Sur l'algorithme de décodage en liste de Guruswami-Sudan sur les anneaux finis

This thesis studies the algorithmic techniques of list decoding, first proposed by Guruswami and Sudan in 1998, in the context of Reed-Solomon codes over finite rings. Two approaches are considered. First we adapt the Guruswami-Sudan (GS) list decoding algorithm to generalized Reed-Solomon (GRS) codes over finite rings with identity. We study in details the complexities of the algorithms for GRS codes over Galois rings and truncated power series rings. Then we explore more deeply a lifting technique for list decoding. We show that the latter technique is able to correct more error patterns than the original GS list decoding algorithm. We apply the technique to GRS code over Galois rings and truncated power series rings and show that the algorithms coming from this technique have a lower complexity than the original GS algorithm. We show that it can be easily adapted for interleaved Reed-Solomon codes. Finally we present the complete implementation in C and C++ of the list decoding algorithms studied in this thesis. All the needed subroutines, such as univariate polynomial root finding algorithms, finite fields and rings arithmetic, are also presented. Independently, this manuscript contains other work produced during the thesis. We study quasi cyclic codes in details and show that they are in one-to-one correspondence with left principal ideal of a certain matrix ring. Then we adapt the GS framework for ideal based codes to number fields codes and provide a list decoding algorithm for the latter.Cette thèse porte sur l'algorithmique des techniques de décodage en liste, initiée par Guruswami et Sudan en 1998, dans le contexte des codes de Reed-Solomon sur les anneaux finis. Deux approches sont considérées. Dans un premier temps, nous adaptons l'algorithme de décodage en liste de Guruswami-Sudan aux codes de Reed-Solomon généralisés sur les anneaux finis. Nous étudions en détails les complexités de l'algorithme pour les anneaux de Galois et les anneaux de séries tronquées. Dans un deuxième temps nous approfondissons l'étude d'une technique de remontée pour le décodage en liste. Nous montrons que cette derni're permet de corriger davantage de motifs d'erreurs que la technique de Guruswami-Sudan originale. Nous appliquons ensuite cette même technique aux codes de Reed-Solomon généralisés sur les anneaux de Galois et les anneaux de séries tronquées et obtenons de meilleures bornes de complexités. Enfin nous présentons l'implantation des algorithmes en C et C++ des algorithmes de décodage en liste étudiés au cours de cette thèse. Tous les sous-algorithmes nécessaires au décodage en liste, comme la recherche de racines pour les polynômes univariés, l'arithmétique des corps et anneaux finis sont aussi présentés. Indépendamment, ce manuscrit contient d'autres travaux sur les codes quasi-cycliques. Nous prouvons qu'ils sont en correspondance biunivoque avec les idéaux à gauche d'un certain anneaux de matrices. Enfin nous adaptons le cadre proposé par Guruswami et Sudan pour les codes à base d'ideaux aux codes construits à l'aide des corps de nombres. Nous fournissons un algorithme de décodage en liste dans ce contexte

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric

We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in the sum-rank metric. The speed-ups are achieved by reducing the core of the underlying computational problems of the decoders to one common tool: computing left and right approximant bases of matrices over skew polynomial rings. To accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm for matrices over usual polynomials. This captures the bulk of the work in multiplication of skew polynomials, and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity. The new faster algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational problems related to the remainder-evaluation of skew polynomials

PERFORMANCE COMPARISON OF NON-INTERLEAVED BCH CODES AND INTERLEAVED BCH CODES

This project covers the research about the BCH error correcting codes and the performance of interleaved and non-interleaved BCH codes. Both long and short BCH codes for multimedia communication are examined in an A WGN channel. Algorithm for simulating the BCH codes was also being investigated, which includes generating the parity check matrix, generating the message code in Galois array matrix, encoding the message blocks, modulation and decoding the message blocks. Algorithm for interleaving that includes interleaving message, including burst errors and deinterleaving message is combined with the BCH codes algorithm for simulating the interleaved BCH codes. The performance and feasibility of the coding structure are tested. The performance comparison between interleaved and noninterleaved BCH codes is studied in terms of error performance, channel performance and effect of data rates on the bit error rate (BER). The Berlekamp-Massey Algorithm decoding scheme was implemented. Random integers are generated and encoded with BCH encoder. Burst errors are added before the message is interleaved, then enter modulation and channel simulation. Interleaved message is then compared with noninterleaved message and the error statistics are compared. Initially, certain amount of burst errors is used. "ft is found that the graph does not agree with the theoretical bit error rate (BER) versus signal-to-noise ratio (SNR). When compared between each BCH codeword (i.e. n = 31, n = 63 and n = 127), n = 31 shows the highest BER while n = 127 shows the lowest BER. This happened because of the occurrence of error bursts and also due to error frequency. A reduced size or errors from previous is used in the algorithm. A graph similar to the theoretical BER vs SNR is obtained for both interleaved and non-interleaved BCH codes. It is found that BER of non-interleaved is higher than interleaved BCH codes as SNR increases. These observations show that size of errors influence the effect of interleaving. Simulation time is also studied in terms of block length. It is found that interleaved BCH codes consume longer simulation time compared to non-interleaved BCH codes due to additional algorithm for the interleaved BCH codes

PERFORMANCE COMPARISON OF NON-INTERLEAVED BCH CODES AND INTERLEAVED BCH CODES

This project covers the research about the BCH error correcting codes and the performance of interleaved and non-interleaved BCH codes. Both long and short BCH codes for multimedia communication are examined in an A WGN channel. Algorithm for simulating the BCH codes was also being investigated, which includes generating the parity check matrix, generating the message code in Galois array matrix, encoding the message blocks, modulation and decoding the message blocks. Algorithm for interleaving that includes interleaving message, including burst errors and deinterleaving message is combined with the BCH codes algorithm for simulating the interleaved BCH codes. The performance and feasibility of the coding structure are tested. The performance comparison between interleaved and noninterleaved BCH codes is studied in terms of error performance, channel performance and effect of data rates on the bit error rate (BER). The Berlekamp-Massey Algorithm decoding scheme was implemented. Random integers are generated and encoded with BCH encoder. Burst errors are added before the message is interleaved, then enter modulation and channel simulation. Interleaved message is then compared with noninterleaved message and the error statistics are compared. Initially, certain amount of burst errors is used. "ft is found that the graph does not agree with the theoretical bit error rate (BER) versus signal-to-noise ratio (SNR). When compared between each BCH codeword (i.e. n = 31, n = 63 and n = 127), n = 31 shows the highest BER while n = 127 shows the lowest BER. This happened because of the occurrence of error bursts and also due to error frequency. A reduced size or errors from previous is used in the algorithm. A graph similar to the theoretical BER vs SNR is obtained for both interleaved and non-interleaved BCH codes. It is found that BER of non-interleaved is higher than interleaved BCH codes as SNR increases. These observations show that size of errors influence the effect of interleaving. Simulation time is also studied in terms of block length. It is found that interleaved BCH codes consume longer simulation time compared to non-interleaved BCH codes due to additional algorithm for the interleaved BCH codes

Multi-non-binary turbo codes

International audienceThis paper presents a new family of turbo codes called multi-non-binary turbo codes (MNBTCs) that generalizes the concept of turbo codes to multi-non-binary (MNB) parallel concatenated convolutional codes (PCCC). An MNBTC incorporates, as component encoders, recursive and systematic multi-non-binary convolutional encoders. The more compact data structure for these encoders confers some advantages on MNBTCs over other types of turbo codes, such as better asymptotic behavior, better convergence, and reduced latency. This paper presents in detail the structure and operation of an MNBTC: MNB encoding, trellis termination, Max-Log-MAP decoding adapted to the MNB case. It also shows an example of MNBTC whose performance is compared with the state-of-the-art turbo code adopted in the DVB-RCS2 standard

Weight-2 input sequences of 1/n convolutional codes from linear systems point of view

Convolutional codes form an important class of codes that have memory. One natural way to study these codes is by means of input state output representations. In this paper we study the minimum (Hamming) weight among codewords produced by input sequences of weight two. In this paper, we consider rate 1/n and use the linear system setting called (A,B,C,D) input-state-space representations of convolutional codes for our analysis. Previous results on this area were recently derived assuming that the matrix A, in the input-state-output representation, is nonsingular. This work completes this thread of research by treating the nontrivial case in which A is singular. Codewords generated by weight-2 inputs are relevant to determine the effective free distance of Turbo codes.The research of the second author was supported by Spanish I+D+i project PID2019-108668GB-I00 of MCIN/AEI/10.13039/501100011033

A survey of digital television broadcast transmission techniques

This paper is a survey of the transmission techniques used in digital television (TV) standards worldwide. With the increase in the demand for High-Definition (HD) TV, video-on-demand and mobile TV services, there was a real need for more bandwidth-efficient, flawless and crisp video quality, which motivated the migration from analogue to digital broadcasting. In this paper we present a brief history of the development of TV and then we survey the transmission technology used in different digital terrestrial, satellite, cable and mobile TV standards in different parts of the world. First, we present the Digital Video Broadcasting standards developed in Europe for terrestrial (DVB-T/T2), for satellite (DVB-S/S2), for cable (DVB-C) and for hand-held transmission (DVB-H). We then describe the Advanced Television System Committee standards developed in the USA both for terrestrial (ATSC) and for hand-held transmission (ATSC-M/H). We continue by describing the Integrated Services Digital Broadcasting standards developed in Japan for Terrestrial (ISDB-T) and Satellite (ISDB-S) transmission and then present the International System for Digital Television (ISDTV), which was developed in Brazil by adopteding the ISDB-T physical layer architecture. Following the ISDTV, we describe the Digital Terrestrial television Multimedia Broadcast (DTMB) standard developed in China. Finally, as a design example, we highlight the physical layer implementation of the DVB-T2 standar

Decoding and constructions of codes in rank and Hamming metric

As coding theory plays an important role in data transmission, decoding algorithms for new families of error correction codes are of great interest. This dissertation is dedicated to the decoding algorithms for new families of maximum rank distance (MRD) codes including additive generalized twisted Gabidulin (AGTG) codes and Trombetti-Zhou (TZ) codes, decoding algorithm for Gabidulin codes beyond half the minimum distance and also encoding and decoding algorithms for some new optimal rank metric codes with restrictions. We propose an interpolation-based decoding algorithm to decode AGTG codes where the decoding problem is reduced to the problem of solving a projective polynomial equation of the form q(x) = xqu+1 +bx+a = 0 for a,b ∈ Fqm. We investigate the zeros of q(x) when gcd(u,m)=1 and proposed a deterministic algorithm to solve a linearized polynomial equation which has a close connection to the zeros of q(x). An efficient polynomial-time decoding algorithm is proposed for TZ codes. The interpolation-based decoding approach transforms the decoding problem of TZ codes to the problem of solving a quadratic polynomial equation. Two new communication models are defined and using our models we manage to decode Gabidulin codes beyond half the minimum distance by one unit. Our models also allow us to improve the complexity for decoding GTG and AGTG codes. Besides working on MRD codes, we also work on restricted optimal rank metric codes including symmetric, alternating and Hermitian rank metric codes. Both encoding and decoding algorithms for these optimal families are proposed. In all the decoding algorithms presented in this thesis, the properties of Dickson matrix and the BM algorithm play crucial roles. We also touch two problems in Hamming metric. For the first problem, some cryptographic properties of Welch permutation polynomial are investigated and we use these properties to determine the weight distribution of a binary linear codes with few weights. For the second one, we introduce two new subfamilies for maximum weight spectrum codes with respect to their weight distribution and then we investigate their properties.Doktorgradsavhandlin

Fast and Efficient Hardware Implementation of HQC

This work presents a hardware design for constant-time implementation of the HQC (Hamming Quasi-Cyclic) code-based key encapsulation mechanism. HQC has been selected for the fourth round of NIST\u27s Post-Quantum Cryptography standardization process and this work presents the first, hand-optimized design of HQC key generation, encapsulation, and decapsulation written in Verilog targeting implementation on FPGAs. The three modules further share a common SHAKE256 hash module to reduce area overhead. All the hardware modules are parametrizable at compile time so that designs for the different security levels can be easily generated. The design currently outperforms the other hardware designs for HQC, and many of the fourth-round Post-Quantum Cryptography standardization process, with one of the best time-area products as well. For the combined HighSpeed design targeting the lowest security level, we show that the HQC design can perform key generation in 0.09ms, encapsulation in 0.13ms, and decapsulation in 0.21ms when synthesized for an Xilinx Artix 7 FPGA. Our work shows that when hardware performance is compared, HQC can be a competitive alternative candidate from the fourth round of the NIST PQC competition