An algorithm for list decoding number field codes

We present an algorithm for list decoding codewords of algebraic number field codes in polynomial time. This is the first explicit procedure for decoding number field codes whose construction were previously described by Lenstra [12] and Guruswami [8]. We rely on a new algorithm for computing the Hermite normal form of the basis of an OK -module due to Biasse and Fieker [2] where OK is the ring of integers of a number field K

Polynomial root finding over local rings and application to error correcting codes

International audienceThis article is devoted to algorithms for computing all the roots of a univariate polynomial with coefficients in a complete commutative Noetherian unramified regular local domain, which are given to a fixed common finite precision. We study the cost of our algorithms, discuss their practical performances, and apply our results to the Guruswami and Sudan list decoding algorithm over Galois rings

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

On Quasi-Cyclic Codes as a Generalization of Cyclic Codes

In this article we see quasi-cyclic codes as block cyclic codes. We generalize some properties of cyclic codes to quasi-cyclic ones such as generator polynomials and ideals. Indeed we show a one-to-one correspondence between l-quasi-cyclic codes of length m and ideals of M_l(Fq)[X]/(X^m-1). This permits to construct new classes of codes, namely quasi-BCH and quasi-evaluation codes. We study the parameters of such codes and propose a decoding algorithm up to half the designed minimum distance. We even found one new quasi-cyclic code with better parameters than known [189, 11, 125]_F4 and 48 derivated codes beating the known bounds as well.Comment: (18/08/2011

Modular SIMD arithmetic in Mathemagix

Modular integer arithmetic occurs in many algorithms for computer algebra, cryptography, and error correcting codes. Although recent microprocessors typically offer a wide range of highly optimized arithmetic functions, modular integer operations still require dedicated implementations. In this article, we survey existing algorithms for modular integer arithmetic, and present detailed vectorized counterparts. We also present several applications, such as fast modular Fourier transforms and multiplication of integer polynomials and matrices. The vectorized algorithms have been implemented in C++ inside the free computer algebra and analysis system Mathemagix. The performance of our implementation is illustrated by various benchmarks

A Lifting Decoding Scheme and its Application to Interleaved Linear Codes

Abstract—In this paper we design a decoding algorithm based on a lifting decoding scheme. This leads to a unique decoding algorithm for Reed-Solomon codes over Galois rings with a very low complexity, and a list decoding algorithm. We show that, using erasures in our algorithms, allows to decode more errors than half the minimum distance with a high probability. Finally we apply these techniques to interleaved linear codes over a finite field and obtain a decoding algorithm that can recover more errors than half the minimum distance. Index Terms—Algorithm design and analysis, Decoding, Error correction, Reed-Solomon codes, Interleaved codes

ISSAC software The decoding Library for List Decoding

Reed-Solomon (RS) codes form an important and well-studied family of codes. They were first proposed in 1960 by Reed and Solomon in their original paper [RS60]. They are widely used in practice [WB99]. RS codes can be efficiently unique decoded [Gao02] and [Jus76]. Sudan’s 1997 breakthrough on list decoding of RS codes [Sud97], further improved by Guruswami and Sudan in [GS98], showed that RS codes ar

On the Algorithms of Guruswami-Sudan List Decoding over Finite Rings

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 contexteThis 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 latterPALAISEAU-Polytechnique (914772301) / SudocSudocFranceF

The Complexity of Fungal β-Glucan in Health and Disease: Effects on the Mononuclear Phagocyte System

β-glucan, the most abundant fungal cell wall polysaccharide, has gained much attention from the scientific community in the last few decades for its fascinating but not yet fully understood immunobiology. Study of this molecule has been motivated by its importance as a pathogen-associated molecular pattern upon fungal infection as well as by its promising clinical utility as biological response modifier for the treatment of cancer and infectious diseases. Its immune effect is attributed to the ability to bind to different receptors expressed on the cell surface of phagocytic and cytotoxic innate immune cells, including monocytes, macrophages, neutrophils, and natural killer cells. The characteristics of the immune responses generated depend on the cell types and receptors involved. Size and biochemical composition of β-glucans isolated from different sources affect their immunomodulatory properties. The variety of studies using crude extracts of fungal cell wall rather than purified β-glucans renders data difficult to interpret. A better understanding of the mechanisms of purified fungal β-glucan recognition, downstream signaling pathways, and subsequent immune regulation activated, is, therefore, essential not only to develop new antifungal therapy but also to evaluate β-glucan as a putative anti-infective and antitumor mediator. Here, we briefly review the complexity of interactions between fungal β-glucans and mononuclear phagocytes during fungal infections. Furthermore, we discuss and present available studies suggesting how different fungal β-glucans exhibit antitumor and antimicrobial activities by modulating the biologic responses of mononuclear phagocytes, which make them potential candidates as therapeutic agents

On Generalized Reed-Solomon Codes Over Commutative and Noncommutative Rings

International audienceIn this paper we study generalized Reed-Solomon codes (GRS codes) over commutative, noncommutative rings, show that the classical Welch-Berlekamp and Guruswami-Sudan decoding algorithms still hold in this context and we investigate their complexities. Under some hypothesis, the study of noncommutative generalized Reed-Solomon codes over finite rings leads to the fact that GRS code over commutative rings have better parameters than their noncommutative counterparts. Also GRS codes over finite fields have better parameters than their commutative rings counterparts. But we also show that given a unique decoding algorithm for a GRS code over a finite field, there exists a unique decoding algorithm for a GRS code over a truncated power series ring with a better asymptotic complexity. Moreover we generalize a lifting decoding scheme to obtain new unique and list decoding algorithms designed to work when the base ring is for example a Galois ring or a truncated power series ring or the ring of square matrices over the latter ring.Dans cet article on étudie les codes de Reed-Solomon généralisés (codes GRS) sur des anneaux commutatifs, non commutatifs, on montre que les algorithmes classiques de décodage Welch-Berlekamp et Guruswami-Sudan restent valides dans ce contexte et on donne leurs complexités. Sous certaines hypothèses, l'étude des codes de Reed-Solomon généralisés sur les anneaux finis montre que les codes GRS sur les anneaux commutatifs ont des meilleurs paramètres que leurs équivalents non commutatifs. De même, les codes GRS sur les corps finis ont de meilleurs paramètres que leurs équivalents sur les anneaux commutatifs. Mais on montre aussi qu'étant donné un algorithme de décodage unique pour un code GRS sur un corps fini, il existe un algorithme de décodage unique pour un code GRS sur l'anneaux des séries tronquées avec une meilleure complexité asymptotique. De plus on généralise un schéma de remontée pour obtenir de nouveau algorithmes de décodage unique et en liste pour les codes GRS sur certains anneaux tels que les anneaux de Galois ou les anneaux de séries tronquées ou les anneaux de matrices sur ces derniers