Johansen | Animal and translational models of SARS-CoV-2 infection and COVID-19[END_REF]. This natural resistance can be overcome either by expressing the human ACE2 receptor by transgenesis [START_REF] Winkler | SARS-CoV-2 infection of human ACE2-transgenic mice causes severe lung inflammation and impaired function[END_REF][START_REF] Yinda | K18-hACE2 mice develop respiratory disease resembling severe COVID-19[END_REF] or by transduction [START_REF] Hassan | A SARS-CoV-2 Infection Model in Mice Demonstrates Protection by Neutralizing Antibodies[END_REF][5][START_REF] Ku | Intranasal vaccination with a lentiviral vector protects against SARS-CoV-2 in preclinical animal models[END_REF][START_REF] Sun | Generation of a Broadly Useful Model for COVID-19 Pathogenesis, Vaccination, and Treatment[END_REF], or by mutating the virus in the receptor binding domain (RBD) of the spike protein to improve its binding to mouse ACE2 [8][9][10][11][12]. These strategies have yielded models with diverse clinical presentations. Upon SARS-CoV-2 infection, transgenic K18-hACE2 mice develop severe symptoms characterized by rapid body weight loss and death in 5-8 days [START_REF] Winkler | SARS-CoV-2 infection of human ACE2-transgenic mice causes severe lung inflammation and impaired function[END_REF][START_REF] Yinda | K18-hACE2 mice develop respiratory disease resembling severe COVID-19[END_REF]. Transduction of hACE2 in the respiratory tract confers permissiveness to viral replication to high titers but infection remains essentially asymptomatic with mild and transient body weight loss [5,[START_REF] Ku | Intranasal vaccination with a lentiviral vector protects against SARS-CoV-2 in preclinical animal models[END_REF]. Mouseadapted viral strains have been produced either by reverse genetics to modify specific residues of the RBD [8], by serial passaging in vitro or in vivo to select variants with improved replication performances [9,11,12], or by a combination of both [10]. Infection with these variants, which carry different sets of mutations, results in a broad range of models from mildly symptomatic to severe with lethality, depending on the genetics and age of mice.

Serial passaging to adapt the virus

In an attempt to infect mice with SARS-CoV-2, we inoculated intranasally one of the first viral isolates (IDF0372) to young adult mice of C57BL/6 and of three strains of the Collaborative Cross (CC), a collection of inbred strains with large genetic diversity [13]. While none of the C57BL/6 and CC002 mice showed significant viral loads in their lungs 4 days post-infection (dpi), one in four CC001 and one in four CC071 mice had viral loads above the limit of detection (LOD), and two in three CC002 and two in four CC071 mice had detectable viral RNA (yet below 100 eqPFU/g tissue). Histological analysis of the lungs was normal in C57BL/6 mice, while the virus-positive CC mice showed peribronchiolar inflammatory infiltrates with epithelial erosion (Fig S1). This observation opened a way to attempt adapting SARS-CoV-2 to mice using young adults of CC strains. We proceeded by serial passaging first on CC071 alone and later in combination with CC001 and C57BL/6 genetic backgrounds (Fig 1A). After an initial increase, lung viral load stabilized in later generations (Fig 1B). Of note, since mice were inoculated with fresh lung homogenates before viral load was quantified, the inoculation dose was variable between passages. After 15 and 16 passages, two viral variants named MACo1 and MACo2 were isolated. Sequencing revealed a shared mutation Q493R (A22040G) located in the RBD region that directly interfaces with the receptor (Fig 1C). MACo2 was used for further passaging on BALB/c mice with the aim of favoring variants which would more efficiently replicate in standard laboratory strains. After 11 passages, another variant was isolated, cloned and named MACo3. Sequencing revealed the presence of a second mutation in the RBD: Q498R (A23055G). Two other aminoacid substitutions and one small deletion were also identified (Fig 1C). Mutations at residues 493 and 498 have been reported in other mouse-adapted strains (Fig 1D), suggesting that these positions can modulate efficient interaction between viral spike and mouse ACE2 (mACE2). MACo3 has retained the capacity to infect VeroE6 cells (suggesting preserved interaction with hACE2) and can also infect mouse cells expressing mACE, unlike the original virus (Fig 1E).

Infection of standard laboratory strains with MACo3

To assess the potential of MACo3 to induce morbidity and pathological features characteristic of COVID-19 in standard laboratory mice, we inoculated intranasally young (9 week-old) and aged (12-18 month-old) BALB/c and C57BL/6 mice with 10 5 inoculum. In young BALB/c mice, inoculation with MACo3 led to asymptomatic infection with a transient reduction of body weight (Fig 2A) and to moderate viral loads in the lung which decreased by two log10 units between dpi 3 and 6 (Fig 2B). In contrast, infection of one year-old BALB/c mice led to a rapid and durable body weight loss (peaking at 12% on dpi [START_REF] Yinda | K18-hACE2 mice develop respiratory disease resembling severe COVID-19[END_REF][START_REF] Hassan | A SARS-CoV-2 Infection Model in Mice Demonstrates Protection by Neutralizing Antibodies[END_REF]Fig 2C), with mice showing symptoms of ruffled fur, hunched back posture, reduced mobility and breathing difficulties that peaked at dpi 3-6 and slowly resolved until dpi 12. Consistently, higher titers of virus were detected in the lungs of aged mice at both day 3 and 6 post inoculation (Fig 2D ), compatible with slower resolution of the infection. We did not detect significant viral dissemination to other organs (Fig S2). Histopathological analysis of the lung revealed minimal broncho-interstitial pneumonia in young mice (Fig 2E), and more severe lesions (mild to moderate) in aged mice (Fig 2F). Alterations included interstitial inflammation, sometimes centered on blood vessels (perivasculitis), alteration and necrosis of bronchial and bronchiolar epithelial cells, and, in aged mice especially, endothelial cell injury and inflammation (endothelitis).

Infection of young C57BL/6 mice similarly led to an asymptomatic infection with moderate body weight loss peaking at dpi 2, but the difference with the mock-infected group persisted longer than in BALB/c mice. Weight decrease was more pronounced in aged C57BL/6 mice (peaking at 14% on dpi [START_REF] Yinda | K18-hACE2 mice develop respiratory disease resembling severe COVID-19[END_REF][START_REF] Hassan | A SARS-CoV-2 Infection Model in Mice Demonstrates Protection by Neutralizing Antibodies[END_REF] but mice developed only mild and transient symptoms (ruffled fur, hunched back posture and breathing difficulties). Viral replication in the lungs was less homogenous than in BALB/c mice, with a similar pattern of higher and longer detection of virus with increased age (Fig 3D). Like in BALB/c mice, the virus did not disseminate to other organs although low viral loads were detected in the feces of three out of four aged C57BL/6 mice at dpi 3 (Fig S2). Histopathological analysis revealed lesions similar to those observed in BALB/c mice. However, in comparison with BALB/c mice, inflammatory lesions were slightly more severe in young animals (Fig 3E), and the severity of the lesions was more heterogeneous in aged mice (Fig 3F).

MACo3 is a novel mouse-adapted SARS-CoV-2 strain which carries a small number of mutations. Its ability to replicate in mice is likely due to amino acid changes at positions 493 and 498 within the spike RBD. Changes at either or both these positions have been repeatedly detected in SARS-CoV-2 viruses passaged in mice [8,[10][11][12], and, in addition to N501Y [9,14], appears to constitute parsimonious evolutionary paths to productive interaction with the murine version of ACE2 (Fig 1D). Concordantly, Huang and colleagues similarly reported a mouse adapted strain carrying changes at positions 493 and 498, but to other basic amino-acids (K and H respectively). Retrospective analysis revealed that the Q493R mutation was fixed in early passages (Fig 1C ), enabling consistent replication in BALB/c mice (Fig 1B), prior to the fixation of Q498R. In other reports, amino acid changes at position 493 appeared secondarily, and were associated with increased pathogenicity for two SARS-CoV-2 strains: MA to MA10 [10] and MASCp6 to MASCp36 [12]. Of note, like other previously reported mouse adapted stains, MACo3 also carries changes outside of the spike which may contribute to the virus fitness and pathogenic profile in mice.

MACo2 and MACo3 expand the collection of mouse-adapted variants which are instrumental to investigate the molecular mechanisms of species-specific interactions between SARS-CoV-2 spike and the ACE2 receptor. Compared to other mouse-adapted variants, inoculation with MACo3 results in mild to moderate symptoms in the two laboratory mouse strains tested, despite high replication in the lungs. We previously reported similar observation upon infection with the beta (B.1.351) and gamma (P.1) variants of concern [14]. Overall, C57BL/6 mice were less susceptible to MACo3 inoculation than BALB/c mice, a difference which has been previously reported with another mouse-adapted SARS-CoV-2 strain [10]. Although more severe mouse models (in particular the K18-hACE2 transgenic model) have been used to test vaccine or antibody efficacy by assessing protection against mortality [15,16], body weight loss, viral replication in the respiratory tract and lung pathology are hallmarks of SARS-CoV-2 infection which are sufficient to test countermeasures efficacy [START_REF] Ku | Intranasal vaccination with a lentiviral vector protects against SARS-CoV-2 in preclinical animal models[END_REF]17]. We have successfully used MACo3 to demonstrate the protection conferred by a measles-vectored COVID-19 vaccine [18].

CC strains CC071 and CC001 were instrumental to adapting the virus to mice. They showed some permissiveness to the replication of an early isolate of SARS-CoV-2 (Pango lineage B), unlike C57BL/6 mice in which no virus was detected upon infection (Fig S1). Although others were successful, we also attempted twice to adapt a B.1 virus (carrying the D614G mutation) on aged BALB/c mice, without success. Most interestingly, CC001 has inherited its Ace2 allele from 129S1/SvImJ, while CC071 carries a NOD/ShiLtJ allele (http://csbio.unc.edu/CCstatus/CCGenomes/). Both alleles encode the same protein sequence as the C57BL/6 allele, suggesting that Ace2 is not responsible for the permissiveness of CC001 and CC071. As they can replicate in any mouse genetic background, mouse-adapted SARS-CoV-2 viruses such as MACo3 will be useful to identify host genetic and non-genetic factors influencing the severity of SARS-CoV-2 infections in mice. A: Body weight variations in 9-week-old C57BL/6 females after inoculation with PBS (n=8) or with 10 5 PFU of MACo3 (n=8). B: Viral load and viral titer in the lung of 9-week-old C57BL/6 females 3 days (n=6) or 6 days (n=6) after inoculation with 10 5 PFU of MACo3. C: Body weight variations in 50-week-old C57BL/6 males after inoculation with PBS (n=7) or with 10 5 PFU of MACo3 (n=7). B: Viral load and viral titer in the lung of 58-week-old C57BL/6 males 3 days (n=2) or 6 days (n=3) after inoculation with 10 5 PFU of MACo3. The dotted line represents the limit of detection (LOD), and undetected samples are plotted at half the LOD. E: H&E staining of lung section from young C57BL/6 mice. Top: Mild broncho-interstitial pneumonia with slightly more severe inflammation than in young BALB/c mice. Bottom: Endothelitis and perivasculitis. F: Aged C57BL/6 mice. Top: More severe broncho-interstitial pneumonia (in two out of four mice, the other mice displayed only minimal lesions). Bottom: Necrosis of bronchiolar epithelial cells (left) and endotheliitis (right, arrowheads). *: p<0.05; **: p<0.01; ***: p<0.001 mice. The first pair (P17) was inoculated with 4.4x10 4 PFUs of MACo2. Passages were performed as described above until P27 from which viruses were isolated and cloned. One of the clones, called MACo3, was amplified and sequenced. This clone was used for all infection experiments.

Infection experiments on BALB/c and C57BL/6 mice were performed as above with 10 5 PFU of MACo3 (sex, age and number of mice are indicated for each mouse group in the figure legends). Clinical signs of disease and weight loss were monitored daily. Mice were euthanized by ketamine/xylazine overdose at the end of the observation period (dpi 12) or at indicated time points (dpi 3 or 6). Samples for viral load (right lung lobe, internal organs and feces) and histopathological analyses (left lung lobe) were collected, immediately frozen in dry ice and stored at -80°C. The left lung lobe was fixed by submersion in 10% phosphate buffered formalin for 7 days prior to removal from the BSL3 for processing. The right lung lobe was placed on a 70μ cell strainer (Falcon), minced with fine scissors and ground with a syringe plunger using 400 μl of PBS. Lung homogenates were used for virus quantification.

Virus quantification

Tissue samples (brain, liver, spleen, heart, jejunum) and feces collected in the colon were weighed and homogenized in PBS using Lysing Matrix D tubes (MP biomedicals) and MP biomedicals tissue homogenizer. After centrifugation, RNA was extracted from 200 µl of lysates using NucleoMag Pathogen kit (Macherey Nagel). Viral RNA quantification was performed by quantitative reverse transcription PCR (RT-qPCR) using the IP4 set of primers and probe as described on the WHO website (https://www.who.int/docs/default-34source/coronaviruse/realtime-rt-pcr-assays-for-the-detection-of-sars-cov-2-institut-35pasteurparis.pdf?sfvrsn=3662fcb6_2) and the Luna Universal Probe One-Step RT-qPCR Kit (NEB).

For plaque assay, 10-fold serial dilutions of samples in DMEM were added onto VeroE6 monolayers in 24 well plates. After one-hour incubation at 37°C, the inoculum was replaced with 2% FBS DMEM and 1% Carboxymethylcellulose. Three days later, cells were fixed with 4% formaldehyde, followed by staining with 1% crystal violet to visualize the plaques.

Histological analysis

Histological analysis was performed on paraffin-embedded 4μm-thick sections used for hematoxylin-eosin (H&E) staining.

Statistical analysis

Body weight loss was compared between control and MACo-3-infected mice by one-way ANOVA. Data were analyzed and graphed using R software.

Figure 1 :

 1 Figure 1: Production and characterization of the MACo3 SARS-CoV-2 strain A: Two CC071 mice received an intranasal inoculation of 10 5 PFU of the SARS-CoV-2 IDF0372 isolate. Their lungs were collected at dpi 3 and individual lung homogenates were pooled and used to infect mice of the next passage. Further passages involved CC071, CC001 and C57BL/6 mice until the MACo1 and MACo2 variants were isolated and characterized. Starting with MACo2, 11 more passages were performed on BALB/c mice and led to the isolation of another variant which was cloned and named MACo3. B: Viral load in individual lung homogenates increased after the first passages but remained stable later (note that the inoculation dose was variable between

Figure 2 :

 2 Figure 2: MACo3 replicates in BALB/c mouse lungs and induces age-dependent pathology.

Figure 3 :

 3 Figure 3: MACo3 replicates in C57BL/6 mouse lungs and induces age-dependent pathology

Figure S1 :

 S1 Figure S1: Viral replication and lung lesions after infection of C57BL/6 and three CC strains with wildtype IDF0372 SARS-CoV-2.A: Viral load in the lungs of 9-16-week-old C57BL/6, CC001, CC002 and CC071 males inoculated with 4x10 4 PFUs of IDF0372. The dotted line represents the limit of detection (LOD), and undetected samples are plotted at half the LOD. B: H&E staining of lung sections from one of the C57BL/6 (virus-negative) mice (left, with no significant histological lesions) and the virus-positive CC001 (center) and CC071 (right) mice which both exhibit broncho-interstitial pneumonia.

Figure S2 :

 S2 Figure S2: Viral loads in samples from young and aged C57BL/6 and BALB/c mice Samples were collected on dpi 3 or 6, on the same mice as in Figures 2B, 2D, 3B and 3D, respectively. Viral load (eqPFU per g of tissue) was determined by qRT-PCR. The dotted line represents the limit of detection (LOD), and undetected samples are plotted at half the LOD.

Acknowledgements

We are grateful to Dr Luis Enjuanes (National Center for Biotechnology, Spain) for the gift of the DBT cells expressing mACE2. We thank Magali Tichit and David Hardy (Histology platform, Institut Pasteur) and Hélène Huet and Jean-Luc Servely (Histology and Pathology Unit, EnvA) for the technical processing of histological samples and the histology pictures, and the staff of the BSL-3 mouse facility. We also thank the team of the core facility P2M (Institut Pasteur) for genomic sequencing.

Funding

This work was supported by the « URGENCE COVID-19 » fundraising campaign of Institut Pasteur), the French Government's Investissement d'Avenir program, Laboratoire d'Excellence Integrative Biology of Emerging Infectious Diseases (Grant No. ANR-10-LABX-62-IBEID), the Agence Nationale de la Recherche (Grant No. ANR-20-COVI-0028-01) and the RECOVER project funded by the European Union's Horizon 2020 research and innovation programme under grant agreement No. 101003589. ESL acknowledges funding from the INCEPTION programme (Investissements d'Avenir grant ANR-16-CONV-0005).

Author contribution

XM and ESL designed and coordinated the study. MP, LL and LC performed in vitro experiments and viral quantification. XM, LC and JJ performed in vivo experiments. GJ and ERG performed histopathological analysis. FD and MA produced the IDF0372 viral isolate under the supervision of SvdW. XM and ESL wrote and revised the manuscript with input from all authors.

Competing interests:

The authors declare no competing interests.

Data and materials availability:

All data are available in the main text or the supplementary materials. Sequence data that support the findings of this study have been deposited in the European Nucleotide Archive under study PRJEB46205.

Supplementary Materials

Materials and Methods

Ethics

All animal work was approved by the Institut Pasteur Ethics Committee (projects dap 200008 and dap 210050) and authorized by the French Ministry of Research under #24613 and #31816 in compliance with the European and French regulations on the protection of live vertebrates.

Cells and viruses

VeroE6 cells (African green monkey kidney cells from ATCC (CCL-81)) and DBT-mACE2 cells (a kind gift of Dr Luis Enjuanes) were maintained in Dulbecco's Modified Eagle's Medium (DMEM) supplemented with 10% fetal bovine serum (FBS) and penicillin/streptomycin. DBT-mACE2 cells were supplemented with 800 μg/ml of G418 to maintain the plasmid expressing murine ACE2.

The B (Pango lineage) strain: hCoV-19/France/IDF-0372-isl/2020, GISAID accession Id: EPI_ISL_410720; and B.1 strain: hCoV-19/France/GES-1973/2020, GISAID accession Id: EPI_ISL_414631; were supplied by the National Reference Centre for Respiratory Viruses hosted by Institut Pasteur (Paris, France) and headed by Pr. S. van der Werf. The human sample from which the IDF-0372 strain was isolated has been provided by X. Lescure and Y. Yazdanpanah from the Bichat Hospital. Viruses were amplified and titrated by standard plaque forming assay on VeroE6 cells. The sequence of the stocks was verified by RNAseq on the mutualized platform for microbiology (P2M). All work with infectious virus was performed in biosafety level 3 containment laboratories at Institut Pasteur.

Mouse strains BALB/cJRj (BALB/c) and C57BL/6JRj (C57BL/6) mice were purchased from Janvier Labs (Le Genest St Isle, France). Collaborative Cross mice were purchased from the Systems Genetics Core Facility, University of North Carolina, and bred at the Institut Pasteur. Mice were maintained under specific-pathogen-free conditions with a 14-h light and 10-h dark cycle and ad libitum food and water in the Institut Pasteur animal facility.

In vivo passaging and infection

Infection studies were performed in animal biosafety level 3 (BSL-3) facilities at the Institut Pasteur, in Paris. To initiate serial passages, 10 5 PFU of IDF0372 were first inoculated to two anesthetized (ketamine/xylazine) 17-week-old CC071 males which were euthanized by ketamine/xylazine overdose at dpi 3 to collect their lungs. Lung homogenate was prepared from each mouse as previously described [14]. Twenty microliters of a 1:1 mix of the two homogenates were immediately inoculated to two other CC071 mice. The viral load was determined in each homogenate by RT-qPCR. The procedure was repeated on CC071, CC001 and C57BL/6 mice as described on Fig 1A . Lungs were collected at dpi 2 or 3. At every passage, mice received a mix of lung homogenates from all mice of the previous passage. The populations of viruses present in the homogenates at P15 and P16 were cultured, identified as MACo1 and MACo2, tittered and sequenced. Further mouse adaptation was performed by passaging on 6-18 month-old BALB/c


Victor Guilloux

Thierry Barisien

Frédérick Bernardot

Mathieu Bernard

Florent Margaillan

Silbé Majrab

Ingrid Stenger

Emmanuel Lhuillier

Christophe Testelin

Maria Chamarro

Laurent Legrand
email: laurent.legrand@insp.jussieu.fr

Phonon modes and exciton-phonon interactions in CsPbCl3 single nanocrystals

Keywords: perovskite nanocrystals, CsPbCl3, acoustic phonons, optical phonons, excitonphonon couplings, single object emission, quantum dots

come

Introduction

All-inorganic halide perovskite nanocrystals (NCs) attract great attention due to their exceptional optical properties since their first synthesis [1]. They are considered as a new generation of promising semiconductor candidates for a wide variety of optoelectronic technologies such as light-emitting diodes [2,3], lasing devices [4][5][6], photo-detectors [7][8][9][10] or also for photovoltaic solar cells [11]. Among these materials whose emission can be easily tuned from near-infrared to violet by changing the composition and/or the NCs size, CsPbCl3

possesses the widest bandgap energy [12,13]. In the context of the development of shortwavelength light-emitting diodes, this Cl-based perovskite appears as an ideal emitter for fabricating efficient ultrapure blue LEDs [14].

In this work, we address the exciton-phonon interaction in a single CsPbCl3 NC, inside which we highlight optical phonon modes. The temperature evolution of the photoluminescence (PL) and its broadening are measured for the first time at the single NC scale in this compound. In particular, this knowledge is essential to control the purity of the blue emission. The spectral lines in the emission are selected thanks to their polarization properties. This study provides a better understanding of the processes at work in the context of the exciton-phonon interaction in a high-energy emitting material with high applicative potential in optoelectronics and confirms general trends already observed in other lower-energy emitting halide perovskites [15][16][17][18][19][20][21][22][23][24].

Materials and methods

Perovskite CsPbCl3

NCs are grown by using a previously described method [1].

The NCs have been characterized by TEM imaging. The NCS have a cubic shape and their typical average side length is close to 8 nm, twice the Bohr diameter value of 3.4 nm (Fig. 1).

A low dilute NC solution was drop-cast onto a copper grid covered with an amorphous carbon film. The grid was degassed overnight under secondary vacuum. Imaging was conducted using a JEOL 2010 transmission electron microscope operated at 200 kV.

Micro-photoluminescence (µ-PL) experiments were performed on highly diluted phases dropcast on a glass slide, as a function of temperature in order to explore the response of individual CsPbCl3 NCs. We used a frequency doubled picosecond Ti:Sapphire laser operating at 82 MHz and tuned in the range 380-400 nm ie in the 3.10-3.27 eV energy range. The average excitation powers are a few μW with a spot diameter close to 1.5 μm. The excitation laser beam is focused using a microscope objective with NA = 0.6. The samples are mounted on the cold finger of a cryostat (Oxford Instruments) designed to compensate for thermal expansion and to maintain the spatial position of the spot as temperature is varied. The luminescence, collected with the same objective attached to a three-axes piezo stage to finely scan the sample spatially, is analyzed in an ACTON SP2760i Roper Scientific-Princeton Instruments spectrometer coupled to a nitrogen-cooled SPEC10 (RS-PI) CCD camera, with an overall spectral resolution of ∼ 50 μeV (δλ ∼ 7 pm) [25]. Polarization measurements are performed by adding a motorized halfwave plate followed by a linear polarizer in front of the spectrometer entrance; the direction of the polarizer is parallel to the grating grooves to optimize the spectrometer response.

Micro-Raman spectroscopy was performed using a HORIBA Jobin-Yvon Labram HR800 system using the 514.5 nm excitation wavelength of an Ar + laser with a power kept below 2 mW. Measurements were conducted while temperature was monitored into a liquid nitrogen cooled Linkam cryostat.

Results and discussions

The vast majority of measured NCs shows two peaks, or doublets, with Lorentzian profiles. In very few cases a set of three lines, triplets, is observed. Each spectral component of the doublets has a linewidth in the range 0.5-1 meV at 5 K and the spacing between the two peaks varies from 1 to 3 meV. By rotating the half-wave plate of the set-up, one obtains polar diagrams for a single NC, thanks to the stability of its emission. were made in other Pb-halide NCs [15][16][17]22,[26][27][28][29][30][31][32]. These doublets or triplets are associated to the splitted exciton fine structure due to an interplay of the electron-hole exchange interaction, the crystal field and the level of shape anisotropy [33,34]. Figure 3(a) shows the µ-PL spectrum of a single CsPbCl3 NC on a larger energy domain as compared to the spectrum given in Fig. 2(a). We highlight the presence of phonon replicas since the energy shifts of the observed peaks with respect to the main excitonic line of the doublet is in good agreement with known Raman scattering CsPbCl3 bulk frequencies [35][36][37]. As TO phonons do not couple through Fröhlich interaction with charges in such a polar system, only LO phonons replicas appear [START_REF] Singh | Electronic and Optoelectronic Properties[END_REF]. TOi (i = 1 to 3) modes as doublets have been moreover observed in bulk CsPbCl3 [35,36]. We identified the longitudinal phonon replicas corresponding to LO1, ELO1 ∼ 14.2 meV, and to LO2 with a mean energy close to 25 meV. More clearly than for LO1 replica, this latter appears as a doublet with an energy splitting of 2 meV exactly as for the exciton doublet shown here. We therefore clearly evidence the existence of a fine-structure splitting in the LO replicas as well. This feature has been nicely evidenced recently in CsPbBr3 NCs by the group of Kanemitsu [22].

We measured the micro-Raman spectrum of CsPbCl3 NCs excited with a continuous excitation at 514.5 nm (2.41 eV) at different temperatures in the range 110-170 K, as seen in Fig. 3(b).

The LO2 mode is observed at 200 cm -1 ie ∼ 25 meV in good agreement with the µ-PL measurements. The LO1 modes expected at ∼ 14 meV ie ∼ 115 cm -1 is covered by the TO3 modes [35]. The inset of Fig. 3 To address the exciton-phonon coupling in single CsPbCl3 NCs, we take advantage of the doublet-polarization properties to follow the temperature dependence of a single homogeneous PL linewidth for several single NCs observed at 5 K. The full width at half maximum (FWHM) of these µ-PL lines has been extracted from a Lorentzian fit of the highest intensity polarized line in a doublet, carefully selected after analysis of the polarization diagrams.

To ensure that the line of a given single NC was unambiguously followed in temperature, we verified that the temperature dependence of the energy peaks of each µ-PL line evolves monotonously and with the same slope, as seen in Fig. 4 for four doublets [START_REF] Varshni | [END_REF]. A blue-shift in energy with respect to the bulk is observed for each line. The slope measured here, αaver. ∼ 0.3 meV/K, is close to the one previously determined in CsPbBr3 NCs [17], which indicates a similar influence of the temperature on the lattice deformation and therefore on the energy gap.

Fig. 5 gathers the temperature dependence of the linewidth, Γ(T), for the selected linearly polarized lines of four doublets. The broadening with temperature is modeled using the expression

𝛤𝛤(𝑇𝑇) = 𝛤𝛤 0 + 𝐴𝐴 𝑎𝑎𝑎𝑎 𝑇𝑇 + 𝐵𝐵 𝐿𝐿𝐿𝐿 𝑛𝑛 𝐿𝐿𝐿𝐿 (𝑇𝑇) (1)
where Γ0 is a residual homogeneous temperature independent linewidth, including mainly spectral diffusion. The second term, with the Aac constant, represents the homogeneous broadening resulting from charge-acoustic phonons scattering; the third one with the Fröhlich BLO constant involves LO phonons [40,41].

𝑛𝑛 𝐿𝐿𝐿𝐿 (𝑇𝑇) = �exp � 𝐸𝐸 𝐿𝐿𝐿𝐿 𝑘𝑘 𝐵𝐵 𝑇𝑇 � -1� -1
is the Bose-Einstein occupation number for the optical phonons, kB is the Boltzmann constant, and ELO is an average representative energy for the LO phonons [START_REF] Yu | Fundamentals of Semiconductors Physics and Materials Properties[END_REF][START_REF] Zhang | [END_REF]. We underline that Eq. ( 1) was used to fit the set of all Γ(T) data of the four lines as a single data set in order to get a better statistics and to highlight an average behavior of the measured broadening. 4). The fit (from Eq. ( 1)) is obtained when considering the four sets of data as a unique one. The inset shows the temperature evolution of one of the selected lines. 1 MA stands for Methylammonium CH3NH3 and FA for Formamidinium HC(NH2)2. Parameters obtained when fitting the temperature emission linewidth broadening of different organic and inorganic perovskite materials under several forms, using the expression 𝛤𝛤(𝑇𝑇) = 𝛤𝛤 0 + 𝐴𝐴 𝑎𝑎𝑎𝑎 𝑇𝑇 + 𝐵𝐵 𝐿𝐿𝐿𝐿 𝑛𝑛 𝐿𝐿𝐿𝐿 (𝑇𝑇) as in this work. The asterisk after the "0" in the third column indicates that this parameter has been set to zero for adjustments. In [20], measurements for MA compounds are done above 80 K. In [23], data are photocurrents measurements till 80 K.

Compounds

In our experiments, as previously observed in halide perovskite materials, the exciton-LO phonons coupling appears more important as compared to the acoustic phonon coupling [15][16][17][18][19][20][21][22]. The linear part due to acoustic phonons scattering is nevertheless clearly evidenced at least in the 5K-100K temperature domain. The coupling constants obtained from the best unique fit are Aac ∼ 4 µeV/K and BLO ∼ 25 meV, using 21 meV or 170 cm -1 for ELO (ELO/kB = 245 K). The residual broadening contribution Γ0 is close to 1.2 meV. This is higher than for CsPbBr3 NCs (0.4 meV) [17]. This contribution is attributed to spectral diffusion during µ-PL measurements due to the appearance of a fluctuating charge distribution with photoexcitation.

The Aac value is comparable to those which are obtained in general in organic or inorganic perovskites under several forms, as seen in Table 1 [15][16][17][18][19][20][21]23,24]. The ELO value used here for CsPbCl3 appears to stand between the values of the LO1 (14 meV) and LO2 (∼ 25 meV) phonon energies phonon energies observed on the main phonon replicas of the µ-PL measurements presented in Fig. 3(a).

For perovskite materials, whether measured on single NCs, ensembles or even bulk materials, there is generally a good agreement between the LO phonon modes energies obtained by adjustment of the temperature linewidth broadenings and the energies measured through PL experiments or calculated for LO modes distribution directly. For example, phonon modes energies compatible with the temperature broadening of the emission lines have been observed in µ-PL experiments in the hybrid materials FAPbI3 single NCs [16] or observed and computed in MAPbI3 single crystal [19,44]. That is why we underline that in all-inorganic or hybrid perovskites the determination of relevant LO phonon energies via the measurement of the temperature dependence of line broadening in µ-PL or even of NCs sets constitutes a reliable and powerful tool. The LO energies are mainly assigned to modes related to internal vibrations of the Pb-halide network [22,45]. As seen in Table 1, the Fröhlich BLO constant measured here in CsPbCl3 as well as those determined in hybrid or all inorganic perovskite materials are almost all of the same order of magnitude, which reflects the polar characteristics of the Pb-halide bonds and the overall ionic character of these materials.

If we extrapolate to room temperature the fit used in Fig. 5, the obtained µ-PL linewidth for a CsPbCl3 NC is of the order of 20 meV, which is in good agreement with the already considered CsPbBr3 case [17].

Conclusions

We have characterized exciton-phonon coupling from 5 to 100 K, in single CsPbCl3 NCs. We have identified optical phonons, with energies in good agreement with the micro-Raman

Fig. 1 .

 1 Fig. 1. TEM image of the CsPbCl3 NCS addressed in this work.

Figure 2 (

 2 Figure 2(a) presents such a single NC doublet. This chosen example corresponds to a splitting of 1.47 meV between the doublet components, a value which is close to the average of the observed splittings. The polar diagram in Fig. 2(b) shows that the two peaks have linear and

Fig. 2 .

 2 Fig. 2. µ-PL of an individual CsPbCl3 NC obtained at 6 K. (a) Spectrum of a doublet. Experimental data were fitted with two single Lorentzian peaks (pink and blue). The black line is the total fit. The splitting in the EFS is here 1.47 meV. (b) Polarization diagram of the doublet. Pink and blue polarization diagrams correspond to each of the two peaks of the µ-PL spectrum.

  (b) shows that the Raman shifts of the three TO doublets (4, 6.5) meV, (8.9, 11.1) meV and(13.6, 15) meV (values at the lowest temperature) are almost temperature independent in the explored temperature domain, with nevertheless a small redshift of the highest frequency for the TO2 doublet and the lowest one of the TO1 modes within the experimental accuracy.

Fig. 3 .

 3 Fig. 3. (a) µ-PL spectrum at 5 K of an excitonic doublet (splitting of 2 meV) and LO phonon replicas of CsPbCl3. The energies of clearly appearing structures are also indicated. The energies of the LO phonons observed are given. The energy reference is the one of the main excitonic line in this doublet. (b) Micro-Raman spectra as a function of temperature with a continuous excitation at 514.5 nm. The inset shows the weak dependences of the Raman shifts with temperature (100 cm -1 corresponds to ≈ 12.4 meV).

Fig. 4 .

 4 Fig. 4. Temperature dependence of the energy of the NC excitonic transitions observed in μ-PL. Each symbol is associated to a given component within the fine structure of the four observed doublets. We find an average value of the slope α aver. ≈ 0.3 meV/K.

Fig. 5 .

 5 Fig. 5. Evolution with temperature of the linewidth of four selectively detected linearly polarized lines corresponding to four different NCs, which are distinguished by different symbols (the same as in Fig.4). The fit (from Eq. (1)) is obtained when considering the four sets of data as a unique one. The inset shows the temperature evolution of one of the selected lines.

Samples type Aac (µeV/K) BLO (meV) ELO (meV) References

	FAPbBr3	single nanocrystals	5 ± 5	52	15.2	[16]
		polycrystalline thin films	0*	61 ± 7	15.3 ± 1.4	[20]
	CsPbBr3	single nanocrystals	8 ± 3	42 ± 13	16	[17]
		Micro/nanowires ensembles	0*	66	19	[24]
		nanocrystals ensembles	8.9	∼ 100	∼ 18	[18]
		single crystal	32	41.7	22.2	[21]
	MAPbBr3	polycrystalline thin films	0*	58 ± 2	∼ 15	[20]
	FAPbI3	polycrystalline thin films	0*	40 ± 5	11.5 ± 1.2	[20]
		single nanocrystals	< 5	27	10.5 ± 0.8	[15]
	MAPbI3	single crystal	0*	12 ± 2	4.2 ± 0.8	[19]
		single crystals	93.5 ± 20.5	57 ± 22	16.1 ± 3.4	[23]
		polycrystalline thin films	0*	40 ± 2	∼ 11	[20]
	CsPbCl3	single nanocrystals	4 ± 1.5	25 ± 2	21	This work
	Table

Acknowledgments

Authors thank Xiang Zhen Xu for performing the TEM imaging. This work was financially supported by the French National Research Agency (ANR IPER-Nano2, ANR-18-CE30). EL thanks the support ERC starting grant blackQD (n°756225).

experiments we performed and with previous works of other groups on bulk CsPbCl3. We deduced the acoustic and optical constants of the exciton-phonon couplings. The extrapolation of our single NC linewidth measurements up to room temperature leads to a linewidth of the order of 20 meV, comparable to the case of equivalent Br-based materials. This value sets a characteristic limit to the spectral width for short-wavelength light emitting diodes using this material at ambient temperature. Further investigations are required to explain the residual broadening contribution attributed to spectral diffusion usually due to the electrostatic environment of the NC.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Keywords: prélèvement, aérosol, tests in vitro, microenvironnement / KEYWORDS: sampling, aerosol, in vitro test, microenvironment

Les prélèvements de particules sur filtre ne permettent pas à l'heure actuelle d'étudier leurs effets toxicologiques correctement. En effet, pour effectuer des analyses in vitro, les particules doivent être désorbées du filtre pour être déposées sur des cellules. Or les étapes de désorption peuvent induire des modifications des propriétés physico-chimiques des particules. Une méthode de prélèvement alternative est donc recherchée. Les préleveurs de bioaérosols représentent de potentiels candidats pour cela, puisqu'ils collectent les particules sur d'autres supports que le filtre (e.g. directement dans un liquide). Deux préleveurs de bioaérosols (i.e. le Coriolis Micro et le Coriolis Compact -Bertin Instruments) ont été testés dans deux microenvironnements différents (i.e. quai et rame de train). Les résultats ont montré qu'il est possible de déployer de tels préleveurs sur le terrain et que les échantillons prélevés sont compatibles avec des tests in vitro. Toutefois, leur efficacité de prélèvement selon le diamètre des particules doit encore être caractérisée.

INTRODUCTION

Un microenvironnement est un espace où la concentration en polluants est considérée comme homogène ou ayant une variation prévisible dans le temps (Duan, 1982), e.g. habitacle de voiture, parking, quai de train. Les caractéristiques des particules rencontrées dans ces environnements peuvent varier fortement (e.g. composition chimique, taille, concentration des particules) (Tran et al, 2021). Selon les trajets empruntés par les individus au cours de leur journée, et leurs temps de passage dans un ou plusieurs microenvironnements, ils subiront différentes formes et intensités de pollution particulaire, avec des conséquences variables sur leur santé (Chen et al, 2020). Il y a donc un besoin fort d'étudier le potentiel toxique des principaux microenvironnements urbains, par exemple par des essais in vitro sur cellules. Pour cela, la méthode la plus utilisée consiste à prélever les particules sur un filtre puis à les désorber du support, afin de les déposer sur des cellules et étudier leurs effets (Goulaouic, 2009). Cependant, les ultrasons et les solvants utilisés pour désorber les particules induisent des modifications sur les propriétés physicochimiques de celles-ci (Marvanova et al, 2018). Dans ce contexte, une méthode de prélèvement alternative a été recherchée pour effectuer des tests ultérieurs in vitro. Les préleveurs de bioaérosols ont été identifiés comme potentiels candidats pour répondre à cette problématique, puisqu'ils présentent l'avantage de prélever des particules sur des supports autres que le filtre (e.g. prélèvement directement dans un liquide, sur un gel, dans un récipient). Cependant, leur efficacité de prélèvement n'est souvent pas bien caractérisée, et reste à étudier.

DEPLOIEMENT EN MICROENVIRONNEMENT DE 2 PRELEVEURS CYCLONIQUES POUR L'ETUDE TOXICOLOGIQUE DE PARTICULES

A. Delater 1). Pour les prélèvements Coriolis Micro dans l'eau Milli Q avec tensioactif, le prélèvement a été dilué dans du milieu de culture 10 fois concentré et déposé sur les cellules. Enfin, pour le Coriolis Compact, le prélèvement a été remis en suspension dans du milieu de culture 1 fois concentré et déposé sur les cellules. Les prélèvements Coriolis Micro dans l'eau Milli Q sans tensioactifs n'ont pas été exposés sur cellules. Les tests in vitro ont été effectués sur un modèle de co-culture composée de cellules épithéliales alvéolaires humaines A549 et de monocytes différenciés en macrophages THP-1. La cytotoxicité a été évaluée par un test métabolique (le presto blue) et un test mesurant l'intégrité de la membrane cellulaire (test LDH), après exposition des cellules aux particules durant 24 heures.

Dans le cadre de la thèse Aerorep1 et du projet TOXinTRANSPORT2 , 2 préleveurs de bioaérosols ont été choisis pour être évalués sur le terrain : (a) le Coriolis Micro, un cyclone en voie liquide et (b) le Coriolis Compact, un cyclone en voie sèche (Bertin Instruments) dans lesquels les particules sont aspirées jusqu'à un cône. L'objectif est de déterminer si ces préleveurs peuvent être détournés de leur utilisation principale (i.e. le prélèvement pour la caractérisation de bioaérosol) pour répondre aux besoins du projet. Les préleveurs devront (i) être déployables en microenvironnement, (ii) prélever des échantillons compatibles avec les tests in vitro et (iii) compatibles avec des méthodes de caractérisations des particules (e.g. détermination de la masse de particules collectées) dans le but d'interpréter les résultats toxicologiques. Dans un second temps, leur efficacité de prélèvement selon la taille des particules sera évaluée.

RESULTATS

Plusieurs constats ont pu être faits lors de l'évaluation des Coriolis sur le terrain. Tout d'abord, les Coriolis peuvent être déployés en microenvironnement sans difficulté grâce à leur petit format et leur légèreté. Ensuite, l'étape de préparation des échantillons avant tests in vitro a montré que l'emploi de tensioactif était nécessaire pour récupérer les particules prélevées en voie liquide. En effet, les particules prélevées dans l'eau Milli Q sans tensioactif (campagne n°1) étaient adsorbées sur les parois du cône. Enfin, le Coriolis Compact (cyclone en voie sèche) s'est révélé être la technologie la plus adaptée pour récupérer les particules, puisque les étapes de récupération étaient moins nombreuses, limitant ainsi les contaminations potentielles de manipulation et les pertes de particules. Des effets d'intensité différente ont été observés sur les cellules lors des analyses in vitro des échantillons des 2 campagnes de terrain. La Figure 2 représente le pourcentage de survie des cellules (colonne de gauche) ainsi que le pourcentage de cytotoxicité (colonne de droite) selon la masse de particules prélevées dans les échantillons, mesurées par le CPS et déposées sur les cellules (les échantillons sont ici dilués à 1:2). Les barres sont colorées selon la durée de prélèvement de l'échantillon.

Figure 2 : Pourcentage de survie des cellules et pourcentage de cytotoxicité sur les cellules en fonction de la masse de particules (en g/L) prélevées (dilution 1:2) et déposées sur les cellules. La 1ère ligne présente les résultats du Coriolis Compact lors de la campagne n°1, la 2 ème ligne présente les résultats du Coriolis Micro lors de la campagne n°1 et la 3 ème ligne présente les résultats du Coriolis Micro lors de la campagne n°2. Tout d'abord, les résultats des concentrations en particules prélevées par les Coriolis ont montré qu'ils permettent de collecter en l'espace de 2h assez de particules pour que celles-ci soient quantifiables (la concentration la plus basse en PM10 mesurée en parallèle des Coriolis par les préleveurs HPEM est 30 µg/m 3 ). Des essais de 10 min réalisés avec le Coriolis Micro à un débit de 300 L/min (au lieu de 100 L/min pour les autres essais) ont montré qu'il est aussi possible d'effectuer des prélèvements bien plus courts que 2h et de quantifier les particules, à condition que l'atmosphère soit assez chargée en particules (la barre rose la plus à droite du graphique représente un prélèvement effectué lors d'un passage en gares souterraines alors que la barre rose la plus à gauche représente un prélèvement effectué lors d'un passage en gares aériennes). Les essais les plus longs ne sont pas toujours les plus chargés en particules. Ce phénomène peut s'expliquer par (i) la concentration en particules dans l'environnement lors du prélèvement mais aussi par (ii) le fait que l'efficacité de rétention (i.e. la fraction des particules collectées qui sont encore retenues dans le fluide de collecte à la fin de l'échantillonnage) d'un préleveur diminue au cours du temps du fait de la ré-aérosolisation des particules et/ou de l'évaporation du liquide (Su et al, 2020). L'efficacité de rétention est propre à chaque préleveur et à la connaissance de l'auteur, celle du Coriolis Micro n'est pas/peu étudiée dans la littérature. A l'avenir, elle devra être évaluée afin de déterminer le temps de collecte optimum du Coriolis Micro.

Enfin, la masse des particules prélevée a été estimée par une analyse optique en voie liquide avec un disque centrifuge, le CPS (CPS Instruments, Inc.). Le CPS mesurait les particules entre 0,02 et 6 µm de diamètre, en prenant comme indice de réfraction celui des oxydes de fer, élément majoritaire en concentration dans les 2 environnements (analyse par ICP-MS de filtres).

REMERCIEMENTS

Les auteurs remercient la société Bertin Instruments pour leur contribution à la réalisation des essais menés dans le cadre de la thèse Aerorep et du projet TOXinTRANSPORT. Pour ce qui est des résultats toxicologiques, on peut constater sur la Figure 2 qu'il y a plus de mortalité cellulaire pour la campagne n°1 que pour la campagne n°2. De plus, on peut voir que les essais les plus chargés en particules ou les plus longs ne sont pas ceux qui induisent le plus de mortalité cellulaire. Pour ce qui est du pourcentage de cytotoxicité, il n'y a pas ou peu d'effets qui ont été observés sur les cellules. Enfin, les résultats des mesures de concentrations particulaires prélevées par les Coriolis ont montré que celles-ci étaient en moyenne supérieures de plus de 2 ordres de grandeur à celles prélevées en parallèle par les méthodes de référence (Figure 3). Cette différence peut s'expliquer (i) par le fait que les appareils de références ne mesurent que les PM2,5 ou PM10 contrairement aux Coriolis qui mesureraient des particules supérieures à 2,5 µm de diamètre et/ou (ii) que les Coriolis surestiment le prélèvement d'une fraction de particules. Ces hypothèses pourront être vérifiées par une analyse plus poussée de la granulométrie des particules prélevées et par une évaluation de l'efficacité des préleveurs en laboratoire.

Figure 1 :

 1 Figure 1 : Déploiement des Coriolis en conditions réelles Le Coriolis Micro (cyclone en voie liquide) a été testé avec différents liquides et différents débits. En parallèle des prélèvements des Coriolis, des méthodes de référence ont été déployées pour mesurer la concentration particulaire. Le Tableau 1 regroupe les paramètres de mesure. Tableau 1 : Paramètres de mesure Paramètres Campagne n°1 Campagne n°2 Liquide de prélèvement (Micro) -Milieu de culture diluée (RPMI + SVF) -Eau Milli Q Eau Milli Q + tensioactifs (0,001% Tween 20) Durée prélèvement 2h à 4h 10 min à 14h Débit Coriolis Micro : 100 L/min Coriolis Compact : 50 L/min Coriolis Micro : 100 L/min et 300 L/min Référence en parallèle TEOM PM2,5 et PM10 Impacteur personnel HPEM PM10 2.1. Présentation des analyses des échantillons Les particules prélevées par les Coriolis ont été récupérées et déposées sur les cellules. Pour les prélèvements Coriolis Micro dans le milieu de culture, le prélèvement a été déposé sur les cellules tel quel.Pour les prélèvements Coriolis Micro dans l'eau Milli Q avec tensioactif, le prélèvement a été dilué dans du milieu de culture 10 fois concentré et déposé sur les cellules. Enfin, pour le Coriolis Compact, le prélèvement a été remis en suspension dans du milieu de culture 1 fois concentré et déposé sur les cellules. Les prélèvements Coriolis Micro dans l'eau Milli Q sans tensioactifs n'ont pas été exposés sur cellules. Les tests in vitro ont été effectués sur un modèle de co-culture composée de cellules épithéliales alvéolaires humaines A549 et de monocytes différenciés en macrophages THP-1. La cytotoxicité a été évaluée par un test métabolique (le presto blue) et un test mesurant l'intégrité de la membrane cellulaire (test LDH), après exposition des cellules aux particules durant 24 heures.

Figure 3 :

 3 Figure 3 : Distribution sous forme de boxplots des concentrations particulaires mesurées en parallèle par les Coriolis et par une méthode de référence (TEOM et HPEM).

TITLE Deployment in microenvironment of 2 cyclone samplers for the toxicological study of particles. RESUME 2. MATERIELS & METHODES 2.1. Présentation des campagnes de terrain Les

  1,3 , B. Berthelot 1 , L. Meunier 1 , S. Fable 1 , M. De Mendonça Andrade 1 , O. Le Bihan 1 , J. Queron 1 , G. Lacroix 1 , M. Plumail 1 , C. Gamez 1 , K. Blazy 1 , M. Floreani 1 , A. Albinet 1 , S. Ngo 2 , G. Brun 2 , H. Carrilho 2 , I. Coll 3 1 INERIS, Institut Nationale de l'Environnement industriel et des RISques, 60550 Verneuil-en-Halatte, France Coriolis ont été déployés lors de 2 campagnes de terrain (2 campagnes pour le Coriolis Micro et 1 campagne pour le Coriolis Compact). La campagne n°1 s'est déroulée sur quai en Enceinte Ferroviaire Souterraine (EFS) et la campagne n°2 en rame de train (Figure

	2 SNCF Voyageurs -Ingénierie du Matériel -Agence d'Essai Ferroviaire, 94407 Vitry-sur-Seine, France
	3 Univ Paris Est Créteil and Université de Paris, CNRS, LISA, F-94010 Créteil, France
	*Courriel de l'orateur : ambre.delater@ineris.fr

« Développement d'une méthode de prélèvement in situ d'AEROsols en vue de l'évaluation des REPonses pulmonaires induites » -Ambre Delater sous la direction d'Isabelle Coll et de Brice Berthelot

2 « Caractérisations TOXicologiques in vitro, chimiques et physiques de particules prélevées dans l'air d'habitacles de TRANSPORT en roulage » -APR IMPACTS ADEME 2018


Hernán Aguirre

Arnaud Liefooghe
email: arnaud.liefooghe@lifl.fr

Sébastien Verel
email: verel@i3s.unice.fr

Kiyoshi Tanaka
email: ktanaka@shinshu-u.ac.jp

Effects of Population Size on Selection and Scalability in Evolutionary Many-objective Optimization

In this work we study population size as a fraction of the true Pareto optimal set and analyze its effects on selection and performance scalability of a conventional multi-objective evolutionary algorithm applied to many-objective optimization of small MNK-landscapes.

Introduction

Conventional multi-objective evolutionary algorithms (MOEAs) [START_REF] Deb | Multi-Objective Optimization using Evolutionary Algorithms[END_REF] are known to scale up poorly to high dimensional objective spaces [START_REF] Ishibuchi | Evolutionary Many-Objective Optimization: A Short Review[END_REF], particularly dominancebased algorithms. This lack of scalability has been attributed mainly to inappropriate operators for selection and variation. The population size greatly influences the dynamics of the algorithm. However, its effects on large dimensional objectives spaces are not well understood. In this work we set population size as a fraction of the true Pareto optimal set and analyze its effects on selection and performance scalability of a conventional MOEA applied to many-objective optimization. In our study we enumerate small MNK-landscapes with 3 -6 objectives, 20 bits, and observe the number of Pareto optimal solutions that the algorithm is able to find for various population sizes.

Methodology

In our study we use four MNK-landscapes [START_REF] Aguirre | Insights on Properties of Multi-objective MNK-Landscapes[END_REF] randomly generated with m = 3, 4, 5 and 6 objectives, n = 20 bits, and k = 1 epistatic bit. For each landscape we enumerate all its solutions and classify them in non-dominated fronts. The exact number of true Pareto optimal solutions P OS T found by enumeration are |P OS T | = 152, 1554, 6265, and 16845 for m = 3, 4, 5, and 6 objectives, respectively. Similarly, the exact number of non-dominated fronts of the landscapes are 258, 76, 29, and 22, respectively.

We run a conventional MOEA for a fixed number of generations. The algorithm uses a population P from which it creates an offspring population Q by recombination and mutation. The population P for the next generation is obtained from the joined population P ∪ Q by survival selection. In this work we use NSGA-II as the evolutionary multi-objective optimizer, set with two point crossover with rate pc = 1.0, and bit flip mutation with rate pm = 1/n.

Once evolution is over, we compare the set of P OS T with the sets of unique non-dominated solutions obtained at each generation after survival selection to determine which are true Pareto optimal solutions, count their number at each generation, and their accumulated number found during evolution.

Experimental Results and Discussion

Let us denote by F 1 the set of non-dominated solutions in population P , and F T 1 the set of solutions in by F 1 that are true Pareto optimal solutions. Fig. 1 shows the number of solutions in F 1 and F T 1 over the generations for m = 3 and 4 objectives, running the algorithm for 100 generations with three different population sizes |P | = 50, 100 and 200.

First we analyze results for m = 3 objectives. When we set population size to |P | = 50 or 100, a value smaller than the number of true Pareto optimal solutions |P OS T | = 152, it can be seen in Fig. 1 (a.1) and (a.2) that after few generations all solutions in the population are non-dominated, |F 1 | = |P |. However, not all solutions in F 1 are true Pareto optimal solutions, i.e.

|F T 1 | < F 1 = |P |. Also, it is important to note that F T
1 fluctuates up and down after an initial increase. On the other hand, when we set the population size to a value larger than the number of true Pareto optimal solutions, |P | = 200 > |P OS T | = 152, it can be seen in Fig. 1 (a.3) that the instantaneous non-dominated set is a subset of the population, F 1 ⊂ P . Also, note that from generation 35 onwards, all nondominated solutions in the population are also true Pareto optimal, F 1 = F T 1 . In this case, the algorithm finds and keeps in P almost all true Pareto optimal solutions, 147 out of 152, during the latest stage of the search.

It is known that the number of true Pareto optimal solutions |P OS T | increases considerably with the number of objectives. However, this is often ignored and the algorithm is set with a very small population size compared to |P OS T |. To study these cases, Fig. 1 the case m = 3 |P | = 200). Some of these solutions are lost from one generation to the next one during the survival selection step of the algorithm. At this step, the algorithm joins the population P with the offspring population Q and ranks individuals with respect to dominance-depth. The best rank is given to true Pareto optimal solutions and also to some others that are not true optimal but appear non-dominated in the combined population. Let us call the set of best ranked non-dominated solutions obtained from P ∪ Q as F P ∪Q

1

. If this set F P ∪Q 1 is larger than the population P , a sample of them P = F 1 ⊂ F P ∪Q 1 is chosen based on crowding distance during the survival step. At this point, some true Pareto optimal solutions are dropped in favor of lest crowded non-optimal solutions. Summarizing, P = F 1 ⊂ F P ∪Q 1 and therefore F T 1 ⊂ F 1 is more likely to occur for population sizes smaller than the number of true Pareto optimal solutions |P OS T |.

Fig. 2 (a) and Fig. 3 (a)-(c) show results for m = 3, 4, 5 and 6 objectives using population sizes that correspond approximately to 1/3, 2/3 and 4/3 of the set P OS T , respectively. From these figures note that increasing population size from 1/3 to 4/3 of P OS T translates into a striking performance scalability of the algorithm, measured on terms of the number of true Pareto optimal solutions found and kept in the population. For population size 4/3 of P OS T the number of AF T 1 = F T 1 ⊂ F 1 and the algorithm can actually find and keep in the population 147 out of 152, 1545 out of 1554, 6248 out of 6265, and 16842 out of 16845 true Pareto optimal solutions for 3, 4, 5 and 6 objectives, respectively.

These results show that the effectiveness of the algorithm in many-objective landscapes depends strongly on the size of the population. However, it should be noted that larger populations demand more computational time and memory. Also, a relatively larger number of solutions need to be evaluated. For example, after 100 generations, using a population size 4/3 of P OS T , the conventional MOEA used in this study evaluates approximately a number of solutions equivalent to 2%, 19%, 76% and 215% of the size of the search space for m = 3, 4, 5, and 6 objectives, respectively. In the future, we would like to analyze the efficiency of MOEAs in many-objective landscapes.

Conclusions

In this work we analyzed the effects of population size on selection and scalability of a conventional dominance-based MOEA for many-objective optimization. We showed that the performance of a conventional MOEA can scale up fairly well to high dimensional objective spaces if a sufficiently large population size is used compared to the size of the true Pareto optimal set.

  (b.1)-(b.3) show results for m = 4 objectives setting population size to the same values used for m = 3 objectives, which are very small compared to |P OS T |, i.e |P | ≤ 200 < |P OS T | = 1554. Note that these settings of population size magnify the difficulties observed for m = 3 with |P | = 50 or |P | = 100. That is, fewer solutions are true Pareto optimal, although the set of non-dominated solutions of the population quickly contains mutually non-dominated solutions only. Also, larger fluctuations are observed in the number of true Pareto optimal solutions F T 1 . In general, if |P | is set to a value smaller than |P OS T |, the algorithm cannot keep all true Pareto optimal solutions in the population. However, we would expect an ideal algorithm to keep as many true Pareto optimal solutions as the size of its population, |F T 1 | = |F 1 | = |P | < |P OS T |. This is not what we observe in our results. To explain this behavior, Fig.2 shows the instantaneous number

1 )

 1 m = 4, |P | = 50

2 )

 2 m = 4, |P | = 100

Fig. 1 .Fig. 2 . 1 and F T 1 ,Fig. 3 .

 12113 Fig. 1. Number of non-dominated F1 and actual number of true Pareto optimal solutions F T 1 in the population over the generations. |P OS T | = 152, and 1554 for m = 3, and 4 objectives, respectively. Population sizes |P | = 50, 100, and 200.


Titouan Gendron
email: titouan.gendron@univ-ubs.fr

Charbel Emmanuel Boutillon
email: emmanuel.boutillon@univ-ubs.fr

Abdel Nour

David Gnaedig
email: david.gnaedig@turboconcept.com

Anrt

CIFRE PhD Turboconcept

C Abdel Nour

Forward Backward Syndrome Computation: A Reduced Complexity CRC Code Decoder

Keywords: error correction, single-bit error, syndrome computation I

This paper proposes a simplified method for correcting a single-bit error event in a message protected by a CRC code thanks to a new decoding method called Forward-Backward Syndrome Computation. The two key ideas of the paper are to show that: 1) syndrome computation can be performed in any direction (Forward and Backward), 2) consequently, it is possible to create a partition of the received message in a set of disjoint syndromes, thus allowing a direct identification of the error position. Evaluation in terms of number of 2-input binary logic functions shows a complexity reduction by a factor of up to 3 compared to conventional methods.

Index Terms-CRC code

classical single-bit error decoder. The first key contribution of this work is to show that syndrome computation can be performed in any direction (Forward or Backward). The "Forward" direction denotes the classical way to compute the syndrome, while the "Backward" direction denotes the bitreversed order. Note that the proposed algorithm is completely different from the well known BCJR algorithm, used to decode a convolutional code and also referred to as the Forward-Backward algorithm. The only commonality resides in the direction of the applied processing schedule.

The other key contribution of this paper is the proof that the computation of a partial syndrome based on any set of M consecutive positions (M being the length of the redundancy part of a codeword) can be used to correct an error within these positions. This work can be seen as a specific and efficient implementation of the more general Information Set Decoding algorithm [START_REF] Prange | The use of Information Sets in decoding Cyclic Codes[END_REF]. It is motivated by the industrial demand of an improved decoder for LTE Turbocode [START_REF] Gendron | Revisiting augmented decoding techniques for LTE Turbo Codes[END_REF].

The rest of the paper is organized as follows. Section II reminds the basics of a CRC code and defines the notations. Section III introduces the proposed FB-SC decoding method. Section IV explains how to use the FB-SC method to correct a single error event, while section V concludes the paper. II. CRC CODES Let K, N and R be the information length, the codeword length and code rate of the polynomial code, respectively. The polynomial code is defined by the generator polynomial g

(x) = x M + g M -1 x M -1 + . . . g 1 x 1 + 1 of degree M = N -K. The message u = (u i ) i=0,1,...,K-1 of length K is expressed as a polynomial of degree K -1 noted u(x) = u K-1 x K-1 + u K-1 x K-2 + . . . u 1 x 1 + u 0 x 0 . The polynomial redundancy r(x) of degree M -1 is computed as the remainder of the polynomial division of u(x)x M by g(x), r(x) = u(x)x M mod g(x), ( 1
)
where mod is the modulo operator. From r(x), the codeword c(x) is generated1 as

c(x) = u(x)x M + r(x). (2)
By construction, c(x) is a multiple of the generator polynomial g(x) and thus c(x) = 0 mod g(x). After transmission on a Binary Symmetric Channel (BSC) with a probability of error p, the received message is

y(x) = c(x) + e(x), (3)
where e(x) represents the error polynomial that contains nonzero coefficients at each error position. From (3), we get s(x) = y(x) mod g(x) = (c(x) + e(x)) mod g(x) and thus, using the linearity of the modulo operation,

s(x) = e(x) mod g(x). (4)
The polynomial s(x) is called the syndrome. It depends only on the error vector e(x) and the generator polynomial g(x) through (4). If s(x) is equal to 0, then an error-free transmission is assumed (when the error sequence e(x) is a codeword, an undetected error event occurs). When the syndrome s(x) differs from zero, then e(x) is a non-null polynomial and its value should be inferred from s(x). In the sequel, the Hamming weight of a polynomial h(x), i.e. its number of non-zero elements will be denoted w(h).

III. THE FB-SC CORRECTION METHOD

This section first discusses the symmetric property of the CRC code, then explains how this symmetry can be used to correct errors. Symmetry refers to the ability to construct a codeword from a message u(x) either in the "Forward" form c 0 (x) = u(x)x M + r 0 (x), or in the "Backward" form c 1 (x) = r 1 (x)X K +u(x). Note that c 1 (x) is a different codeword of the same CRC code and r 1 (x) can be processed from the reverseordered generator polynomial with a method described next.

A. Property of symmetric encoding

Lemma 1: Let a(x) be a polynomial of degree n and (a n a n-1 . . . a 1 a 0 ) its associated binary representation. Then, a polynomial a(x -1 )x n is a polynomial of degree n with a binary representation given in reversed order, i.e. (a 0 a 1 . . . a n-1 a n ).

Proof The i th monomial, i = 0, 1, . . . , n, of a(x), i.e. a i x i , is associated to the (n -i) th monomial of a(x -1 )x n since a i x -i x n = a i x n-i As c(x) is a multiple of g(x), a unique quotient q(x) exists, such that c(x) = q(x)g(x).

By replacing x by x -1 in (5) and by multiplying both sides of the equality by x N -1 , we obtain

c(x -1 )x N -1 = q(x -1 )g(x -1 )x N -1 . (6)
The x N -1 term on the right side of ( 6) can be decomposed into the product x K-1 x N -K , or equivalently, x K-1 x M since M = N -K. Thus, (6) becomes

c(x -1 )x N -1 = (q(x -1 )x K-1 )(g(x -1 )x M ). (7)
From ( 7), we can see that the reversed-order polynomial c(x) = c(x -1 )x N -1 is a multiple of the reversedorder polynomial ḡ(x) = g(x -1 )x M . In other words, c = (c 0 , c 1 , . . . , c N -2 , c N -1 ) is a codeword of the code generated by the polynomial ḡ = (g 0 , g 1 , . . . , g M -1 , g M ). Therefore, it becomes possible to revisit the decoding process in a symmetrical way by reversing the order of all polynomials. If we call "Forward" the natural way to compute a syndrome (the natural bit order), the symmetrical way can then be defined as "Backward" (the bit-reversed order).

Let us define the syndrome s(x) as s(x) = y(x -1 )x N -1 mod ḡ(x). This equation gives y(x -1 )x N -1 + s(x) = 0 mod ḡ(x) and q(x) exists such that

y(x -1 )x N -1 + s(x) = q(x)ḡ(x). (8)
Replacing in [START_REF] Fossorier | Soft decision decoding of Linear Block Codes based on Ordered Statistics[END_REF] x by x -1 and multiplying both terms by x N -1 reverts the equation to the natural order. For example, the term y(x -1 )x N -1 is switched back using y((x -1 ) -1 )(x -1 ) N -1 x N -1 = y(x)). The whole equation gives

y(x) + s(x -1 )x N -1 = q (x)g(x), (9)
with q (x) = q(x -1 )x N -1 . In other words, y(x) + s(x -1 )x N -1 is a codeword of the code generated by the polynomial g(x). The polynomial s(x -1 )x N -1 is of degree N -1, but its K lowest degree coefficients are all equal to zeros.

In the next section, we combine forward and backward remainders to compute a middle remainder and associated middle syndromes.

B. Computation of a middle syndrome

We propose a generalization of the capability of finding a codeword by correcting errors within any adjacent subset of M bits starting from index l and ending at index l + M -1, with l = 0, 1, . . . , N -M -1. More precisely, we present a method that provides a polynomial codeword from a vector y(x) of size N bits by only modifying the subset of coefficients between positions l and l + M -1

To do so, let us decompose y(x) into three vectors as follows (see Fig. 1 and Fig. 2)

y(x) = y f l (x)x l+M + y m l (x)x l + y b l (x) (10)
with y f l (x) corresponding to the polynomial constructed with the N -M -l highest degree coefficients of y(x) (exponent f for "Forward coefficients", this name is justified later by the fact that they are processed in the forward direction), y m l (x) the polynomial constructed with the M coefficients of y(x) between x l and x l+M -1 (exponent m corresponding to "middle coefficients") and y b l (x) the polynomial corresponding to the l lowest coefficients of y(x) (exponent b for "Backward coefficients"). These vectors are defined as

y f l (x) = N -1 i=l+M y i x i-M -l , y m l (x) = l+M -1 i=l y i x i-l and y b l (x) = l-1 i=0 y i x i . (11)
We denote by r f l the partial forward remainder such that

r f l (x) = y f l (x)x M mod g(x) (12)
Multiplying both terms of ( 12) by x l and regrouping the terms on the left side gives

y f l (x)x M +l + r f l (x)x l = 0 mod g(x). ( 13
)
y n-1 , . . . , y l+M y f l (x) y M +l-1 , . . . , y l y m l (x) y l-1 , . . . , y 0 y b l (x) r f l (x) forward compute r b l (x)
backward compute

y f l (x) r f l (x) + r b l (x) y b l (x) codeword N -M -l M l
Fig. 1: Illustration of the FB-SC principle.

For a bit-reversed order processing, we can define the partial backward remainder rb l (x) from the reversed-order polynomial ȳb l (x) = y b l (x -1 )x l multiplied by x M and the reversed-order key ḡ(x) as

rb l (x) = ȳb l (x)x M mod ḡ(x). (14)
Therefore, ȳb l (x)x M +r b l (x) = 0 mod g(x), and thus, there exists q(x) such that

ȳb l (x)x M + rb l (x) = q(x)ḡ(x). (15)
Similarly to the transformation between ( 8) and ( 9), replacing x by x -1 in (15) and multiplying both terms by x M +l gives y b l (x) + r b l (x)x l = 0 mod g(x),

where r b l (x) = rb l (x -1 )x M denotes the reversed-order polynomial of rb l (x). By adding the two terms r f l (x) and r b l (x), we can define the middle (or generalized) remainder r m l (x) as

r m l (x) = r f l (x) + r b l (x) (17)
and the middle syndrome s m l (x) at the level of the bit l within the frame, as

s m l (x) = y m l (x) + r m l (x). ( 18
)
The corresponding process is illustrated in Fig. 1. The sum of the polynomials in ( 13) and ( 16) verifies

y f l (x)x M +l + r m l (x)x l + y b l (x) = 0 mod g(x), (19)
or equivalently, is also a codeword of the polynomial code.

Note that it can be demonstrated that, if y(x) is a codeword, then for all l, s m l (x) = 0, and also its reciprocal property, i.e. if index l exists such that s m l (x) = 0, then y(x) is a codeword. In the following, a method to compute the middle syndrome is presented.

C. Recursive computation of r f l (x) and r b l (x) Following the definition of y f l (x) in [START_REF] Gendron | Revisiting augmented decoding techniques for LTE Turbo Codes[END_REF], we can write

y f l (x) = y f l+1 (x)x + y l+M . ( 20
)
Starting from [START_REF] Gpp | Evolved Universal Terrestrial Radio Access (E-UTRA); Radio Resource Control (RRC)[END_REF], for l = K -1 down to 0 we have

r f l (x) = (y f l+1 (x)x + y l+M )x M mod g(x) (21) = (r f l+1 (x)x + y l+M x M ) mod g(x) (22)
with r f l=K (x) = 0. Note that ( 22) is the mathematical expression used to compute the final redundancy r(x) = r f 0 (x) of the CRC encoding process when y(x) = u(x)x M .

The computation of rb l from l = 0 to K-1 can be performed by applying a symmetrical encoding approach following

rb l (x) = (r b l-1 (x)x + y l x M ) mod ḡ(x), (23)
with r b l=-1 (x) = 0. Without loss of generality, we assume that N is a multiple of M , i.e. N = γM . If it is not the case, y(x) can be padded with dummy null coefficients until its size matches a multiple of M . It is then possible to partition y(x) into γ segments of size M each, and to compute the γ middle syndromes associated to each segment s m kM (x) = r m kM (x) + y m kM (x), with k = 0, 1, . . . , γ -1. To do so, the partial remainders r f kM (x) from ( 22) and r b kM (x) from (23) computed for k = 0, 1, . . . , γ -1 have to be saved periodically every M cycles. Finally, after reordering, the middle remainder denoted by r m kM (x) is computed (see ( 17)) followed by the computation of the middle syndromes s m kM (x) (see ( 18)), with k = 0, 1, . . . , γ -1.

The next section shows how to simplify the correction of a single error event using the partitions of FB syndromes.

IV. CORRECTION OF THE SINGLE-BIT ERROR EVENT

In this section, we first present the standard method to correct a single-bit error event and we evaluate its complexity. Then, we show how to benefit from the set of FB syndromes associated to the segments of y(x) to reduce the complexity of the correction by a factor M .

A. Classical single-bit error correction method

In case of a single-bit error at position a, i.e. e(x) = x a , then, the corresponding syndrome is s

(x) = s (1) a (x), with s (1) a (x) = x a mod g(x), (24)
where the superscript (1) indicates a syndrome for an error event of Hamming weight of 1 (single-bit error) and the subscript a indicates the position of the error. The set of N syndromes S (1) = {s

(1) i (x)} i=0,1,...,N -1 can be precomputed to correct every single-bit error event with a complexity in O(N ). To this end, s(x) is compared to the values of S (1) until finding the value a verifying s(x) = s (1) a (x). Hence, the resulting codeword is obtained from y(x) by flipping the bit at the a th position. When s(x) = 0 and s(x) / ∈ S (1) , then at least 2 errors occurred and the decoding process fails. Note that this method is applicable only if all elements of S (1) are distinct. When g(x) is irreducible over the set of polynomials with binary coefficients GF(2)[x], this hypothesis is fulfilled for N < 2 M . This method is called the table method [START_REF] Habibizad Navin | Data-oriented architecture for double and single bits error correction using Cycle Redundancy Code[END_REF]. Recently, in [START_REF] Boussard | Table-Free Multiple Bit-Error Correction Using the CRC Syndrome[END_REF], another algorithm called "Table-Free Multiple Bit-Error Correction" (TFMBEC) Fig. 2: Computation example of middle remainders was proposed. However, the proposed algorithm involves data dependent iterations and, in the worst case (especially for a hardware implementation since constraints are set during execution time), it has a higher complexity than the table method (see section IV.B).

B. FB-SC correction method

Let us assume that there is only one error at position a, with

a = k a M + δ a (25)
and 0 ≤ δ a < M . In others words, the error belongs to the k a th segment. Then, using the same arguments as the ones used to derive (4), the middle syndromes depend only on the error pattern e(x). Let us focus on the middle syndrome s m kaM . Since e(x) = x a , we have e f kaM (x) = 0, e m kaM (x) = x δa , and e b kaM (x) = 0. Thus, the partial Forward and Backward remainders linked to e(x) are both null, giving s m kaM = x δa , i.e. a weight-1 polynomial. The value δ a identifies the index a = k a M + δ a of the bit to be flipped in y(x) to obtain a codeword. The following toy example illustrates the proposed method. Let us assume M = 7, the generator polynomial g(x) = x 7 + x 3 + 1 and the length K = 28 message u = (0001000111001000000101001111). Based on the recursive equation ( 22), the obtained codeword c is given as c = (00010001110010000001010011111001010). Let us assume an error at position a = 15, the noisy received message becomes y = (00010001110010000000010011111001010). The index a = 15 can be decomposed following (25) such that k a = 2 and δ a = 1. Fig. 2 shows graphically the computation of the 4 remainders (r f 14 , r b 14 ) and (r f 7 , r b 7 ). From this figure, we can deduce that r m 14 = r f 14 + r b 14 = (0000010), and thus, that s m 14 = y m 14 + r m 14 = (0000000) + (0000010) = (0000010), i.e. a weight-1 middle syndrome. Similarly, we get r m 7 = (1011101) and s m 7 = (1011101), i.e. weight-5 syndromes. The other middle syndromes for this example are s m 28 (x) = (1001111), s m 21 (x) = (0010011) and s m 0 (x) = (0001011). As predicted, the only weight-1 syndrome is s m 14 , i.e. the one that is associated with the position of the error. and 3) s j = 1 are met, then s(x) is a weight-1 polynomial and the corrected bit is obtained by flipping the bit in the j th position. If one of these 3 conditions is not fulfilled, then the bit should not be flipped. This operation can be implemented by the Slice Middle Syndrome Correction (SMSC) component given in Fig. 4.a). The serial association of M SMSC, as shown in Fig. 4.b), gives the Middle Syndrome Correction (MSC) components. MSC allows correcting a single-bit error in a middle syndrome. The initial conditions are T u (M ) = 0 and T d (0) = 0. By construction, if the Hamming weight of the syndrome is larger than 1, or equal to 0, the MSC just leaves y(x) unchanged. The MSC is thus used γ times during the decoding process, one for each length-M segment of the partition. Fig. 4.c shows the correction of the middle segments y m 14 and y m 7 given in the example of section IV.B. The complexity per bit can be approximated by 5 binary operations (one XOR and 2 OR operations for the computation of T u (l) and T d (l) and 2 OR for the computation of f (l) = T u (l + 1) OR T u (l -1) OR NOT(s l ), leading to an overall complexity of 5N two-inputs Boolean operations. Note that this estimate is slightly pessimistic. In fact, due to side effects, the first and the last SMSCs of the MSC require only 2 OR gates instead of 4. Table I compares the complexity of the two methods. Although one can argue about the difference in the relative complexity between the hardware of a XOR and an OR operation, the results are intended to provide a simple insight on how FB-SC reduces the decoding complexity compared to the classical method.

The optimized implementation of the TFMBEC algorithm [START_REF] Boussard | Table-Free Multiple Bit-Error Correction Using the CRC Syndrome[END_REF] requires first N (w(g) -1) XOR operations for the syndrome computation. Then, the single-bit error correction requires N iterations. For each decoding iteration, a verification condition is performed using the Hamming weight of a length M vector. If the result is greater than 1, then w(g) XOR operations are performed. By approximating the verification step to 4M gates operations, the worst case complexity is given as N (2w(g) -1 + 4M ) gate operations. This number is greater than both the classical and the FB-SC methods (see Table I).

As an application example, Table I shows also the decoding complexity for the M = 24 CRC codes used in the LTE standard [START_REF] Gpp | Evolved Universal Terrestrial Radio Access (E-UTRA); Radio Resource Control (RRC)[END_REF]. This latter specifies two CRC codes with generator polynomials g a (x) and g b (x) that contain 14 and 6 non-zero values, respectively. The FB-SC method reduces the complexity by a factor 2 for g a (x) and 3 for g b (x).

V. CONCLUSION

The paper presents a simplified method called FB-SC to detect and correct all single-bit error of a CRC code. The principle Fig. 4: Example of architecture for the correction of the 1 bit flip-error event is to simplify the error detection and correction of a singlebit error by computing a partition of middle syndromes. In our future work, we will extend the FB-SC method to correct two-error patterns. Preliminary results confirm the appeal of the proposed method.

TABLE I :

 I Complexity Comparison in Terms of 2-Input Binary Operations Between the Classical and The FB-SC Methods.

			Classical	FB-SC
		Syndrome	N w(g) -N	2N w(g) -N
	General case	Correction	2N M	5N
		Total	N (2M + w(g) -1)	N (2w(g) + 4)
	LTE CRC	ga(x) g b (x)	61N 53N	32N 16N

From c(x), the binary encoded message is given as c = (u K-1 , u K-2 , . . . , u 0 , r M -1 , r M -2 , . . . , r 0 ).

C. Complexity comparison

In this section, we compare the complexities of the classical and the proposed FB-SC methods. Both methods apply a 2-phase approach consisting of syndrome determination followed by error mitigation. The recursive computation of r f l (x) is given in (22). Starting from r f K (x) = 0, the recursion r f l (x) = (r f l+1 (x)x+y l+M x M ) mod g(x) is performed from l = K -1 down to 0. This is a well known Linear Feedback Shift Register (LFSR) structure, similar to the one represented in Fig. 3 for M = 7, g(x) = x 7 + x 3 + 1. In the general case, the number of XOR operations to perform the full recursion is given as N (w(g) -1), where w(g) denotes the number of non-zero values of the generator polynomial g. For the FB-SC method, both forward and backward recursions need to be performed, which doubles this complexity. Moreover, N additional XOR operations are required to compute the middle syndrome (i.e. γ = N/M computations of (17), each requiring M XOR operations). The overall complexity for syndrome computation of the FB-SC method becomes 2N × (w(g) -1) + N XOR operations. While the FB-SC method requires a larger number of computations for the first phase of the algorithm, its complexity is considerably reduced when applying the second phase of single-bit error correction.

The complexity of the classical method is given as the cost to compare the N syndromes of the set S (1) with the actual syndrome s(x) of the received codeword. This is motivated by the fact that the comparison of two vectors of size M (i.e. the size of the syndrome) requires M XOR operations to sum them, then a M -entry NOR gate to test that the resulting vector is equal to the null vector. This latter operation requires a binary tree of M -2 OR gates followed by an NOR gate inverter. The complexity of this method can be approximated to N × M required XOR and OR operations.

Applied during the second phase, the FB-SC decoder checks if one of the middle syndromes has a Hamming weight of one. Several ways can be used to achieve this goal. We propose to derive an upper bound on the corresponding complexity. In order to simplify the notations, we omit the exponent m (denoting middle) as well as the subscript kM of the considered middle syndrome in the following description.

Let us thus consider the received middle section y(x), the associated remainder r(x) and syndrome s(x) = y(x) + r(x), each of size M . If s(x) has a weight 1, then the corrected version is equal to ĉ(x) = y(x) + s(x). For a given index l in the range 0 to M -1, let us define the Boolean variable T u (l) (subscript u for up) equals to one if the set (s M -1 , s M -2 , . . . , s l+1 ) contains at least one value 1, 0 otherwise. Similarly, we define T d (l) (subscript d for down) that equals 1 if the set (s l , s l-1 , . . . , s 0 ) contains at least one value 1, 0 otherwise. By construction, T u (l) = T u (l + 1) OR s l and T d (l) = T d (l -1) OR s l . For a given index j, if the 3 following conditions 1) T u (j + 1) = 0 (all syndrome values between j + 1 and M -1 are equal to 0), 2) T d (j -1) = 0 (all syndrome values between 0 and j -1 are equal to 0)