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Les codes de Reed Solomon: étude et simulation.

Les codes de Reed Solomon: étude et simulation.

La partie théorique a pour but de dégager les fondements théoriques des codes correcteurs d’erreurs en général, à savoir : les théorèmes fondamentaux des codes correcteurs, la classification des codes, les codes linéaires, la matrice génératrice d’un code linéaire, la matrice de contrôle d’un code linéaire, la distance minimale d’un code linéaire, le décodage d’un code linéaire, les codes parfaits et les codes cycliques. Une classe très importante des codes parfaits et cycliques a été présenté dans cette partie qui est la classe des codes de Reed Solomon. Dans cette classe on a présenté le problème principal du codage, une généralité sur les corps finis de Galois, la défin ition d’un code de Reed solomon et ses avantages, la technique de codage, les algorithmes de décodage. Il y a plusieurs algorithmes de décodage des codes de Reed Solomon, mais dans cette partie on a exposé que deux algorithmes qui sont : l’algorithme d’Euclid et l’algorithme de Reed Solomon. Les deux premiers chapitres ont été consacrés à cette partie.
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Complexity Comparison of the Use of Vandermonde versus Hankel Matrices to Build Systematic MDS Reed-Solomon Codes

Complexity Comparison of the Use of Vandermonde versus Hankel Matrices to Build Systematic MDS Reed-Solomon Codes

A must be non-singular. In this paper, in addition to the tra- ditional method for building A, based on Vandermonde ma- trices, we have introduced another approach, based on Han- kel matrices. To the best of our knowledge, this is the first mention to this alternative way of designing systematic Reed- Solomon codes since their introduction in 1985, in [8]. We proved, both theoretically and experimentally, that this al- ternative solution is an order of magnitude simpler than the Vandermonde approach. The A sub-matrix of the genera- tor matrix is produced immediately, instead of having to in- vert a matrix and multiplying this inverted matrix with an- other one. Major speedups, for instance up to 157, can be achieved thanks to the Hankel approach with our software C language Hankel Reed-Solomon codec, derived from a well- known Vandermonde Reed-Solomon codec for the erasure channel. This result is of high importance for all situations where a software RS(n, k) codec needs to generate on the fly an RS code with appropriate dimension and length values.
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Algebraic Soft- and Hard-Decision Decoding of Generalized Reed--Solomon and Cyclic Codes

Algebraic Soft- and Hard-Decision Decoding of Generalized Reed--Solomon and Cyclic Codes

In Chapter 6, we propose four new bounds on the minimum distance of linear cyclic codes, denoted by bound I-a, I-b, II and III. Bound I-a is very close to bound I-b. While bound I-a is based on the association of a rational function, the embedding of a given cyclic code into a cyclic product code is the basis of bound I-b. The idea of embedding a code into a product code is extended by bound II and III. We prove the main theorems for the bounds and give syndrome-based error/erasure decoding algorithms up to bounds I-a, I-b and II. Good candidates for the embedding-technique are discussed and, as a first result, conditions for non-primitive lowest-code-rate binary codes of minimum Hamming distance two and three are given. The work is based on the contributions of Charpin, Tiet¨av¨ainen and Zinoviev [A-CTZ97; A-CTZ99]. Furthermore, we outline how embedding a given cyclic code into a cyclic product code can be extended to the embedding into a cyclic variant of generalized product codes, which has not been defined before.
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Performance of random linear network codes concatenated with reed-solomon codes using turbo decoding

Performance of random linear network codes concatenated with reed-solomon codes using turbo decoding

AP can accomplish network decoding with high probability as long as it has received at least K independent packets. A straightforward solution is to let AP do the network decoding. After getting the original packets, the AP encodes them with a FEC code and sends them to T. The system error rate performance is then determined by the FEC code. Its drawback is that the packet delay in the broadcast network can differ a lot [5] and each AP has to wait for K independent packets before network decoding and wireless transmission. Another problem happens when T moves from AP1 to AP2 during the wireless transmission. It will only receive M packets (M < K) from AP1. In order to recover all K original packets, T has to resort to AP2 either by indicating the (K −M) missing packet numbers through signaling channel or with some central control unit over the APs to let AP2 know the missing packets. This solution suffers from additional cost. In this paper, we consider another solution with high flexi- bility when the end user switches AP during the transmission. The AP starts the wireless transmission of the network coded packets (after FEC encoding) as soon as they are received at
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A comparative evaluation of high-level hardware synthesis using Reed-Solomon decoder

A comparative evaluation of high-level hardware synthesis using Reed-Solomon decoder

that different branches take equal amount of time, while we are trying to exploit the imbalance in branches. To further improve the performance and synthesis results, we made use of common guidelines [14] for code refinement. Adding hierarchy to Berlekamp computations and making its complex loop bounds static by removing the dynamic variables from the loop bounds, required algorithmic modifications to ensure data consistency. By doing so, we could unroll the Berlekamp module to obtain a throughput of 2073 cycles per block. However, as seen in Section V, even this design could only achieve 66.7% of the target throughput and the synthesized hardware required considerably more FPGA resources than the other designs.
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Good coupling between LDPC-Staircase and Reed-Solomon for the design of GLDPC codes for the Erasure Channel

Good coupling between LDPC-Staircase and Reed-Solomon for the design of GLDPC codes for the Erasure Channel

1 I. I NTRODUCTION Application Level Forward Erasure Correction (AL-FEC) codes are now a key component of reliable multicast/broadcast transmission systems. They are the key building block of the FLUTE/ALC (RFC 6726) [1] reliable multicast transport protocol that is used to push any kind of files (e.g. multimedia) for instance over the wireless 3G/4G channels (e.g. they are part of the 3GPP MBMS, DVB-H/SH IPDatacasting, or ISDB- Tmm services). They are also the key building block of robust streaming protocols, like the FECFRAME (RFC 6363) [2] transport protocol, that is also included in the above systems. The codes we consider are AL-FEC codes for the erasure channel, that can be used in FLUTE/ALC and FECFRAME. Among them Low Density Parity Check (LDPC) codes are of particular interest. The LDPC codes have been intensively studied due to their near-Shannon limit performance under iterative Belief-Propagation (BP) decoding [3][4]. A (N, K) LDPC code, where N is the code length and K is its dimension, can be graphically represented as a bipartite graph with N ”variable nodes” (VN) and M = N −K ”check nodes” (CN). Equivalently, LDPC codes can be represented through
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Enhanced Recursive Reed-Muller Erasure Decoding

Enhanced Recursive Reed-Muller Erasure Decoding

III. R ESULTS We have compared our proposal with maximum likelihood decoding using Gaussian Elimination (GE) on an Intel Core 2 Extreme @3.06Ghz on Mac OS X 10.6 in 64-bit mode. Fig. 1 shows the decoding failure probabilities for RM (3, 7) code (k = 64, n = 128), in terms of extra-packets. By extra-packets, we mean the number of received symbols above the source block size k, which corresponds to the erasure channel capacity. We compare ML, which is optimal, with our algorithm and the classical recursive algorithm. In addition, we provide results for the recursive algorithm with only permutation selection and only partial information passing. We can see that our algorithm performs well compared to the optimal GE and requires only 3 extra-symbols in average to match the performances of GE. On the opposite, the raw recursive algorithm is not able to recover anything with up to 20% of extra-symbols. For RM (3, 7), the decoding speed of both algorithms is provided in Table I. For the sake of completeness, the encoding speed for the RM(3,7) code is in the order of 4Gbps, and the decoding speed of a Reed- Solomon code with the same parameters is around 350Mbps [14].
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Enhanced Recursive Reed-Muller Erasure Decoding

Enhanced Recursive Reed-Muller Erasure Decoding

III. R ESULTS We have compared our proposal with maximum likelihood decoding using Gaussian Elimination (GE) on an Intel Core 2 Extreme @3.06Ghz on Mac OS X 10.6 in 64-bit mode. Fig. 1 shows the decoding failure probabilities for RM (3, 7) code (k = 64, n = 128), in terms of extra-packets. By extra-packets, we mean the number of received symbols above the source block size k, which corresponds to the erasure channel capacity. We compare ML, which is optimal, with our algorithm and the classical recursive algorithm. In addition, we provide results for the recursive algorithm with only permutation selection and only partial information passing. We can see that our algorithm performs well compared to the optimal GE and requires only 3 extra-symbols in average to match the performances of GE. On the opposite, the raw recursive algorithm is not able to recover anything with up to 20% of extra-symbols. For RM (3, 7), the decoding speed of both algorithms is provided in Table I. For the sake of completeness, the encoding speed for the RM(3,7) code is in the order of 4Gbps, and the decoding speed of a Reed- Solomon code with the same parameters is around 350Mbps [14].
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Reed-solomon forward error correction (FEC) schemes, RFC 5510

Reed-solomon forward error correction (FEC) schemes, RFC 5510

6.1. Determining the Maximum Source Block Length (B) The finite field size parameter, m, defines the number of non-zero elements in this field, which is equal to: q - 1 = 2^^m - 1. Note that q - 1 is also the theoretical maximum number of encoding symbols that can be produced for a source block. For instance, when m = 8 (default) there is a maximum of 2^^8 - 1 = 255 encoding symbols. Given the target FEC code rate (e.g., provided by the user when starting a FLUTE sending application), the sender calculates: max1_B = floor((2^^m - 1) * CR)

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Simple Reed-Solomon Forward Error Correction (FEC) Scheme for FECFRAME, RFC 6865

Simple Reed-Solomon Forward Error Correction (FEC) Scheme for FECFRAME, RFC 6865

The finite field size parameter m defines the number of non-zero elements in this field, which is equal to: q - 1 = 2^^m - 1. This q - 1 value is also the theoretical maximum number of encoding symbols that can be produced for a source block. For instance, when m = 8 (default) there is a maximum of 2^^8 - 1 = 255 encoding symbols. So: k < n <= 255. Given the target FEC code rate (e.g., provided by the end-user or upper application when starting the FECFRAME instance, and taking into account the known or estimated packet loss rate), the sender calculates:
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Lifted projective Reed–Solomon codes

Lifted projective Reed–Solomon codes

6.1 Information rate In this section, we emphasize how projective lifted codes surpasses projective Reed-Muller codes in terms of code rate (the local correcting capability being fixed). In Figure 1, we present the rate of PRM q ( m, k ) and PLift q ( m, k ) for increasing values of q = 2 e . These codes are comparable since they have same length n = q m q + − 1 − 1 1 , and same local correction features (locality and error tolerance). In each subfigure of Figure 1, four curves are plotted: blue ones represent projective lifted codes and red ones projective Reed-Muller codes. Plain curves correspond to the minimum error tolerance setting, for which local correction admits no error on the line being picked (see Section 4). To compare, dotted curves correspond to a constant fraction of errors tolerated by the local correcting algorithm. Here, the constant has been arbitrarily fixed to 1/32.
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Du code aux modèles, des modèles au code: enseigner les langages dédiés (DSL)

Du code aux modèles, des modèles au code: enseigner les langages dédiés (DSL)

Nous proposons un cours  brique de base  sur 8 semaines, qui pourra servir de fondation à des cours plus avancés via des extensions. L'objectif est de permettre aux étudiants d'acquérir des connaissances et compétences langages indispensables comme la (méta) modélisation, la notion d'abstraction, les choix de conception, et l'outillage théorique et pratique pour la géné- ration de code. Le cours sera également l'occasion d'aborder des thématiques plus spécialisées comme la vérication formelle et la génération de code avancée.

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Sludge drying reed beds for septage treatment: towards design and operation recommendations

Sludge drying reed beds for septage treatment: towards design and operation recommendations

This article focus on the feasibility of septage treatment and disposal on SDRB, based on experiments carried out in Andancette (France) on Cemagrefs’ pilot-scale units fed with septage. Results deal with : (i) septage quality and dewaterability, (ii) performance on sludge drying reed bed pilot-scale experiments fed with septage (result will be compared to those fed with activated sludge), (iii) design and operation conditions improvement in the French context.

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Générateur de code multi-temps et optimisation de code multi-objectifs

Générateur de code multi-temps et optimisation de code multi-objectifs

Les versions statique et LLVM O3 obtiennent les même performances, attei- gnant 66 % performance maximale théorique. Le noyau est produit par la même partie arrière dans les 2 cas, le code résultant est donc ici similaire. La version LLVM & Spe O3 est plus rapide que les versions précédentes. On obtient un gain de 21 % par rapport à la version statique obtenue par une réduction des instructions de chargement pendant les itérations, soit 84 % de la performance maximale théorique. Dans la version statique et LLVM, le compi- lateur ne peut pas légalement déplacer les instructions de chargement en dehors de la boucle car celles-ci peuvent changer (à cause de l’aliasing). Dans cette version, les coefficients sont déplacés hors de la boucle et mis dans un “pool” de constantes. Le “pool” de constantes est une zone mémoire située juste après la définition de la fonction regroupant les constantes ne pouvant tenir sur un immédiat. L’avantage est que les constantes sont regroupées de manière com- pacte et que le compilateur a la garantie de la constance des données et peut donc optimiser en conséquence. À l’exécution, le chargement est donc fait une fois et hors de la boucle, évitant ainsi les effets d’un mauvais ordonnancement des instructions qui faisaient caler l’application.
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A Psycho-Thematic Reading of Solomon Northup’s Twelve Years a Slave (1853)

A Psycho-Thematic Reading of Solomon Northup’s Twelve Years a Slave (1853)

Abstract The history of the United States is rampant with racism. Slavery is one form of it. Accordingly, black people were offensively, cruelly and brutally treated throughout their history in the US. Along their inferior journey, numerous African-American writers defied slavery and racism through a set of literary works, poetry and other artistic and creative tools. Thus, they responded with one of the prominent literary genre called “slave narratives”. Among their creative works is “Twelve Years a Slave” (1853) by Solomon Northup. This research work delves into Solomon Northup’s Psyche and analyses both the historical and literary facts in his story. It reflects upon all the psychological and physical aspects he experienced; from trauma, dehumanization to violence and oppression, which created a sense of otherness and revenge. Despite the disheartening and agony, his determination and willingness directed his fate to freedom.
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Almost Optimal Scaling of Reed-Muller Codes on BEC and BSC Channels

Almost Optimal Scaling of Reed-Muller Codes on BEC and BSC Channels

I. I NTRODUCTION Consider a binary linear code of length N and minimum distance d min . Assume that we transmit over the binary erasure channel (BEC) with parameter ǫ, 0 < ǫ < 1, or the binary symmetric channel (BSC) with parameter ǫ, 0 < ǫ < 1 2 . Assume further that the receiver performs block maximum-a posteriori (block-MAP) decoding. It was shown by Tillich and Zemor [1] that in this case the error probability transitions “quickly” from δ to 1 − δ for any δ > 0 if the minimum distance is large. In particular they showed that the width of the transition is of order O(1/ √ d min ). For codes whose
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Purification performances of common reed beds based on the residence time: Case of Benin

Purification performances of common reed beds based on the residence time: Case of Benin

5685 Picture 1 : The drainpipes Picture 2 : The gravel Picture 3 : The reed bed Diagram 1 : The three layers of gravel Many types of reed grow in the town of Calavi. In this study phragmites were chosen. Young phragmite plants, which naturally grow on the campus, were dug up and transplanted to an experimentation field. The plants were often watered, especially during the dry seasons. After an year, twelve young plants from the experimentation field were dug up and transplanted into the filtration bed. The reed bed was fed for two months with slightly concentrated wastewater that comes from one of the septic tanks of university residence halls. This period is necessary because it allows plants to accustom themselves to the filtration bed and allows the biofilm to develop. The experiment was carried out based on a batch system. According to Molle et al., 2005, the volume
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Vérification de code-octet avec sous-routines par code-certifié

Vérification de code-octet avec sous-routines par code-certifié

Notre technique est inspirée du travail présenté par Rose [RosR98] [Rose02], mais va plus loin en proposant un format de certificat différent qui permet à notre al[r]

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Le code épigénétique des histones

Le code épigénétique des histones

vent dans certains cas s’établir sur de très grandes dis- tances, permettant à la cellule de co-régler plusieurs gènes à l’intérieur de locus chromosomiques. Ce méca- nisme peut également permettre la transmission de l’in- formation aux cours des divisions cellulaires [3] . La découverte initiale qui avait permis de mettre en évi- dence les liens associant les histones acétylées aux régions transcriptionnellement actives du génome est confirmée. Cependant, de nombreuses précautions doi- vent être prises lorsqu’on regarde le code des histones dans son ensemble. Chaque type de modification ne concerne plus exclusivement l’activation ou la répres- sion de la transcription. Le contexte global des diffé- rentes modifications reste encore difficile à saisir car, même si une grande partie des modifications est désor- mais connue, il faut encore caractériser les enzymes res- ponsables de ces modifications et étudier toutes leurs interactions et interdépendances. ◊
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Mesopelagic N2 Fixation Related to Organic Matter Composition in the Solomon and Bismarck Seas (Southwest Pacific)

Mesopelagic N2 Fixation Related to Organic Matter Composition in the Solomon and Bismarck Seas (Southwest Pacific)

Of the observed heterotrophic diazotrophic phylotypes, it is unclear which ones contributed to the N 2 fixation rates measured or in what proportion. Finding cyanobacterial sequences related to Trichodesmium at depths between 200 and 1000 m ( Fig 7 ) was surprising, although similar observations have been made in the Sargasso Sea [ 22 ]. In the surface waters of the same transect, we observed Trichodesmium abundances at up to 10 5 nifH copies L -1 (Berthelot et al., unpublished), which is in agreement with the maximum abundances observed in hotspots of Trichodesmium such as the Northwest Atlantic [ 11 ], and hence corroborates the importance of Trichodesmium in the sunlit layer of these waters. Indeed, several surface accumulations of Tri- chodesmium in the form of ‘slicks’ were visually observed at the surface in the Solomon Sea during this cruise. The presence of Trichodesmium-like nifH genes at mesopelagic depths sug- gests that the collapse of blooms can form aggregates that sink out of the euphotic zone. We discard the possibility that these Trichodesmium-like phylotypes were actively fixing N 2 at these depths that were below the euphotic zone.
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