179 research outputs found
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
Generic interpolation polynomial for list decoding
AbstractWe extend results of K. Lee and M.E. OʼSullivan by showing how to use Gröbner bases to find the interpolation polynomial for list decoding a one-point AG code C=CL(rP,D) on any curve X, where P is an Fq-rational point on X and D=P1+P2+⋯+Pn is the sum of other Fq-rational points on X. We then define the generic interpolation polynomial for list decoding such a code. The generic interpolation polynomial should specialize to the interpolation polynomial for most received strings. We give an example of a family of Reed–Solomon 1-error correcting codes for which a single error can be decoded by a very simple process involving substituting into the generic interpolation polynomial
Skew and linearized Reed-Solomon codes and maximum sum rank distance codes over any division ring
Reed-Solomon codes and Gabidulin codes have maximum Hamming distance and
maximum rank distance, respectively. A general construction using skew
polynomials, called skew Reed-Solomon codes, has already been introduced in the
literature. In this work, we introduce a linearized version of such codes,
called linearized Reed-Solomon codes. We prove that they have maximum sum-rank
distance. Such distance is of interest in multishot network coding or in
singleshot multi-network coding. To prove our result, we introduce new metrics
defined by skew polynomials, which we call skew metrics, we prove that skew
Reed-Solomon codes have maximum skew distance, and then we translate this
scenario to linearized Reed-Solomon codes and the sum-rank metric. The theories
of Reed-Solomon codes and Gabidulin codes are particular cases of our theory,
and the sum-rank metric extends both the Hamming and rank metrics. We develop
our theory over any division ring (commutative or non-commutative field). We
also consider non-zero derivations, which give new maximum rank distance codes
over infinite fields not considered before
Row Reduction Applied to Decoding of Rank Metric and Subspace Codes
We show that decoding of -Interleaved Gabidulin codes, as well as
list- decoding of Mahdavifar--Vardy codes can be performed by row
reducing skew polynomial matrices. Inspired by row reduction of \F[x]
matrices, we develop a general and flexible approach of transforming matrices
over skew polynomial rings into a certain reduced form. We apply this to solve
generalised shift register problems over skew polynomial rings which occur in
decoding -Interleaved Gabidulin codes. We obtain an algorithm with
complexity where measures the size of the input problem
and is proportional to the code length in the case of decoding. Further, we
show how to perform the interpolation step of list--decoding
Mahdavifar--Vardy codes in complexity , where is the number of
interpolation constraints.Comment: Accepted for Designs, Codes and Cryptograph
Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants
We construct s-interleaved linearized Reed-Solomon (ILRS) codes and variants
and propose efficient decoding schemes that can correct errors beyond the
unique decoding radius in the sum-rank, sum-subspace and skew metric. The
proposed interpolation-based scheme for ILRS codes can be used as a list
decoder or as a probabilistic unique decoder that corrects errors of sum-rank
up to , where s is the interleaving order, n the
length and k the dimension of the code. Upper bounds on the list size and the
decoding failure probability are given where the latter is based on a novel
Loidreau-Overbeck-like decoder for ILRS codes. The results are extended to
decoding of lifted interleaved linearized Reed-Solomon (LILRS) codes in the
sum-subspace metric and interleaved skew Reed-Solomon (ISRS) codes in the skew
metric. We generalize fast minimal approximant basis interpolation techniques
to obtain efficient decoding schemes for ILRS codes (and variants) with
subquadratic complexity in the code length. Up to our knowledge, the presented
decoding schemes are the first being able to correct errors beyond the unique
decoding region in the sum-rank, sum-subspace and skew metric. The results for
the proposed decoding schemes are validated via Monte Carlo simulations.Comment: submitted to IEEE Transactions on Information Theory, 57 pages, 10
figure
Interpolation-Based Decoding of Folded Variants of Linearized and Skew Reed-Solomon Codes
The sum-rank metric is a hybrid between the Hamming metric and the rank
metric and suitable for error correction in multishot network coding and
distributed storage as well as for the design of quantum-resistant
cryptosystems. In this work, we consider the construction and decoding of
folded linearized Reed-Solomon (FLRS) codes, which are shown to be maximum
sum-rank distance (MSRD) for appropriate parameter choices. We derive an
efficient interpolation-based decoding algorithm for FLRS codes that can be
used as a list decoder or as a probabilistic unique decoder. The proposed
decoding scheme can correct sum-rank errors beyond the unique decoding radius
with a computational complexity that is quadratic in the length of the unfolded
code. We show how the error-correction capability can be optimized for
high-rate codes by an alternative choice of interpolation points. We derive a
heuristic upper bound on the decoding failure probability of the probabilistic
unique decoder and verify its tightness by Monte Carlo simulations. Further, we
study the construction and decoding of folded skew Reed-Solomon codes in the
skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with
different block sizes that come along with an efficient decoding algorithm.Comment: 32 pages, 3 figures, accepted at Designs, Codes and Cryptograph
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 the Complexity of the Generalized MinRank Problem
We study the complexity of solving the \emph{generalized MinRank problem},
i.e. computing the set of points where the evaluation of a polynomial matrix
has rank at most . A natural algebraic representation of this problem gives
rise to a \emph{determinantal ideal}: the ideal generated by all minors of size
of the matrix. We give new complexity bounds for solving this problem
using Gr\"obner bases algorithms under genericity assumptions on the input
matrix. In particular, these complexity bounds allow us to identify families of
generalized MinRank problems for which the arithmetic complexity of the solving
process is polynomial in the number of solutions. We also provide an algorithm
to compute a rational parametrization of the variety of a 0-dimensional and
radical system of bi-degree . We show that its complexity can be bounded
by using the complexity bounds for the generalized MinRank problem.Comment: 29 page
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