Marie Bourdin

Guillaume Salek

Alexandre Fargues

Sandra H Messaddeq

Younes Messaddeq

Thierry Cardinal

Manuel Gaudon

Investigation on the coloring and bleaching processes of WO3-x photochromic thin films

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Photochromism is the ability for a material to change its coloration in a reversible way under light irradiation. The photochromic materials can be divided into two main groups: organic materials, such as spiropyrans, azobenzenes or diarylethenes [1][2][3][4] , and inorganic materials such as transition metal oxides, metal halides or rare earth complexes [5][6][7][8] . Several chemical processes, induced by irradiation, can lead to the color change of the compound. For inorganic materials, photochromic phenomena are essentially linked to a photo-induced redox process. Inorganic photochromic materials are generally transition metal oxides. These oxides are large gap semiconductors (between 2.5 and 3.5 eV) like the molybdenum, niobium, vanadium or tungsten compounds: MoO3 6,9-12 , Nb2O5 13,14 , V2O5 15,16 or WO3 5,[17][18][19][20][21] . Regarding more particularly the tungsten oxide, Deb was the first to highlight the photochromic properties of the WO3 thin films, in 1973 5 . Although WO3 is not the preferred material for working on photochromism, compared to MoO3, which exhibits better photochromic effect in terms of contrast and kinetic. Indeed, despite of the weak optical contrast between colored and bleached states as well as its very slow and incomplete bleaching process (lack of adequate reversibility for many applications), WO3 oxide is more and more studied. Indeed, from the optimization of crystallite morphologies, recent studies have been carried out showing a particular interest for producing photosensitive and reprintable inks 22,[START_REF] Wei | ACS Applied Material Interfaces[END_REF] based on WO3 as nanoparticles. The mechanisms governing photochromism in transition metal oxides are complex and are not fully understood yet, at least they remain a subject to controversies. However, the literature reports agreed on the fact that the photochromism mechanism is induced for irradiation wavelengths allowing the transfer through the semiconductor forbidden band with the creation of an exciton, i.e. for light energy in UV range for WO3 oxide since the gap between valence and conduction band is located at the border between UV and visible range for this oxide. In 1973 5 , Deb has proposed a complete model explaining the photochromic effect in WO3 films under UV irradiation. Deb proposed that the photochromism is linked to the creation of colored centers during the preparation of the film. These colored centers would trap the photo-induced free electrons leading to the coloration of the material. Several studies were carried out [20][21][22][START_REF] Wei | ACS Applied Material Interfaces[END_REF][START_REF] Gavrilyuk | [END_REF] , it is now well established that the blue color in WO3 issued from the photo-reduction of the material is based on the transfer of electrons between tungsten ions having different valence states (W 6+ , W 5+ ) via an intervalence charge transfer (IVCT). The photo-redox process follows the next steps. When the WO3 film is irradiated with UV light, electrons (e -) and holes (h + ) are generated (excitons). Then, sufficiently slow charge depletion occurs, allowing separate reactions of the initiated electrons and holes, additionally to photo-excitons recombination which remains the main. Thus, few electrons are then trapped by some tungsten ions, leading to W 5+ ion formation, so IVCT phenomenon and blue coloration. IVCT is associated with the creation of an absorption site with energy corresponding to the frontier between visible range and near infrared, and with the blue coloration of the material 20,[START_REF] Gavrilyuk | [END_REF] In literature, it is often suggested that the holes would then react with the water adsorbed on the WO3 film surface to form the H + ions (water oxidation); these protons would then diffuse into the material, thus creating a tungsten bronze with HxWO3 chemical formula. This model explains the stabilization of the reduction of a fraction of the W 6+ ions into W 5+ ions thanks to their association with the integration of protons inside the WO3 crystalline network. Nonetheless, the latter model is in default to explain the bleaching process while maintaining the material in dark, which runs like the reverse redox equation and so implies the deprotonation. Although most of the studies devoted to the photochromism of WO3 agree and re-propose the Deb's process, all the parameters necessary to see the appearance of photochromism are not certain. For illustration, recent studies have shown that photochromism only appears if the WO3 oxide previously exhibits oxygen vacancies 25,26 . Gérard et al. have shown that perfectly stoichiometric WO3 films have a very weak photochromic effect compared to films that are slightly deficient in oxygen 27 . Other authors have shown that water must be initially chemisorbed on the film' s surface 28 or even directly incorporated as crystalline water groups into the material such as in the WO3,0.33H2O 29 to observe efficient photochromism. With the aim to determine which parameters could lead to an improvement in the photochromic performances of WO3, a new generation of studies have focused the effort on the evaluation of many factors which can influence photochromic behavior, such as heat treatment, the presence of doping ions 30 or adsorbed water 28,31 . The different film preparation techniques can also lead to slight modification on the forbidden band energy, the chemical compositions and the film microstructures, thus leading to different photochromic performances. It has also been shown that particle size plays a predominant role on the photochromism intensity. Indeed, particles of small sizes and therefore having a large specific surface are shown to be more photo-reactive 32 . The impact of the WO3 particle shape has also been studied: nanosheets or nanoflocons, with large specific surfaces, have a good photoreactivity with faster coloration process 22,33 . The comparison of photochromic contrast obtained from the different available crystal structures for WO3 oxide: cubic, hexagonal, monoclinic systems 31,34 , have shown that the optical contrast is linked to the structural feature. In this article, we propose to study the photochromic properties of thin films of WO3 obtained by dip-coating from suspended nanoparticles in ethanol. In the first part, the discussion focuses on the film elaboration. In a second part, the photochromic behavior of two types of film is compared and deeply analyzed: on the one hand a WO3 pristine film, and on the other hand a sandwich film constituted by a WO3 sub-layer covered with a Ta2O5 top-layer. Finally, the last section is devoted to the reversibility aspect of the photochromism of the two previous films.

Experimental

2.1. Synthetic procedures WO3-x film elaboration. Chemical reagents were purchased from Sigma Aldrich and used as received. Tungsten (VI) chloride was used as tungsten source and diethylene glycol (DEG) as solvent. WCl6 (7,2g) was added to 100mL of DEG and 20mL of distilled water. The mixture was heated at 180°C with continuous stirring and refluxed for 3h. At the end of the reaction, a deep blue precipitate was obtained. The precipitate was washed and centrifuged several times with ethanol to remove any trace of solvent and dried in an oven at 80°C and then was washed in HCL solution (1 mol.L -1 ) for surface functionalization with protons. The functionalized WO3-x nanoparticles (NPs) were then dispersed in ethanol to prepare the suspensions (weight content in NPs is 86 g.L -1 ) which are dipcoated on standard glass substrate to obtain pristine WO3-x thin films. The dip-coating parameters are (i) the dip-coating speed: 133mm.min -1 and (ii) for multilayers, successive dip coating steps are operated with between two steps 30 min drying in an oven (90°C).

Ta2O5 and film elaboration. Chemical reagents were purchased from Sigma Aldrich and used as received. Tantalum ethoxide was used as Ta source and anhydrous ethanol as solvent. The sol is obtained from tantalum ethoxide, anhydrous ethanol, 37% hydrochloric acid and distilled water. The protocol consists in preparing two separate solutions before mixing to obtain a stable sol. In solution A, 1 g of tantalum ethoxide is added to 4.92 ml of anhydrous ethanol, and stirred for 10 min. In solution B, 2.46 mL of ethanol is mixed with 0.0664 mL of distilled water and 0.01 mL of hydrochloric acid. After mixing, by adding solution B, drop by drop, into solution A, the asprepared sol is stirred for 1 hr. Once the sol was prepared, films were made by dip-coating with a deposition rate fixed at 133 mm.min -1 . The layers are dried after each deposition for 30 min. at 90 ° C. Deposition of one to four successive layers have been made. Ta2O5 / WO3-x composite film elaboration. "sandwich design" with WO3-x film covered by a Ta2O5 as top-layer are discussed.

Characterization techniques

Chemical, structural and morphological characterizations. WO3-x chemical composition of the as-prepared powder can be determined thanks to chemical titration. The chemical titration was made after dissolution of the powder in NaOH aqueous solution within presence of KMnO4 and KI salt dissolved in large excess in regard of the tungsten trioxide concentration. Redox process between MnO4 -and W 5+ ions and then between the rest of MnO4 -and I -lead to the formation of I2 species which can be titrated by Na2S2O3 solution in order to get W 5+ concentration, and then to deduce the oxygen substoichiometry in WO3-x compounds (x equals half time the W 5+ concentration). Zeta potential, which corresponds to the potential of the particles located at the sliding plane between the layer of Stern and the diffuse layer, is measured basing on the electrophoretic mobility of the particles in aqueous solution with different pH values with a Zetasizer NanoZS (Malvern). With the same apparatus, Dynamic light scattering (DLS) was used to determine the size distribution profile of the WO3-x particles in the suspensions used for dip-coating process. The NPs and the film structures were characterized by X-ray diffraction analysis (PANanalytical X'Pert Pro instrument Cu K1 = 1.54056 Å, K2 = 1.54439 Å and 2θ range from 8° to 80°). The morphology of the as-prepared films was studied using Scanning electron microscopy (SEM) (Quanta 3D FEG) working at 5 kV acceleration voltage. Film thickness was measured using an optical profilometer WYCO NT1100 (Veeco). Irradiation condition. Irradiations were performed using a UV Hg-lamp (Vilbert Lourmat 8.LC) at a monochromatic wavelength of 365 nm. The sample is placed at 10 cm from 8 W tubes: the power per surface unit received by the sample is equal to 5.6 W/m 2 , i.e. irradiance close to the UV-daylight irradiance in London. The use of such low fluence is to be near the real condition, in regard of life-conditions the smart windows or smart textiles are exposed. Optical characterization. Film transmission spectra were recorded at room temperature from 200 to 2500 nm on a Cary 5000 spectrophotometer using an integration sphere (spectral resolution: 1 nm and band width: 2 nm).

Results and Discussion

3.1. Film elaboration WO3-x suspensions. The WO3-x nanoparticles, with a pale blue color, were already described in terms of morphology, structure and chemical composition in our previous paper devoted to the comparison of the photochromic efficiency of various WO3-type powders 35 . The amount of W 5+ within the raw material, was determined using an iodometric titration. The tests have given an average x value of 0.018, leading to the formula of the raw powder W 6+ 0.982W 5+ 0.036O2.982. The structure was shown to be pseudo-cubic with a unit-cell parameter about 3.80 Å. The particle sizes exhibit a narrow distribution centred around 4-5 nm diameter. Preliminary works to obtain some stable WO3-x NPs suspensions are firstly conducted (Fig. 1). The suspension obtained with the particles extracted from the precipitation medium (DEG) with simple washing with ethanol were sedimenting very quickly. It was necessary to develop a strategy for the preparation a more stable suspension. First, the determination of the isoelectric point of the WO3-x particles, was performed. Zeta potential values of the particles obtained as a function of the pH of the aqueous solution are shown in Fig. 1a. The isoelectric point of our particles is at a very acidic pH of 1.5 as already listed in the literature 36,37 . Note that this can be interpreted as characteristic of a particle surface very rich in cations (W 6+ , W 5+ ), the surface therefore has a very clear tendency for hydroxylation at the expense of protonation. Cation-rich surfaces are in good agreement with the WO3-x oxygen-deficient stoichiometry: a segregation of the oxygen vacancies at the surface may also be considered. Furthermore, zeta measurements show that the particles sediment easily for suspensions of pH between 1 and 2.2 (generally a suspension is said to be stable when the potential, in absolute value, is greater than 20 mV); suspension is stable for pH between 2.2 and 6. In this range, the particles exhibit negatively-charged surfaces due to the formation of pending bonds between the surface tungsten ions and OH -hydroxyl groups. For pH values higher than 5-6 the particles are gradually dissolved since the paradoxical rise in surface potential between pH between 4 and 6 (the surface charge becomes less negative) can be seen as a sign precursor to dissolution, with transformation of the nature of the dispersed particles: transition from tungsten oxides to tungsten hydroxides. This measurement in aqueous solution allowed us to develop a protocol for stabilization of our particle suspensions in ethanol. Once the ethanol washes carried out, the particles were washed two additional times in distilled water, the goal being to negatively charge the surface of our particles (hydroxyl group fixation). The particles were subsequently re-suspended in ethanol. The final suspension obtained is blue and much more stable against sedimentation, even over long periods of time (more than 4 weeks). In order to illustrate the influence of such an hydroxyl functionalization of the WO3-x NPs surfaces on the particles, a quasi-elastic light scattering (DLS) measurement was performed on two suspensions, respectively with particles which are waterwashed and not water-washed (results reported in Fig. 1b).

Figure 1: a) Evolution of the zeta potential of the WO3-x particles according to the pH. b) Size distribution of WO3-x particles in suspension with (ii) or without (i) hydroxyl group fixation.

WO3-x film elaboration. The WO3-x films are prepared from 86 g.L -1 suspension with hydroxylated particles. The photographs and transmission spectra show that the films from the modified suspension present a good visible transmission with a low scattering coefficient (Fig. 2a). In the near infrared, the (i) (ii) transmission at 1200 nm varies from 80% for 1-layer film to 62% for 6-layer film (1-, 2-, 3-, 4-, 5-and 6-layer films were studied). From the 3-layer film, a distinct absorption band appears centred at 1200 nm; this absorption band is linked to the presence of W 5+ ions due to the intervalence charge transfer (IVCT) between W 5+ and W 6+ ions in the structural network of the WO3-x particles. As the photochromism is related to the increase of these IVCT charge transfer band, the thickness reached from the 3-layer coating is enough to carry out photochromism studies. The thickness measurements performed with optical profilometer allow assessing the thickness evolution of the films versus the number of successive coatings. The thickness variation is nearly linear with the number of deposited layers (Fig. 2b). The SEM images of our coatings (top views), reported in Fig. 3, show that the surface of the films has an adequate density; however, the presence of cracks, in particular on films for the 6 layers film, is evidenced. Ta2O5 film elaboration. The films obtained from 1 to 4 successive coating in Ta2O5 sols were characterized with the same procedure as the WO3-x films i.e. by transmission spectra, photographs, thickness measurements and SEM (top views) images. X-ray diffraction shows no diffraction peaks, the Ta2O5 films are amorphous. The transmission spectra (Fig. 4a) show that the four films studied are transparent with a transmission in the visible range of more than 80%; even with the increase the number of successive deposits. In addition, the appearance of interference fringes, (with a single very wide fringe for films from one or two successive deposits and two clearly visible fringes for films from three or four deposits) is also a sign that the surface of the films is very smooth and the film thicknesses are very regular. Indeed, the interference fringes only appear when the film roughness is low, and the thickness is enough regular to allow consistent interference between the multiple reflections appearing at the different film / air interfaces. The thickness evolution versus the number of successive deposits is roughly linear (Fig. 4b), and SEM images resolution is not sufficiently to evidence any defects (cracks or roughness): films are smooth and crack-free. From these data and using the method proposed by Swanepoel 38 (illustration figure 4c) it is possible to estimate the value of the refractive index of the film of Ta2O5 from the envelope which includes the interference fringes, i.e. from the tm minimal and TM maximal transmission values in the interference regime and the substrate index from equations: 𝑛 = (N+(N 2 -𝑛𝑠 2 ) 1/2 ) 1/2 (nS = 1.52) with N = 2×ns×(TM-tm)/(TM×tm)+(ns 2 +1)/2. Thus, the effective refractive index of the Ta2O5 film can be estimated to 1.9 (at a wavelength of 600 nm). In regard of the theoretical refractive index of tantalum oxide: 2.23 39 , the estimated effective refractive index indicates that quite dense films are obtained, with about 15-20% of porosity (considering that the effective index can be calculated as a mixing law of porosity and Ta2O5 particle linear susceptibilities). WO3-x / Ta2O5 composite film elaboration. The method chosen to make composite films is to prepare "sandwich-type" films, i.e. on a 1 to 6-layer WO3-x film, a single top-layer of Ta2O5 is then coated. The idea is to fill surface defects of the WO3-x films with the Ta2O5 additional layer, in order to limit the film scattering phenomenon thanks to the proximity of the refractive indices of these two compounds. In addition, the top layer of tantalum oxide also constitutes a protective layer of the film of tungsten oxide particles (particularly regarding oxidative attack: hydration, from the outside medium).

From the transmission spectra (Fig. 5a), several elements deserve discussion. The good transparency in the visible area of films, already demonstrated by the photographs of these films, is confirmed by the transmission spectra. The appearance of interference fringes, implying multireflection effect within the film, indicates that the optical quality (roughness) of the film surface is greatly improved by adding the Ta2O5 layer. Moreover, the absorption band relative to the presence of W 5+ ions (IVCT), centred at 1200 nm, is now visible for films made of successive deposition steps greater than 3 layers. Analysis of the surface of the films by scanning electron microscopy (Fig. 5b), also shows that the surface roughness of the films is improved by the additional layer of Ta2O5 oxide. Nonetheless, a significant roughness and small cracks linked to the tungsten oxide bottom layer (especially visible on high magnification images) remain visible. For less than 2 successive coatings of tungsten oxide, the cracks are no longer visible. On the other side, there is also deadhesion of the films from the substrate for films issued from 5 and 6 successive coatings of tungsten oxide. This delamination phenomenon can be linked to a higher chemical affinity of the WO3-x layer with Ta2O5 than with the silica glass substrate. Constraints created during gelation of the of Ta2O5 top-layer (during the sol-gel transition), can spread to the interface between the WO3-x film and the substrate. Recent studies performed in our laboratory have shown that this delamination can be cancelled by depositing as a first primer, a layer of Ta2O5. Such Ta2O5 / WO3-x / Ta2O5 architecture is in prospect. In the following, the 4-layer WO3-x recovered with Ta2O5 top-layer, representing a good compromise between a strong IVCT absorption and low scattering will be studied.

Photochromic properties

Photo-induced colouring of WO3-x film. The photochromic study has firstly focused on the 4-layer WO3-x pristine film for which the IVCT in the near infrared region is already visible before irradiation. The film was submitted to different irradiation durations (cumulated) (Fig. 6a).

We notice firstly that the irradiation makes the film "reacting" in the visible and near-infrared ranges with higher reactivity in the near-infrared region. Nevertheless, the absorption of a part of the red light under irradiation produces a pale blue coloration of the films. This well suggests a significant increase of W 5+ ions. The photo-redox process is herein the origin of a "huge photochromic phenomenon" for such a thin film in regard of the literature devoted to WO3 photochromism. Indeed, an important increase of the absorption at 1200 nm (1 eV) with a loss of more than 8% of transmission after only 10 seconds of irradiation is observed (when this decrease is only 0.25% at 600 nm (2 eV), showing a poor photochromism in the visible range). Similarly, after 2.5 hours of irradiation, the transmission decreased by 62% at 1200 nm.

Considering that the initial absorbance of the film as the reference, the transmission curves were converted in relative gained absorption of the WO3-x film (Fig. 6b). This conversion of the transmission of the film-substrate sandwich into the selective absorption of the film allows a de-summation in Gaussian peak of the large absorption IVCT band (Fig. 6c).

To try to assign each phenomenon to a specific region of the spectrum we have de-summed the spectra with Gaussians to determine the number of phenomena involved in the photochromism of WO3-x oxide films. For this de-summation, data are plotted vs energy, then treated between 0.62 eV and 2 eV (2000 nm -620 nm) to include the whole part of the spectrum reacting to UV radiation. The absorption bands are simulated by Gaussian according to the following relation:

𝑦 = 𝐴 × exp {-½ × [(𝑥 -𝑥𝑐)/𝑤] 2 },
with y is the absorbance, A is the magnitude of a Gaussian contribution in absorbance unit, x the energy in eV, xc the position of the Gaussian curve in energy, w the width of Gaussian curve in energy. To obtain satisfying simulation, i.e. a fit of the experimental absorbance envelope with low reliability factor, three Gaussian curves, partially convoluted, must be used. The 3 Gaussians have the following positions (see Table 1): the main one (Gaussian curve #1) is centred on 1.14 eV (1090 nm); this main peak is surrounded by two satellite signals, located respectively at lower energy, around 0.78 eV (1600 nm), and at higher energy, around 1.53 eV (810 nm). The hypothesis put forward to justify the consideration of 3 Gaussians is difficult to definitively establish. As far as we know, none report of such deep investigation can be found in the literature. The partial convolution of three absorption bands means three different absorption phenomena. The absorption can come from polaronic transfers (IVCT band), but also from the formation of an electronic gas (plasmonic band linked to Drude's theory). Here, the appearance of the high energy component (band located on 1.5 eV), potentially could be attributed to plasmonic effect related to the appearance of free electrons. The existence of the other two sub-bands can be attributed to two different polaronic transfers. One may propose that these two bands correspond respectively to bulk and surface polaronic transfers (nonetheless, it can be also related to a polaronic transfer involving W 5+ ion neighbouring oxygen vacancies get a specific energy). The magnitude A of these bands roughly evolves versus irradiation time with quite the same kinetics. Nonetheless, some slight differences, especially the Gaussian curve #3 evolves with a lower kinetic than the two other ones, explain the slight blue shift of the maximum of the absorption envelop versus the irradiation time. Thereafter the plot of the evolution of the absorbance at 1 eV (1200 nm), i.e. corresponding to the central position of the complex absorption phenomenon appearing under irradiation is used to investigate more accurately the photo-redox kinetic. It can be noted that for all the kinetic studies that will follow, the choice to work on absorbance spectra rather than transmission spectra is justified by the fact that the absorbance phenomenon is directly related to the concentrations of chromophore species. Overall, absorbance seems to be changing rapidly in the first minutes of irradiation, then seems to slow down after a few tens of minutes until almost no longer evolve after 2 hours. From a macroscopic point of view, the evolution in term of amplitude of the photo-induced absorbance signal shows an asymptotic evolution. To go further, the absorbance evolution at 1 eV versus the irradiation time is depicted on Figure 7. The experimental plot has been fitted by trend curves (considering a combination of two 1 st order kinetic laws), according to the equation:

𝑦 = 𝐴1 × [1 -exp (-𝑡/𝑘1)] + 𝐴2 × [1 -exp (-𝑡/𝑘2)],
With two characteristic times (k1 and k2) associated to the phenomena with each their own amplitude (A1 and A2). Table 2 summarizes the four working parameters (A1, k1, A2, k2). The evolution is composed of a fast-kinetic answer (k2 is about the minute), and a slow-kinetic answer (k1, of predominant impact for long irradiation times, is about the hour). The fast and the slow phenomena get roughly the same amplitude (A1 and A2 are around 0.3 in absorbance unit). For the energy considered (1 eV) the k2 parameter monitors mainly the answer of the material during the first minutes of irradiation and is significantly smaller than the parameter k1.

The independent plot of each fast-kinetic and slow-kinetic exponential contribution allows the schematization of this complex kinetics: the pink curve is associated with (A2, k2) exponential, it increases very quickly in the first seconds and then stabilizes after a while 5-10 minutes of irradiation. Blue plot for (A1, k1) exponential continues to increase even for long-time irradiation; this 2 nd phenomenon has not reached saturation (asymptotic value) even after 150 hours. These two different evolutions (fast-and slow-kinetics) can be associated to two different photochromic phenomena : during the first minutes a very pronounced reaction from the surface of the film (the charge transfer being the rate-limiting step)

and with for longer times, a reaction rate-limited by the cationic renewal of the WO3 particle surfaces (kinetics of a diffusional nature). Indeed, the quick photo-reduction of surface cationic species (photo-reduction from W 6+ to W 5+ ions) would saturate with the increase of the W 5+ surface concentration ; to go further, the surface must "desaturate" by ionic diffusion of the W 5+ surface ions towards the bulk, which leads to slower reaction speed. The photoredox reaction so proceeds with two reaction speeds is depicted in Figure 8.

W 6+ W 6+ W 6+ W 6+ W 6+ W 5+ W 6+ W 6+ W 5+ W 5+ W 6+ W 6+ W 6+ W 6+ W 6+ W 5+ W 5+ W 6+ W 6+ W 5+ W 6+ W 6+ W 5+ W 5+ k 2 k 2 k 2 k 1 k 1 k 1 2k 1 k 1 : diffusion mechanism k 2 : charge transfer surface bulk h
It can be seen that taking into account a very simplified model with two cations on surface and two cations in material bulk, different phenomena can occur during the irradiation to produce a photo-redox reaction with the reduction of the two W 6+ ions and charge migration to the bulk. Finally, the observation of equivalent amplitudes for the two phenomena (fast and slow), palpitates for bulk and surface attribution. Indeed, particles of 5 of diameter like those constituting the as-prepared films get roughly 50% of ions located on surface. Photo-induced colouring of WO3-x / Ta2O5 composite film. The photochromic study has then focused on the 4-layer WO3-x recovered by a Ta2O5 top-coat to evaluate the impact of the tantalum oxide top-layer on the photochromic effect. Indeed, this quite dense and glassy film could act like a barrier and avoid the participation of the atmospheric water molecules to the photo-redox reaction associated to the photochromism. Using same irradiation conditions and source, the same investigations than for the WO3-x pristine films were carried out on the composite films, transmission versus irradiation, absorbance versus irradiation time and de-summation of the absorbance bands are reported in Figure 9. We notice that the composite film reacts in the near infrared but less markedly than for the WO3-x pristine film. In fact, after 2 hours of irradiation, the maximal decrease in transmission at 1200 nm is only 28% as compared to the 62% drop observed in the case of the WO3-x film alone. Similarly, the film reacts truly little in the range of visible, the film is not coloured by the radiation. It seems that the phenomenon at high energy (which was attributed to the creation of an electronic gas: plasmonic: Gaussian curve #3) is not observed by adding the Ta2O5 top layer. Such phenomenon is probably causally linked to the lower amplitude of the photochromic effect: the concentration of free carriers (or W 5+ ions) remains probably insufficient to produce the creation of free electron gas. As previously made for the WO3-x pristine film, the spectra were de-summed up in the range 2000 nm -620 nm (0.62 eV -2 eV). The de-summation of the transmission spectra allows the determination of the number of convoluted Gaussians and therefore the number of photo-induced chromophores occurring in the composite film for the sandwich configuration. Two Gaussian curves are only necessary; the Gaussian curve parameters are reported in the table 3. Adding a layer of Ta2O5 the surface of the WO3-x film well limits the amount of photo-reduced tungsten VI ions. The Ta2O5 film being transparent to the irradiation lengths, it has, a priori, none effect on the creation of the concentration of exciton pairs in the tungsten oxide semi-conductor. Therefore, the addition of a Ta2O5 surface film could limit the charge depletion (between valence band h + holes and conductive band free e -) and acts in favour of the exciton rapid recombination (thus, the photochromic effect is decreased). The photo-reduction of W 6+ to W 5+ ions on one side, and/or the oxidation which is necessary to loop the redox process on the other side, are partially impeded. We believe that the toplayer contribution acts on this latter oxidation step: either in avoiding holes recombination with air water molecules (basing on Deb's mechanism), or in avoiding the surface O 2-anions departure in the form of dioxygen (what we assume to be the half-oxidation to compensate for tungsten photo-reduction, tests have shown that the humidity of the surrounding atmosphere is of a negligible impact). Moreover, the kinetics of the photochromic phenomenon was also investigated; the absorbance evolution at 1 eV versus the irradiation duration is depicted on Figure 10. As compared to the previous results obtained on WO3-x pristine films, the main new phenomenon to note for the WO3-x / Ta2O5 composite films is the increase of the absorption up to 25 minutes of irradiation followed by its gradual decrease after 45 minutes. This overly complex kinetic cannot be fitted considering standard exponential law. The bleaching effect occurring for long irradiation time, acting as a "negative photochromic effect", was never reported, as far as we know, in literature devoted to inorganic oxide's photochromism. This negative photochromism be explained by the re-oxidation of the tungsten ions (W 5+ to W 6+ from an anionic diffusion coming from the Ta2O5 film. Indeed, Ta2O5 would behave as a potential source of O 2-anions which facilitates the reoxidation of WO3-x films. Bleaching and cycling. After studying the colouring behaviour of the film as a function of duration of irradiation, we investigated its reversibility. Data from the literature report that the reversibility kinetics are very slow when the product is only left in a dark environment to relax and it is often necessary to heat the film at 80°C-100°C in air or under ozone to obtain a complete relaxation 20,22,40 . Herein, the pristine or composite films were previously irradiated for 2 hours (slightly beyond saturation time) at room temperature and let in dark for self-bleaching, directly in the spectrometer chamber in order to record the evolution of the transmission spectra: an analysis is automatically performed every hour. The calculated absorbance spectra are shown in Figure 11 and Figure 12, respectively for the WO3-x pristine film and the WO3-x / Ta2O5 composite film. The time necessary for a total return seems very long (out of the investigated time domain), the gap between the spectrum obtained after 16 hours darkening and spectrum of the initial state is still important: the maximal absorption after 16 hours darkening being respectively around 1200 nm equal to about 0.3 and 0.2 for WO3-x pristine film and the WO3-x / Ta2O5 composite film, respectively. The absorbance evolution at 1 eV (1200 nm) versus darkening time are plotted in Figure 11b and The variation of the absorbance has been simulated using the following equation: 𝑦 = 𝐴1' × exp (-𝑡/ 𝑘1') + 𝐴2' × exp (-𝑡/𝑘2'), i.e. as for the colouring effect under irradiation, a double kinetic (slow-vs fast-process) for the bleaching process.

Fitting parameters are reported in Table 4. The trends of the two plots and so for the extracted kinetic parameters for WO3-x pristine film and the WO3-x / Ta2O5 composite film are nearly similar. Indeed, for both films, the parameters k2' and k1'corresponding to the relaxation rates of the fast-and slow-bleaching processes are about 3-5 hours for the fast relaxation and of the order of the day (20-22 hours) for the slow phenomenon. Also, the ratio between fast and slow phenomenon amplitude: the A2'/A1' ratio, is about 1/3 in both cases. Time necessary for the film to return to its initial state is however very long, but the quality of the fit with a bleaching equation as a convolution of two decreasing exponential terms, show that 100% bleaching is predicted (i.e. no positive value as final asymptotic point has to be taken into consideration in our kinetic equation). The phenomenon of reoxidation of W 5+ to W 6+ by the oxygen in the air is therefore very slow (k times about few hours) compared to photoreduction of W 6+ to W 5+ under UV irradiation (k times of few minutes). Furthermore, the occurrence of slow and fastcolouring processes has been previously interpreted by a fast charge transfer circumscribed to the film surface followed by surface renewal from ionic diffusion (slow process). Logically, the re-oxidation should follow the same pathways with a rapid process involving the W 5+ ions located on the film surface followed by a slow process dealing with W 5+ species initially located in bulk progressively diffusing to the surface before to be re-oxidized. Finally, cycling test were carried out on n the WO3-x pristine film design (Figure 13). Several irradiation steps were applied (2 hours irradiation, allowing the colouring process saturation), with sufficient time in dark in between two successive irradiation steps to let the compound fully self-bleached (allowing the film to return to its "initial" state between each irradiation step). The relaxation time between the first and the second cycle is so fixed to 72 h, and 120 h was necessary between the second and the third irradiation. For the 3 first cycles, the absorbance at 1 eV (1200 nm) was recorded during the 12 hours following the colouring saturation under irradiation only. The cycling study demonstrates that the shapes of the bleaching curves of the first, the second, as well as the third cycle are all almost identical. These results tend to show the good performance, in terms of stability of the film under successive cycles (colouring under irradiation+ bleaching under dark).

Conclusions

Huge photochromic properties of WO3-x films obtained by dipcoating from suspended non-stoichiometric tungsten oxide nanopowder synthetized by a polyol chemical route have been investigated. The kinetic of the photochromism was studied. Several phenomena are responsible for two distinct kinetics of photochromism, the photoreduction of W 6+ ions at the particle surface leading to intervalence charge transfer and the diffusion of the W 5+ species within the bulk of the particles. The deposition of a layer of tantalum oxide on the WO3-x films is decreasing the photochromic effect. In this later case, the top layer is proposed to limit the holes recombination with air water molecules or the surface O 2-anions departure. Cycling effect has been also demonstrated. This study opens the door to possible applications of non-stoichiometric tungsten oxide as smart windows or in smart textiles, which could progressively coloured during the day due to day-light

Figure 1 :

 1 Figure 1: a) Transmission spectra and photographs of WO3-x films, b) evolution of the thickness of the films according to the number of layers.

Figure 2 :

 2 Figure 2: SEM pictures of the films.

Figure 4 :

 4 Figure 4: a) Transmission spectra and photographs of Ta2O3-x films, b) evolution of the thickness of the films according to the number of layers, c) Illustration of the Swanepoel plot for the evaluation of the refractive index.

Figure 5 :

 5 Figure 5: a) Transmission spectra and photographs of WO3-x / Ta2O5 composite films, and b) SEM pictures of the films.

Figure 6 :

 6 Figure 6: Evolution of a) the transmission of the WO3-x film coated on glass substrate and b) the absorbance of the WO3-x film during irradiation. c) De-summation of the WO3-x film absorbance spectrum by three Gaussian curves (for 2h30 of irradiation).

Figure 7 :

 7 Figure 7: Absorbance evolution at 1eV (1200nm) during irradiation

Figure 9 :

 9 Figure 9: Evolution of a) the transmission of the WO3-x / Ta2O5 film on glass substrate and b) the absorbance of film during irradiation. c) De-summation of the film absorbance spectrum by three Gaussian curves (for 2h30 of irradiation).

Figure 10 :

 10 Figure 10: Absorbance evolution of WO3-x / Ta2O5 film during irradiation at 1eV (1200nm)

  exp(-x*k1)) + A2*(1-ex p(-x*k2))+ A3*exp(-x*k3)+A4*

Figure 3 :Figure 4 :

 34 Figure 3: a) Bleaching of the WO3-x pristine film versus time in dark and b) corresponding absorbance evolution at 1eV

Figure 13 :

 13 Figure 13: Absorbance variation at 1 eV after several hours without irradiation, study completed on 3 different cycles.

Table 1 : Results of the of the absorbance of the WO3-x with three Gaussian curves.

 1

	2h30 irr.	Gauss 1	Gauss 2	Gauss 3
	Abs. (A) (a.u.)	0.49	0.19	0.27
	Width (w) (eV)	0.21	0.3	0.2
	Position (xc) (eV)	1.14	1.53	0.78

Table 2 : Kinetic parameters extracted from the fit of the absorbance evolution during irradiation at 1eV

 2

	A1 (a.u.)	k1(min)	A2(a.u.)	k2 (min)
	0.29	35.13	0.32	1.65

Table 3 : Results of the fit of the absorbance of the WO3-x / Ta2O5 film with two Gaussian curves.

 3

	2h30 irr.	Gauss 1	Gauss 2
	Abs. (A) (a.u.)	0.10	0.05
	Width (w) (eV)	0.21	0.25
	Position (xc) (eV)	0.95	1.32

Table 4 : Kinetic parameters extracted from the fit of the self- bleaching evolution at 1eV after 2h irradiation.

 4

		WO3-x pristine film
	A1' (a.u.)	k1' (min)	A2' (a.u.)	k2' (min)
	0.56	21.78	0.23	3.78
		WO3-x / Ta2O5 composite film
	A1' (a.u.)	k1' (min)	A2' (a.u.)	k2' (min)
	0.37	20.57	0.11	4.82

Acknowledgements

The Ph.D. grant of M. Bourdin was supported by idex of Bordeaux, the support of the LUMAQ International Associate Laboratory (LIA) between the universities of Laval and BordeauxThe authors are also grateful to the Natural Sciences and Engineering Research Council of Canada (NSERC), the ANR (ANR-17-CE08-0042-01), the Rise FunGlass and the French Region Nouvelle Aquitaine for the financial support.

The authors would like to thank Sonia Buffière for SEM images.

Conflicts of interest

There are no conflicts to declare.


Aicha Batoul

Delphine Boucher

Ranya Djihad Boulanouar

A construction of self-dual skew cyclic and negacyclic codes of length n over F p n

The aim of this note is to give a construction and an enumeration of self-dual θ-cyclic and θ-negacyclic codes of length n over Fpn where p is a prime number and θ is the Frobenius automorphism over Fpn . We use the notion of isodual codes to achieve this construction.

Introduction

Isodual codes ( [START_REF] Rains | Self-dual codes, Handbook of coding theory[END_REF]) have been recently studied on many aspects ( [START_REF] Batoul | Repeated-root isodual cyclic codes over finite fields, Codes, cryptology, and information security[END_REF], [START_REF] Batoul | Cyclic Isodual and Formally Self-dual Codes over F q + vF q[END_REF], [START_REF] Alahmadia | Isodual cyclic codes over finite fields of odd characteristic[END_REF]). Meanwhile, in [START_REF] Boucher | Construction and number of self-dual skew codes over F p 2[END_REF], a construction and an enumeration formula for self-dual θ-cyclic and θ-negacyclic codes of even length n over F p 2 were given in the case when p is a prime number and θ is the Frobenius automorphism over F p 2 . The aim of this note is to give a construction and an enumeration formula for self-dual θ-cyclic and θ-negacyclic codes of length n over F p n when θ is the Frobenius automorphism over F p n . To this end, we will use and develop the notion of (θ, ν)-isodual codes which form a subfamily of the family of isodual codes. Lastly we will consider the construction of some self-dual Gabidulin evaluation codes.

The text is organized as follows. In Section 2 we define the notion of (θ, ν)-isodual codes over F q where θ is an automorphism of F q and ν belongs to F * q . We recall the definitions of (θ, a)-constacyclic, θ-cyclic and θ-negacyclic codes and some generalities on the dual of a (θ, a)constacyclic code. Then we characterize (θ, ν)-isodual θ-cyclic and θ-negacyclic codes thanks to an equation satisfied by the skew check polynomials of the codes. In Section 3 we consider the special case when q is equal to p n where p is a prime number and θ is the Frobenius automorphism over F p n . After having given a necessary and sufficient condition for the existence of (θ, ν)-isodual θ-cyclic and θ-negacyclic codes, we give a construction and an enumeration formula for (θ, ν)isodual and self-dual θ-cyclic and θ-negacyclic codes. In Section 4, we consider a subclass of self-dual θ-cyclic codes over F p n which are self-dual Gabidulin codes. We parametrize this family by a parameter which satisfies a polynomial system.

Some generalities on isodual skew codes

We first recall that a linear code C of length n and dimension k over F q is a subspace of dimension k of F n q . A generator matrix G of C is a k × n matrix with coefficients in F q and rank k such that

C = {m G | m ∈ F k q }. Furthermore the dual of C is C ⊥ = {x ∈ F n q | ∀c ∈ C, <
x, c >= 0} where for x = (x 0 , . . . , x n-1 ), y = (y 0 , . . . , y n-1 ) in F n q , < x, y >:= n-1 i=0 x i y i is the Euclidean scalar product of x and y. Isodual codes ( [START_REF] Rains | Self-dual codes, Handbook of coding theory[END_REF]) have been recently studied on many aspects ( [START_REF] Batoul | Repeated-root isodual cyclic codes over finite fields, Codes, cryptology, and information security[END_REF], [START_REF] Batoul | Cyclic Isodual and Formally Self-dual Codes over F q + vF q[END_REF], [START_REF] Alahmadia | Isodual cyclic codes over finite fields of odd characteristic[END_REF]). Definition 1 ( [17] page 199). A code C with generator matrix G is isodual if it is equivalent to its dual. That means that there exists a monomial matrix D such that G • D is a generator matrix of the dual C ⊥ of C.

In what follows, we define a special class of isodual codes which are parameterized by an automorphism θ of F q and an element ν of

F * q . Definition 2. Consider n ∈ N * , ν ∈ F * q and θ ∈ Aut(F q ). A linear code C of length n and generator matrix G is a (θ, ν)-isodual code if G • D is a generator matrix of C ⊥ where D is the n × n diagonal matrix with diagonal coefficients ν, θ(ν), . . . , θ n-1 (ν). Remark 1. A code C is self-dual if and only if there exists ν fixed by θ such that C is (θ, ν)-isodual.
Recall that if θ is an automorphism of F q , the skew polynomial ring R is defined as R = F q [X; θ] under usual addition of polynomials and where multiplication is defined by the commutation law : ∀a ∈ F q , X •a = θ(a)X ( [START_REF] Ore | Theory of non-commutative polynomials[END_REF]). The ring R is noncommutative unless θ is the identity automorphism on F q . The ring R is right-Euclidean and left-Euclidean. For f = a i X i in R and α in F q , the evaluation f (α) of f at α is the remainder in the right division of f by X -α. We have [START_REF] Lam | Vandermonde and Wronskian matrices over division rings[END_REF]). Recall also that if q = p n and θ is the Frobenius automorphism, then the center of R is

f (α) = i a i N i (α) where N i (x) := xθ(x) • • • θ i-1 (x) (see
F p [X n ] .
For a in F * q and θ in Aut(F q ), a (θ, a)-constacyclic code C of length n and dimension

k is a left R-submodule Rg/R(X n -a) ⊂ R/R(X n -a)
where g is a monic skew polynomial of degree n -k right-dividing X n -a in R ( [START_REF] Boucher | Codes as modules over skew polynomial rings, Cryptography and coding[END_REF]). That means that a word c = (c 0 , . . . , c n-1 ) ∈ F n q belongs to C if and only if the skew polynomial g right-divides the skew polynomial c

0 + c 1 X + • • • + c n-1 X n-1 in R.
The skew polynomial g is called the skew generator polynomial of C. The monic skew polynomial h defined by

Θ n (h) • g = X n -a (1) is called skew check polynomial of C. The (θ, a)-constacyclic code C is denoted C = (g) a n,θ . If a = 1, the code is θ-cyclic and if a = -1, the code is θ-negacyclic. A generator matrix of C is G =       g 0 g 1 . . . . . . 1 0 . . . 0 0 θ(g 0 ) θ(g 1 ) . . . . . . 1 . . . . . . . . . . . . . . . . . . 0 0 . . . 0 θ k-1 (g 0 ) θ k-1 (g 1 ) . . . . . . 1       . ( 2
)
The

skew reciprocal polynomial of h = Σ k i=0 h i X i ∈ R of degree k is h * = Σ k i=0 θ i (h k-i )X i . If h 0 = 0, the left monic skew reciprocal polynomial of h is h = 1 θ k (h0) h * .
The following technical lemma will be useful later. We will use the application Θ : R → R given by

k i=0 a i X i → k i=0 θ(a i )X i . Lemma 1 (Lemma 1 of [8]). Consider θ ∈ Aut(F q ), R = F q [X; θ], h and g in R. Then (h • g) * = Θ deg(h) (g * ) • h * . Example 1. Consider n = 4, F 2 4 = F 2 (a) with a 4 + a + 1 = 0 and R = F 2 4 [X; θ]. We have X 4 + 1 = (X 2 + a 5 X + a 5 ) • (X 2 + a 5 X + a 10 )
therefore the skew polynomial g 1 = X 2 + a 5 X + a 10 generates a θ-cyclic code C 1 of length 4 and dimension 2 over F 2 4 . As Θ 4 is the identity over F 2 4 , the skew check polynomial of the code is

h 1 = X 2 + a 5 X + a 5 .
We have

X 4 + 1 = (X 2 + aX + a 14 ) • (X 2 + a 4 X + a)
therefore the skew polynomial g 2 = X 2 + a 4 X + a generates a θ-cyclic code C 2 of length 4 and dimension 2 over F 2 4 with skew check polynomial h 2 = X 2 + aX + a 14 .

The following proposition describes the dual of a (θ, a)-constacyclic code.

Proposition 1 (Theorem 1 and Lemma 2 of [START_REF] Boucher | A note on the dual codes of module skew codes[END_REF], Proposition 1 of [START_REF] Boucher | Autour de codes définis à l'aide de polynômes tordus[END_REF]). Consider n ∈ N * , a ∈ F * q , θ ∈ Aut(F q ) and C a (θ, a)-constacyclic code of length n with skew generator polynomial g and skew check polynomial h. Then the dual C ⊥ of C is a (θ, 1/a)-constacyclic code with skew generator polynomial h .

Proof. (proof of Proposition 1 of [START_REF] Boucher | Autour de codes définis à l'aide de polynômes tordus[END_REF]) We consider the equality (1) in R = F q [X; θ] and we multiply both members of this equality by h on the right. We get Θ n (h) • g • h = (X n -a) • h and we deduce from this equality that Θ n (h) • (X n -g • h) = a • h. As the skew polynomials Θ n (h) and a • h have the same degrees, the skew polynomial X n -g • h is a constant that we will denote λ and

Θ n (h) • λ -a • h = 0. As the leading coefficient of Θ n (h) • λ -a • h is equal to θ k (λ) -a, we get that λ = θ -k (a).
Furthermore, as Θ n (h)•g = X n -a, according to Lemma 1, we have -

1 a Θ k-n (g * )•h * = X n -1 a . Therefore h right-divides X n -1
a and is the skew generator polynomial of a (θ, 1 a )-constacyclic code of length n.

A quick computation gives that for all (i, j) in {0, . . . , k -1} × {0, . . . , n -k -1}, the Euclidean scalar product of the words associated to X i • g and X j • h * is equal to θ i ((g • h) j-i+k ). Therefore the scalar product is equal to 0 and the words of the code (g) a n,θ are orthogonal to the words of the code (h )

1/a n,θ .
In what follows, we characterize (θ, a)-constacyclic codes which are (θ, ν)-isodual.

Proposition 2. Consider k ∈ N * , n = 2k, ν ∈ F * q , a ∈ F q , θ ∈ Aut(F q ), R = F q [X; θ], h ∈ R monic.
The (θ, a)-constacyclic code of length n, dimension k and skew check polynomial h is (θ, ν)-isodual if and only if

Θ n (h) • θ k (ν) • h • 1 ν = X n -a. (3)
In this case we have a 2 = θ n (ν)/ν.

Proof. Consider C = (g) a n,θ the (θ, a)-constacyclic code of length n = 2k, dimension k, skew check polynomial h and skew generator polynomial g. According to (1), we have Θ n (h)

• g = X n -a. Therefore, the relation (3) is satisfied if and only if h = g where g = θ k (1/ν) • g • ν. Let us prove that C is (θ, ν)-isodual if and only if h = g. As g right-divides X n -a, g right-divides X n -ã where ã = a ν θ n (ν)
. Therefore we can consider the (θ, ã)-constacyclic code of length n and skew generator polynomial g. Furthermore, according to Proposition 1, C ⊥ is a (θ, 1/a)-constacyclic code of length n with skew generator polynomial h .

Let us prove that C is (θ, ν)-isodual if and only if (h )

1/a n,θ = (g) ã n,θ . Denote g = n-k i=0 g i X i = k i=0 g i X i and g = k i=0 gi X i . We have gi = θ k (1/ν)g i θ i (ν) for all i in {0, . . . , k}. Therefore a generator matrix of (g) ã n,θ is G =       g0 g1 . . . . . . 1 0 . . . 0 0 θ(g 0 ) θ(g 1 ) . . . . . . 1 . . . . . . . . . . . . . . . . . . 0 0 . . . 0 θ k-1 (g 0 ) θ k-1 (g 1 ) . . . . . . 1       = G • D
where G is given by ( 2) and D is the diagonal matrix with diagonal elements ν, θ(ν), . . . , θ n-1 (ν).

According to Definition 2, the code C is (θ, ν)-isodual if and only if a generator matrix of

C ⊥ = (h ) 1/a n,θ is G • D. As G • D = G is a generator matrix of (g) ã n,θ , we obtain that C is (θ, ν)-isodual if and only if (h ) 1/a n,θ = (g) ã n,θ . Lastly as g right-divides X n -a ν θ n (ν) and h right-divides X n -1 a
, we obtain a 2 = θ n (ν)/ν. In the case when θ is the Frobenius automorphism over F q and q = p n where p is prime and n is the length of the code we obtain the following corollary that will be useful in next section.

Corollary 1. Consider k ∈ N * , n = 2k, p a prime number, ν ∈ F * p n , a ∈ F p n , θ : x → x p ∈ Aut(F p n ), R = F p n [X; θ], h ∈ R monic. The (θ, a)-constacyclic code of length n and skew check polynomial h is (θ, ν)-isodual if and only if h • θ k (ν) • h • 1 ν = h • 1 ν • h • θ k (ν) = X n -a. (4)
Furthermore, a 2 = 1.

Proof. As θ is the Frobenius automorphism over F p n , the order of θ is equal to n. Therefore Θ n (h) = h and θ n (ν) = ν. According to Proposition 2, the (θ, a)-constacyclic code of length n and skew check polynomial

h is (θ, ν)-isodual if and only if h • θ k (ν) • h • 1 ν = X n -a. In this case a 2 = 1. Therefore X n -a is central in R, and we have h•θ k (ν)•h • 1 ν = X n -a = h • 1 ν •h•θ k (ν). Example 2. (Example 1 continued) The left monic skew reciprocal polynomial of h 1 = X 2 + a 5 X + a 5 is h 1 = X 2 + a 5 X + a 10 and X 4 + 1 = (X 2 + a 5 X + a 5 ) • (X 2 + a 5 X + a 10 ) = (X 2 + a 5 X + a 10 ) • (X 2 + a 5 X + a 5 ), therefore the θ-cyclic code C 1 with skew check polynomial h 1 is self-dual.
The left monic skew reciprocal polynomial of

h 2 = X 2 + aX + a 14 is h 2 = X 2 + a 6 X + a 4 . Furthermore, X 4 +1 = (X 2 +aX +a 14 )•(X 2 +a 4 X +a) = (X 2 +aX +a 14 )• 1 a 4 •(X 2 +a 6 X +a 4 )• 1 a 14 , therefore the θ-cyclic code C 2 with skew check polynomial h 2 is (θ, a 14 )-isodual.
Lastly, we consider below a technical lemma which will be useful later and which deals with the factorization of skew polynomials right-dividing X n ± 1 in F p n [X; θ] where θ is the Frobenius automorphism. These skew polynomials belong to a wide class of skew polynomials, called Wedderburn polynomials, which have been extensively studied (see Theorem 6.4 of [START_REF] Delenclos | Noncommutative symmetric functions and W -polynomials[END_REF] for the factorizations of these skew polynomials). Lemma 2 can be directly deduced from Theorem 6.4 of [START_REF] Delenclos | Noncommutative symmetric functions and W -polynomials[END_REF] as well as from Proposition 2.2.2. of [START_REF] Caruso | A new faster algorithm for factoring skew polynomials over finite fields[END_REF]. We propose here a proof very specific to our special case.

Lemma 2. Consider n ∈ N * , p a prime number, θ : x → x p ∈ Aut(F p n ), R = F p n [X; θ], f in R of degree d and ∈ {-1, 1} such that f right-divides X n -in R. Then f is the product of d linear factors right-dividing X n -and #{(α 1 , . . . , α d ) ∈ F d p n | f = (X + α 1 ) • • • (X + α d )} = d i=1 p i -1 p -1 .
Proof. Consider y 1 , . . . , y n in F p n linearly independent over F p . Consider ξ in F p n such that X -ξ right-divides X n -, which means N n (ξ) = . Denote α 1 := ξ θ(y1) y1 , . . . , α n = ξ θ(yn) yn . According to [START_REF] Lam | Vandermonde and Wronskian matrices over division rings[END_REF], the least common left multiple of X -α 1 , . . . , X -α n is lclm 1≤i≤n (X -α i ) = X n -. As f right-divides X n -, according to Theorem 4 of [START_REF] Ore | Theory of non-commutative polynomials[END_REF], there exist β 1 , . . . , β d in F p n such that f = lclm 1≤i≤d (X -β i ). Furthermore N n (β i ) = = N n (ξ). Therefore according to Theorem 27 of [START_REF] Jacobson | The Theory of Rings[END_REF], there exists z i in F * p n such that β i = ξ θ(zi) zi . According to [START_REF] Lam | Vandermonde and Wronskian matrices over division rings[END_REF], z 1 , . . . , z d are linearly independent over

F p . Denote f = d i=0 a i X i , we have {α ∈ F p n | f (α) = 0} = {α = ξ θ(y) y = ξy p-1 ∈ F p n | L(y) = 0} where L(y) := d i=0 a i N i (ξ)θ i (y).
As the equation L(y) = 0 has d solutions (z 1 , . . . , z d ) in F p n linearly independent over F p , there are p d -1 nonzero y in F p n such that L(y) = 0. Therefore {α ∈ F p n | X -α right-divides f } has (p d -1)/(p -1) elements. We conclude using an inductive argument.

Remark 2. It could be noted that the number

d i=1 p i -1 p -1
is the size of the general linear group GL(d, p) modulo diagonal matrices, and corresponds to choosing a set of 1-dimensional vector spaces spanning a d-dimensional vector space.

Example 3. (Example 1 continued) The skew polynomial h 1 = X 2 +a 5 X +a 5 has 3 factorizations into the product of linear monic skew polynomials, namely h 1 = (X+a 14 )•(X+a 6 ) = (X+a 11 )•(X+ a 9 ) = (X +1)•(X +a 5 ). The skew polynomial h 2 = X 2 +aX +a 14 has also 3 factorizations into the product of linear monic skew polynomials, namely

h 2 = (X + a 7 ) • (X + a 7 ) = (X + a 11 ) • (X + a 3 ) = (X + a 10 ) • (X + a 4 ).
3 Construction and enumeration of (θ, ν)-isodual θ-cyclic and θ-negacyclic codes of length n over F p n

The aim of this section is to construct and to enumerate self-dual θ-cyclic and θ-negacyclic codes of length n over F p n where θ is the Frobenius automorphism. Note that in this setting, θ-cyclic codes are called Gabidulin p-cyclical codes (page 6 of [START_REF] Gabidulin | Theory of codes with maximum rank distance, Problemy Peredachi Informatsii[END_REF]).

To achieve this construction, we will consider θ-cyclic and θ-negacyclic codes which are (θ, ν)isodual.

We introduce some notation. Consider R = F p n [X; θ]. We will denote, for ∈ {-1, 1} and ν ∈ F * q :

H ν, := {h ∈ R | h monic, h • 1 ν • h • θ k (ν) = X n -}.
According to Corollary 1, the set H ν, is the set of the skew check polynomials of (θ, ν)-isodual (θ, )-constacyclic codes. Following Remark 1, the set H 1, is the set of the skew check polynomials of self-dual (θ, )-constacyclic codes.

3.1 Necessary and sufficient existence condition for (θ, ν)-isodual θ-cyclic and θ-negacyclic codes of length n over F p n

In [START_REF] Boucher | A note on the existence of self-dual skew codes over finite fields[END_REF] a necessary and sufficient condition for the existence of self-dual (θ, )-constacyclic codes over a finite field F q was derived where θ is an automorphism of F q and ∈ {-1, 1}. In what follows we give a necessary and sufficient condition for the existence of (θ, ν)-isodual (θ, )-constacyclic codes of length n over F p n where p is a prime number and θ is the Frobenius automorphism.

Proposition 3. Consider k ∈ N * , n = 2k, p a prime number, θ : x → x p ∈ Aut(F p n ), ∈ {-1, 1}.
(i) If p = 2, then there exists a (θ, ν)-isodual θ-cyclic code of length n for all ν in F * p n .

(ii) If p is odd, then for ν ∈ F * p n , there exists a (θ, ν)-isodual (θ, )-constacyclic code of length 2k if and only if ν

p n -1 2 = -(-1) k p-1 2 .
Proof. According to Corollary 1, there exists a (θ, ν)-isodual (θ, )-constacyclic code of length n if and only if the set H ν, is nonempty.

• Assume that p = 2 (therefore = 1) and

ν ∈ F * 2 n . Consider α in F 2 n such that α 2 = 1/ν 2 k -1 = ν θ k (ν) and h = X k + α. We have h • 1 ν • h • θ k (ν) = X k + 1 θ k (α) • X k + θ k (ν)α ν = X 2k + θ k 1 α + α θ k (ν) ν X k + θ k (ν)α θ k (α)ν . As ν/θ k (ν) = α 2 , we obtain h • 1 ν • h • θ k (ν) = X 2k + 1 αθ k (α)
.

As θ(α) = α 2 = ν θ k (ν) , we obtain θ k+1 (α) = θ k (ν) ν , θ(α)θ k+1 (α) = 1 and αθ k (α) = 1. Therefore the skew polynomial h belongs to H ν, .

• Assume that p is odd and ν

p n -1 2 = -(-1) k p-1 2 . We have (-θ k (ν) ν ) (p n -1)/2 = (-1) (p n -1)/2 θ k (ν (p n -1)/2 ) ν (p n -1)/2
= (-1) (p n -1)/2 = 1 (because p is odd, therefore p 2 ≡ 1 (mod 4) and p n ≡ 1 (mod 4)).

Therefore

-ν θ k (ν) is a square. Consider α such that -ν θ k (ν) = α 2 . Consider h = X k + α in R. h • 1 ν • h • θ k (ν) = X k + 1 θ k (α) • X k + θ k (ν)α ν = X 2k + θ k 1 α + α θ k (ν) ν X k + θ k (ν)α θ k (α)ν .
As ν/θ k (ν) = -α 2 , we obtain

h • 1 ν • h • θ k (ν) = X 2k + θ k (ν)α θ k (α)ν . Furthermore θ k (ν)α θ k (α)ν = θ k (ν) ν -θ k (ν) ν p k -1 2 (because -θ k (ν)/ν = 1/α 2 ) = (-1) p k -1 2 ν p n -1 2 = -(because ν p n -1 2 = -(-1) k p-1 2 ). Therefore h • 1 ν • h • θ k (ν) = X n -.
• Assume that p is odd and that there exists a (θ, )-constacyclic code of length 2k which is (θ, ν)-isodual. Consider h its skew check polynomial and h 0 the constant term of h. Necessarily the degree of h is equal to k.

As the code is (θ, )-constacyclic of length 2k, h right-divides X 2k -. As the code is defined over

F p 2k with θ : x → x p , X 2k -is central of degree 1 in F p [X 2k
] and h is the product of k linear skew polynomials X + α 1 , . . . , X + α k right-dividing X 2k -. Therefore h 0 is the product of the k constant terms α 1 , . . . , α k . As N 2k (-α i ) = , we have :

N 2k (h 0 ) = k = N k (h 0 )N k (θ k (h 0 )).
As the code is (θ, ν)-isodual, we have h

• 1 ν • h • θ k (ν) = X n -and h 0 θ k (ν) + θ k (h 0 )ν = 0. As N k (h 0 )N k (θ k (h 0 )) = k , we obtain N k (h 0 ) 2 (-) k N 2k (ν) N k (ν) 2 = k and N 2k (ν) = (-1) k N k (ν/h 0 ) 2 . Furthermore, θ(N k (ν/h 0 )) = N k (ν/h 0 ) θ k (ν/h0) ν/h0
= N k (ν/h 0 )(-). Therefore we have

N k (ν/h 0 ) p-1 = -. To conclude, if p is odd, ν p n -1 2 = N 2k (ν) (p-1)/2 = (-1) k p-1 2 N k (ν/h 0 ) p-1 = -(-1) k p-1 2 . Remark 3. Consider k ∈ N * , n = 2k, p an odd prime number, θ : x → x p ∈ Aut(F p n ), ∈ {-1, 1}.
According to Proposition 3 and Remark 1, there exists a self-dual (θ, )-constacyclic code of length n = 2k over F p n if and only if 1 = -(-1) k p-1 2 . We therefore obtain the previous result of Proposition 5 of [START_REF] Boucher | A note on the existence of self-dual skew codes over finite fields[END_REF]: if p is odd, there exists a self-dual θ-cyclic code of dimension k if and only if p ≡ 3 (mod 4) and k is odd; there exists a self-dual θ-negacyclic code of dimension k if and only if p ≡ 1 (mod 4) or p ≡ 3 (mod 4) and k even.

The existence result of Proposition 3 is not constructive and the aim of what follows is to design a construction of the set H ν, based on the construction of the sets Hµ, defined for µ ∈ F *

p n by Hµ, := {h ∈ R | h ∈ H h0/µ, }.
Thanks to factorization properties of the elements of Hµ, , we will give both a construction and an enumeration formula.

Construction and enumeration formula for the set Hµ,

The following technical lemma (Lemma 3) will be useful for the construction of Hµ, (Proposition 4).

Lemma 3. Consider k ∈ N * , n = 2k, p a prime number, θ : x → x p ∈ Aut(F p n ), R = F p n [X; θ],
P ∈ R and ∈ {0, . . . , k} such that Θ k+ (P * ) = P and deg(P ) = 2k -2 . If X + α right-divides P then there exists Q ∈ R satisfying the two following properties :

1. P = Θ k--1 ((X + α) * ) • Q • (X + α); 2. Θ k+ +1 (Q * ) = Q. Proof. Consider f ∈ R such that P = f • (X + α).

Let us prove that

P = Θ k-1-((X + α) * ) • Θ k+ (f * ) and that X + α right-divides Θ k+ (f * ). As P = f • (X + α), according to Lemma 1, we have P * = Θ 2k-2 -1 ((X + α) * ) • f * . Therefore P = Θ k+ (P * ) = Θ k--1 ((X + α) * ) • Θ k+ (f * ). Denote K = k -and f = 2K-1 j=0 a j X j , then f * = 2K-1 j=0 θ 2K-1-j (a j )X 2K-1-j . Consider β = -θ K (α), then X + α right-divides Θ k+ (f * ) if and only if f * cancels at β.

Let us prove that

K-1 j=0 θ 2K-1-j (a j )N 2K-1-j (β) = - 2K-1 j=K θ 2K-j-1 (a j )N 2K-j-1 (β). As P = f • (X + α) = Θ K-1 ((X + α) * ) • Θ K+2 (f * ), we obtain : ∀j ∈ {1, . . . , K}, a j-1 = j i=1 N i-1 (-α) N j (-α) θ K (θ 2 +i-1 (a 2K-i ) + αθ 2 +i-1 (a 2K-i+1 )).
Therefore,

K-1 j=0 θ 2K-1-j (a j )N 2K-1-j (β) = K j=1 θ 2K-j (a j-1 )N 2K-j (β) = - K j=1 N 2K-j (β)θ 2K-j j i=1 N i-1 (-α) N j (-α) θ K (θ 2 +i-1 (a 2K-i ) + αθ 2 +i-1 (a 2K-i+1 )) = - K j=1 N 2K-j (β) j i=1 θ K-j N i-1 (β) N j (β) θ 3K-j θ 2 +i-1 (a 2K-i ) + αθ 2 +i-1 (a 2K-i+1 ) = - K i=1 K j=i N 2K-j (β)θ K-j N i-1 (β) N j (β) θ i+K-j-1 (a 2K-i ) - K-1 i=1 K j=i+1 N 2K-j (β)θ K-j N i (β) N j (β) θ i+K-j (a 2K-i )θ 3K-j (α) = - K i=1 N K (β) N i-1 (β) N K (β) θ i-1 (a 2K-i ) - K-1 i=1 K-1 j=i N 2K-j (β)θ K-j Ni-1(β) Nj (β) θ i+K-j-1 (a 2K-i ) +N 2K-(j+1) (β)θ K-(j+1) Ni(β) Nj+1(β) θ i+K-j-1 (a 2K-i ) θ 3K-j-1 (α) = - K i=1 N i-1 (β)θ i-1 (a 2K-i ) - K-1 i=1 K-1 j=i N 2K-j-1 (β) θ K-j-1 N i (β) N j+1 (β) θ i+K-j-1 (a 2K-i ) (θ 2K-j-1 (β) + θ 3K-j-1 (α)) .
As β = -θ K (α), we obtain θ 2K-j-1 (β) + θ 3K-j-1 (α) = 0. Therefore

K-1 j=0 2K-1-j (a j )N 2K-1-j (β) = - K i=1 N i-1 (β)θ i-1 (a 2K-i ) = - 2K-1 j=K θ 2K-j-1 (a j )N 2K-j-1 (β).
We conclude that f * (β) = 0 and that X +α right-divides Θ k+ (f * ). We deduce the existence of a skew polynomial

Q such that Θ k+ (f * ) = Q • (X + α) and we obtain P = Θ k-1-((X + α) * ) • Q • (X + α).

Let us prove that

Q = Θ k+1+ (Q * ). According to Lemma 1, as Θ k+ (f * ) = Q • (X + α), we obtain f = Θ k-1-((X + α) * ) • Θ k+1+ (Q * ). Therefore P = Θ k-1-((X + α) * ) • Q • (X + α) = Θ k-1-((X + α) * ) • Θ k+1+ (Q * ) • (X + α) and Q = Θ k+1+ (Q * ).
We are now giving a construction and an enumeration formula for the sets Hµ, (Proposition 4 and Algorithm 1).

Proposition 4. Consider k ∈ N * , n = 2k, p a prime number, θ : x → x p ∈ Aut(F p n ), R = F p n [X; θ], ∈ {-1, 1}, µ ∈ F * p n and P k = -µ (X 2k -) ∈ R.
The set Hµ, is nonempty if and only if θ k (µ) + µ = 0. In this case we have Algorithm 1 Construction of the set Hµ, Require: : µ in F * p n such that θ k (µ) + µ = 0 Ensure: : Hµ,

1: P k ← -µ(X n -) 2: S ← ∅ 3:
Construct the sequences (α i , P i-1 ) i=k,...,1 such that

• X + α i right-divides P i ; • P i-1 is the quotient in the left-division of Q i-1 by Θ i-1 ((X + α i ) * ) where Q i-1
is the quotient in the right-division of P i by X + α i .

4: for each sequence (α 1 , . . . , α k ) do 5:

S ← S ∪ {(X + α 1 ) • • • (X + α k )} 6: end for 7: return S Ha 10 ,1 = {X 2 + a 10 X + a 10 , X 2 + a 13 X + a 7 , X 2 + a 12 X + a 8 , X 2 + a 9 X + a 11 , X 2 + X + a 5 , X 2 + a X + a 4 , X 2 + 1, X 2 + a 6 X + a 14 , X 2 + a 3 X + a 2 , X 2 + a 9 , X 2 + a 4 X + a, X 2 + a 12 , X 2 + a 7 X + a 13 , X 2 + a 3 , X 2 + a 6 }.

Enumeration of (θ, ν)-isodual and self-dual θ-cyclic and θ-negacyclic codes

From Proposition 4, we deduce the number of (θ, )-constacyclic codes of length 2k which are (θ, ν)-isodual. Let us first consider the particular case when k = 1 :

Example 5. Consider k = 1, n = 2. If p = 2 and ν ∈ F * 2 2 , then H ν, = {X + α} where α 2 = ν. If p is odd and ν p 2 -1 2 = -(-1) p-1 2 , then H ν, = {X + α | α 2 = -ν p-1 }. Namely, consider h = X + α ∈ R, then h • ν • h • 1 θ(ν) = (X + 1 θ(α) ) • (X + να θ(ν) ) = X 2 + ν + θ(ν)θ(α 2 ) θ(α)ν X + να θ(να)
.

Therefore h belongs to H ν, if and only if

α 2 = -θ(ν)/ν. Proposition 5. Consider k ∈ N * , n = 2k, p a prime number, θ : x → x p ∈ Aut(F p n ), ∈ {-1, 1} and ν ∈ F * p n . If ν p n -1 2 = -(-1) k p-1 2 then the number of (θ, ν)-isodual (θ, )-constacyclic codes of length n over F p n is N k-1 i=1 (p i + 1)
where N = 1 if p = 2 and N = 2 if p is odd.

Proof. We first prove that for ν in F * p n , if H ν, is nonempty, then its cardinality does not depend on ν. Consider ν, ν in F * p n such that

ν p n -1 N = (ν ) p n -1 N = -(-1) k p-1 N .
Then according to Proposition 3, H ν, and H ν , are nonempty. Consider ξ = ν ν and a a square root of θ k (ξ). As ξ p n -1 N = 1, a is well defined. The application H a 8 ,1 = {X 2 + a 7 X + a 8 , X 2 + a 12 X + a 13 , X 2 + a 3 }, H a 7 ,1 = {X 2 + a 8 X + a 7 , X 2 + a 12 , X 2 + a 3 X + a 2 }, H a 6 ,1 = {X 2 + a 14 X + a 11 , X 2 + a 4 X + a, X 2 + a 6 }, H a 5 ,1 = {X 2 + 1, X 2 + X + a 10 , X 2 + a 10 X + a 5 }, H a 4 ,1 = {X 2 + a 11 X + a 4 , X 2 + a 9 , X 2 + a 6 X + a 14 }, H a 3 ,1 = {X 2 + a 2 X + a 8 , X 2 + a 7 X + a 13 , X 2 + a 3 }, H a 2 ,1 = {X 2 + a 3 X + a 7 , X 2 + a 13 X + a 2 , X 2 + a 12 }, H a,1 = {X 2 + a 14 X + a, X 2 + a 9 X + a 11 , X 2 + a 6 }.

We check that

µ =µ ( Hµ,1 ∩ Hµ ,1 ) = ν =ν (H ν, ∩ H ν , ) = {X 2 + 1, X 2 + a 9 , X 2 + a 3 , X 2 + a 12 , X 2 + a 6 }, H1,1 ∩ Ha 5 ,1 ∩ Ha 10 ,1 = ν =ν =ν (H ν, ∩H ν , ∩H ν , ) = {X 2 +1, X 2 +a 9 , X 2 +a 3 , X 2 +a 12 , X 2 +a 6 }, ∅ = ν =ν =ν =ν (H ν, ∩ H ν , ∩ H ν , ∩ H ν , ).
In [START_REF] Boucher | Construction and number of self-dual skew codes over F p 2[END_REF], a formula for the number of self-dual (θ, )-constacyclic codes of length n is given over

F p 2 ⊂ F p n when 2 = 1.
In what follows, we deduce from Proposition 5 the number of self-dual (θ, )-constacyclic codes of length n over F p n . Ensure: : H ν, : the set of the skew check polynomials h of (θ, ν)-isodual (θ, )-constacyclic codes of length n and dimension k over F p n . 1: S ← ∅ 2: for µ in F * p n such that θ k (µ) + µ = 0 do 3:

P k ← -µ(X n -) 4:
Construct the sequences (α i , P i-1 ) for i = k, . . . , 1 such that

• X + α i right-divides P i ; • P i-1 is the quotient in the left-division of Q i-1 by Θ i-1 ((X + α i ) * ) where Q i-1
is the quotient in the right-division of P i by X + α i ;

• P 0 = µ/ k i=1 α i = ν. 5:
for each (α 1 , . . . , α k ) do 6:

S ← S ∪ {(X + α 1 ) • • • (X + α k )} 7:
end for 8: end for 9: return S Example 7. There are 3 self-dual θ-cyclic codes of length 4 over F 2 4 , given in Example 6 by H 1,1 . There are 15 self-dual θ-cyclic codes of length 6 over F 2 6 = F 2 (a) where a 6 + a 4 + a 3 + a + 1 = 0. Their skew check polynomials are : X 3 + a 52 X 2 + a 23 X + a 54 , X 3 + X 2 + a 9 X + a 36 , X 3 + a 36 X 2 + a 36 X +a 45 , X 3 +1, X 3 +a 19 X 2 +a 29 X +a 27 , X 3 +a 41 X 2 +a 46 X +a 45 , X 3 +a 18 X 2 +a 18 X +a 54 , X 3 +a 21 X 2 +a 42 X +1, X 3 +a 26 X 2 +a 43 X +a 27 , X 3 +a 9 X 2 +a 9 X +a 27 , X 3 +a 42 X 2 +a 21 X +1, X 3 +a 13 X 2 +a 53 X +a 45 , X 3 +a 38 X 2 +a 58 X +a 54 , X 3 +X 2 +a 18 X +a 9 and X 3 +X 2 +a 36 X +a 18 .

A construction of self-dual θ-cyclic codes over F p n which are Gabidulin evaluation codes

We consider here a special subfamily of self-dual θ-cyclic codes of length n over F p n which are in fact Gabidulin evaluation codes and therefore Maximum Rank Distance (MRD) codes (see [START_REF] Gabidulin | Theory of codes with maximum rank distance, Problemy Peredachi Informatsii[END_REF] for the theory of MRD codes). In previous sections we have provided a construction of self-dual codes based on the factorization of the skew check polynomials into the product of linear skew polynomials. Here we change the point of view by writting the skew generator polynomials as least common left multiples (lclm) of special linear skew polynomials, namely skew polynomials of the form X -θ i (α).

G =      y 1 y 1 . . . y n θ(y 1 ) θ(y 2 ) . . . θ(y n ) . . . θ k-1 (y 1 ) θ k-1 (y 2 ) . . . θ k-1 (y n )      . (7)
The following Proposition 7 can be found in [START_REF] Boucher | Linear codes using skew polynomials with automorphisms and derivations[END_REF] (Proposition 3, part 1).

Proposition 7 (Proposition 3 of [9]). Consider k ≤ n ∈ N * , p a prime number, θ : x → x p ∈ Aut(F p n ), R = F p n [X; θ], α ∈ F p n , = N n (α)
. Assume that 2 = 1 and that 1, α, N 2 (α), . . ., N n-1 (α) are linearly independent over F p . The θ-cyclic code of length n and skew generator polynomial g = lclm 0≤i≤k-1 (X -θ i (α)) is the dual of the Gabidulin evaluation code of length n, dimension k and support (1, α, N 2 (α), . . . , N n-1 (α)).

Proof. Consider c = (c 0 , . . . , c n-1 ) ∈ F n p n . The word c belongs to the θ-cyclic code generated by g if and only if for all i in {0, . . . , k -1}, X -θ i (α) right-divides g. As the remainder in the right division of c by X -θ i (α) is n-1 j=0 c j N j (θ i (α)) = n-1 j=0 c j θ i (N j (α)), we obtain that a check matrix for the code is the matrix G given by [START_REF] Boucher | Codes as modules over skew polynomial rings, Cryptography and coding[END_REF] where y 1 = 1, y 2 = α, . . . , y n = N n-1 (α).

According to Theorem 4.10 of [START_REF] Nebe | On self-dual MRD codes[END_REF], if a Gabidulin evaluation code of length 2k and dimension k is self-dual then p ≡ 3 (mod 4) and k is odd. In what follows we construct a family of self-dual Gabidulin evaluation codes parameterized by an element α of F p n . Proposition 8. Consider k ∈ N * , n = 2k, p a prime number, θ : x → x p ∈ Aut(F p n ), R = F p n [X; θ], α ∈ F p n such that N n (α) = 1 and 1, α, N 2 (α), . . . , N n-1 (α) are linearly independent over F p . The θ-cyclic code of length n and skew generator polynomial g = lclm 0≤i≤k-1 (X -θ i (α)) is self-dual if and only if n-1 i=0 N i (α) 1+p = 0, ∀ ∈ {0, . . . , k -1}. Proof. The code is self-dual if and only if the lines of G are pairwise orthogonal i.e. G G T = 0 where G is the matrix defined by [START_REF] Boucher | Codes as modules over skew polynomial rings, Cryptography and coding[END_REF] with y 1 = 1, y 2 = α, . . . , y n = N n-1 (α). i=0 N i (α) 10 = 0 has 18 solutions α : a 580 , a 406 , a 378 , a 436 , a 124 , a 8 , a 648 , a 126 , a 388 , a 216 , a 42 , a 72 , a 14 , a 284 , a 24 , a 372 , a 488 , a 490 and we get 18 self-dual θ-cyclic codes generated by the skew polynomials g = lclm(Xα, X -θ(α), X -θ 2 (α)). These codes are self-dual Gabidulin evaluation codes (and therefore selfdual MRD codes). For example, take α = a 8 , then g = lclm(X -a, X -θ(a), X -θ 2 (a)) = X 3 + a 185 X 2 + a 383 X + a 322 generates a self-dual θ-cyclic code of length 6 which is the Gabidulin evaluation code of dimension 3 and support (1, a, . . . , N 5 (a)) = (1, a 8 , a 32 , a 104 , a 320 , a 240 ).

An open question is to determine the number of self-dual θ-cyclic codes generated by g = lclm 0≤i≤k-1 (X -θ i (α)) over F p n with n = 2k. More generally, it could be interesting to construct and count self-dual θ-cyclic codes which are MRD.

Conclusion

This note was devoted to the construction of self-dual θ-constacyclic codes of length n over F p m when m is equal to n and θ is the Frobenius automorphism over F p n . This work completes previous works on self-dual θ-constacyclic codes over F p m when m = 1 (then θ is the identity and codes are classical constacyclic codes) and when m = 2. As a further work, it could be interesting to study the cases when 2 < m < n and to have a more general classification only based on the order of the automorphism θ in Aut(F p n ).

Proposition 6 . 2 = - 1 ,

 621 Consider k ∈ N * , n = 2k, ∈ {-1, 1}, p a prime number and θ :x → x p ∈ Aut(F p n ). If p = 2, there are k-1 i=1 (p i + 1) self-dual θ-cyclic codes of length n over F p n . If p = 2, and if (-1) k p-1 there are 2 k-1 i=1 (p i + 1) self-dual (θ, )-constacyclic codes of length n over F p n .Proof. Self-dual (θ, )-constacyclic codes are (θ, 1)-isodual (θ, )-constacyclic codes. The result follows from Proposition 5. Algorithm 2 (θ, ν)-isodual (θ, )-constacyclic codes of length n over F p n Require: : k ∈ N * , n = 2k, p, prime number, ∈ {-1, 1}, ν ∈ F * p n

Example 8 . 6 i=0

 86 For p = 3 and k = 3, according to Proposition 5, there are 80 self-dual θ-cyclic codes of length 6 over F 3 3 . The polynomial system   N i (α) 2 = 0 6 i=0 N i (α) 4 = 0 6

  Definition 3.[START_REF] Gabidulin | Theory of codes with maximum rank distance, Problemy Peredachi Informatsii[END_REF] Consider k ≤ n ∈ N * , p a prime number, θ : x → x p ∈ Aut(F p n ), y 1 , . . . , y n ∈ F p n linearly independent over F p . The Gabidulin evaluation code of length n, dimension k and support (y 1 , . . . , y n ) is the code with generator matrix

Hµ, = {(X + α 1 ) • • • (X + α k ) | X + α k right-divides P k X + α k-1 right-divides P k-1 (α k ) . . . X + α 1 right-divides P 1 (α 2 , . . . , α k )}

where for i = k, k -1, . . . , 2, P i-1 (α i , . . . , α k ) is the quotient in the left-division of Q i-1 by Θ i-1 ((X + α i ) * ) and Q i-1 the quotient in the right-division of P i (α i+1 , . . . , α k ) by X + α i :

Furthermore Hµ, has k i=1 (p i + 1) elements.

Proof. Consider h in Hµ, and

the skew polynomial h is the product of k linear factors right-dividing X n -and therefore

As the skew polynomial X + α k right-divides P k , according to Lemma 3 applied to P = P k and = 0, there exists

We conclude using an inductive argument and Lemma 3.

Conversely, consider h = (X+α 1 ) • • • (X+α k ) in R such that X+α i right-divides P i (α i+1 , . . . , α k ) defined by [START_REF] Boucher | Construction and number of self-dual skew codes over F p 2[END_REF]. According to Lemma 3, Θ k+1 (P * k-1 ) = P k-1 , . . . , Θ 2k-1 (P * 1 ) = P 1 ; furthermore, as X + α 1 right-divides P 1 , there exist P 0 such that

. Therefore, we obtain h * • P 0 • h = P k . In particular, the constant term of both polynomials is

Let us determine the cardinality of Hµ, . According to Lemma 2, the number of factorizations (as a product of k linear monic factors) of any monic skew polynomial of degree k right-dividing

p-1 . Furthermore, as P i has degree 2i and right-divides X n -, the number of

The number of elements of Hµ, is therefore

Example 4. Consider F 2 4 = F 2 (a) with a 4 + a + 1 = 0 and n = 4. For µ ∈ {1, a 5 , a 10 }, Hµ,1 has 15 elements : H1,1 = {X 2 + a 14 X + a, X 2 + a X + a 14 , X 2 + a 11 X + a 4 , X 2 + a 2 X + a 13 , X 2 + a 13 X + a 2 , X 2 + a 4 X + a 11 , X 2 + a 5 X + a 10 , X 2 + 1, X 2 + a 8 X + a 7 , X 2 + a 9 , X 2 + a 7 X + a 8 , X 2 + a 12 , X 2 + a 3 , X 2 + a 6 , X 2 + a 10 X + a 5 },

and

Therefore f is bijective and all nonempty sets H ν, have the same number of elements, M : for all ν in F p n such that ν

N , the number of (θ, ν)-isodual (θ, )-constacyclic codes of length 2k is M . Furthermore, we have

Therefore

Now consider the union of the intersections Hµ, ∩ Hµ , for µ = µ and Hµ, , Hµ , nonempty. We have, according to (6) :

Similarly, we get

where the involved sets H ν, and Hµ, are nonempty. Furthermore, #(∪ µ Hµ, ) = µ # Hµ, -

Lastly, according to Proposition 4, the p k -1 nonempty sets Hµ, all have k i=1 (1+p i ) elements. As the p 2k -1 N nonempty sets H ν, all have M elements, we obtain

Example 6. Consider the (θ, ν)-isodual θ-cyclic codes of length 4 over F 2 4 = F 2 (a) where a 4 + a + 1 = 0. For µ ∈ {1, a 5 , a 10 }, Hµ,1 has 15 elements (Example 4) and Hµ,1 = H ν, has 35 elements. For ν ∈ F * 2 4 , H ν, has 3 elements : there are 3 (θ, ν)-isodual θ-cyclic codes : H 1,1 = {X 2 + 1, X 2 + a 10 X + a 10 , X 2 + a 5 X + a 5 }, H a 14 ,1 = {X 2 + a X + a 14 , X 2 + a 9 , X 2 + a 6 X + a 4 }, H a 13 ,1 = {X 2 + a 2 X + a 13 , X 2 + a 12 X + a 8 , X 2 + a 3 }, H a 12 ,1 = {X 2 + a 8 X + a 2 , X 2 + a 13 X + a 7 , X 2 + a 12 }, H a 11 ,1 = {X 2 + a 9 X + a, X 2 + a 6 , X 2 + a 4 X + a 11 }, H a 10 ,1 = {X 2 +1, X 2 +X +a 5 , X 2 +a 5 X +a 10 }, H a 9 ,1 = {X 2 +a X +a 4 , X 2 +a 11 X +a 14 , X 2 +a 9 },