9 research outputs found
Row reduced representations of behaviors over finite rings
Row reduced representations of behaviors over fields posses a number of useful properties. Perhaps the most important feature is the predictable degree property. This property allows a finite parametrization of the module generated by the rows of the row reduced matrix with prior computable bounds. In this paper we study row-reducedness of representations of behaviors over rings of the form , where is a prime number. Using a restricted calculus within we derive a meaningful and computable notion of row-reducedness
An iterative algorithm for parametrization of shortest length shift registers over finite rings
The construction of shortest feedback shift registers for a finite sequence
S_1,...,S_N is considered over the finite ring Z_{p^r}. A novel algorithm is
presented that yields a parametrization of all shortest feedback shift
registers for the sequence of numbers S_1,...,S_N, thus solving an open problem
in the literature. The algorithm iteratively processes each number, starting
with S_1, and constructs at each step a particular type of minimal Gr\"obner
basis. The construction involves a simple update rule at each step which leads
to computational efficiency. It is shown that the algorithm simultaneously
computes a similar parametrization for the reciprocal sequence S_N,...,S_1.Comment: Submitte
On the key equation over a commutative ring
We define alternant codes over a commutative ring R and a corresponding key equation.
We show that when the ring is a domain, e.g. the p-adic integers, the error–locator polynomial
is the unique monic minimal polynomial (shortest linear recurrence) of the syndrome sequence
and that it can be obtained by Algorithm MR of Norton.
When R is a local ring, we show that the syndrome sequence may have more than one (monic) minimal polynomial, but all the minimal polynomials coincide modulo the maximal ideal of R. We characterise the minimal polynomials when R is a Hensel ring. We also apply these results to decoding alternant codes over a local ring R: it is enough to find any monic minimal polynomial over R and to find its roots in the residue field. This gives a decoding algorithm for alternant codes over a finite chain ring, which generalizes and improves a method of Interlando et. al. for BCH and Reed–Solomon codes over a Galois ring
On the Hamming distance of linear codes over a finite chain ring
Let R be a finite chain ring (e.g. a Galois ring), K its residue field and C a linear code
over R. We prove that d(C), the Hamming distance of C, is d((C : α)), where (C : α) is a
submodule quotient, α is a certain element of R and — denotes the canonical projection
to K. These two codes also have the same set of minimal codeword supports. We explicitly
construct a generator matrix/polynomial of (C : α) from the generator matrix/polynomials
of C. We show that in general d(C) ≤ d(C) with equality for free codes (i.e. for free R-
submodules of Rn) and in particular for Hensel lifts of cyclic codes over K. Most of the codes
over rings described in the literature fall into this class.
We characterise MDS codes over R and prove several analogues of properties of MDS codes
over finite fields. We compute the Hamming weight enumerator of a free MDS code over R
MDS 2D convolutional codes with optimal 1D horizontal projections
Two dimensional (2D) convolutional codes is a class of codes that generalizes standard one-dimensional (1D) convolutional codes in order to treat two dimensional data. In this paper we present a novel and concrete construction of 2D convolutional codes with the particular property that their projection onto the horizontal lines yield optimal [in the sense of Almeida et al. (Linear Algebra Appl 499:1–25, 2016)] 1D convolutional codes with a certain rate and certain Forney indices. Moreover, using this property we show that the proposed constructions are indeed maximum distance separable, i.e., are 2D convolutional codes having the maximum possible distance among all 2D convolutional codes with the same parameters. The key idea is to use a particular type of superregular matrices to build the generator matrix
Propriedades das distâncias dos códigos convolucionais sobre Z pr
Doutoramento em Matemática e AplicaçõesNesta tese consideramos códigos convolucionais sobre o anel polinomial
[ ] r p
′ D , onde p é primo e r é um inteiro positivo. Em particular, focamo-nos
no conjunto das palavras de código com suporte finito e estudamos as suas
propriedades no que respeita às distâncias. Investigamos as duas
propriedades mais importantes dos códigos convolucionais, nomeadamente, a
distância livre e a distância de coluna.
Começamos por analisar e solucionar o problema de, dado um conjunto de
parâmetros, determinar a distância livre máxima possível que um código
convolucional sobre [ ] r p
′ D pode atingir. Com efeito, obtemos um novo limite
superior para esta distância generalizando os limites obtidos no contexto dos
códigos convolucionais sobre corpos finitos. Além disso, mostramos que esse
limite é ótimo, no sentido em que não pode ser melhorado. Para tal,
apresentamos construções de códigos convolucionais (não necessariamente
livres) que permitem atingir esse limite, para um certo conjunto de parâmetros.
De acordo com a literatura chamamos a esses códigos MDS.
Definimos também distâncias de coluna de um código convolucional. Obtemos
limites superiores para as distâncias de coluna e chamamos MDP aos códigos
cujas distâncias de coluna atingem estes limites superiores. Além disso,
mostramos a existência de códigos MDP. Note-se, porém, que os códigos
MDP apresentados não são completamente gerais pois os seus parâmetros
devem satisfazer determinadas condições.
Finalmente, estudamos o código dual de um código convolucional definido em
(( )) r p
′ D . Os códigos duais de códigos convolucionais sobre corpos finitos
foram exaustivamente investigados, como é refletido na literatura sobre o
tema. Estes códigos são relevantes pois fornecem informação sobre a
distribuição dos pesos do código e é neste sentido a inclusão deste assunto no
âmbito desta tese. Outra razão importante para o estudo de códigos duais é a
sua utilidade para o desenvolvimento de algoritmos de descodificação quando
consideramos um erasure channel. Nesta tese são analisadas algumas
propriedades fundamentais dos duais. Em particular, mostramos que códigos
convolucionais definidos em (( )) r p
′ D admitem uma matriz de paridade. Para
além disso, apresentamos um método construtivo para determinar um
codificador de um código dual.
keywords
Convolutional codes, finite rings, free distance, column distance, MDS, MDP,
dual code
abstract
In this thesis we consider convolutional codes over the polynomial ring [ ] r p
′ D ,
where p is a prime and r is a positive integer. In particular, we focus in the
set of finite support codewords and study their distances properties. We
investigate the two most important distance properties of convolutional codes,
namely, the free distance and the column distance.
First we address and fully solve the problem of determining the maximum
possible free distance a convolutional code over [ ] r p
′ D can achieve, for a
given set of parameters. Indeed, we derive a new upper bound on this distance
generalizing the Singleton-type bounds derived in the context of convolutional
codes over finite fields. Moreover, we show that such a bound is optimal in the
sense that it cannot be improved. To do so we provide concrete constructions
of convolutional codes (not necessarily free) that achieve this bound for any
given set of parameters. In accordance with the literature we called such codes
Maximum Distance Separable (MDS).
We define the notion of column distance of a convolutional code. We obtain
upper-bounds on the column distances and call Maximum Distance Profile
(MDP) the codes that attain the maximum possible column distances.
Furthermore, we show the existence of MDP codes. We note however that the
MDP codes presented here are not completely general as their parameters
need to satisfy certain conditions.
Finally, we study the dual code of a convolutional code defined in (( )) r p
′ D .
Dual codes of convolutional codes over finite fields have been thoroughly
investigated as it is reflected in the large body of literature on this topic. They
are relevant as they provide value information on the weight distribution of the
code and therefore fit in the scope of this thesis. Another important reason for
the study of dual codes is that they can be very useful for the development of
decoding algorithms of convolutional codes over the erasure channel. In this
thesis some fundamental properties have been analyzed. In particular, we
show that convolutional codes defined in (( )) r p
′ D admit a parity-check matrix.
Moreover, weIn this thesis we consider convolutional codes over the polynomial ring [ ] r p
′ D ,
where p is a prime and r is a positive integer. In particular, we focus in the
set of finite support codewords and study their distances properties. We
investigate the two most important distance properties of convolutional codes,
namely, the free distance and the column distance.
First we address and fully solve the problem of determining the maximum
possible free distance a convolutional code over [ ] r p
′ D can achieve, for a
given set of parameters. Indeed, we derive a new upper bound on this distance
generalizing the Singleton-type bounds derived in the context of convolutional
codes over finite fields. Moreover, we show that such a bound is optimal in the
sense that it cannot be improved. To do so we provide concrete constructions
of convolutional codes (not necessarily free) that achieve this bound for any
given set of parameters. In accordance with the literature we called such codes
Maximum Distance Separable (MDS).
We define the notion of column distance of a convolutional code. We obtain
upper-bounds on the column distances and call Maximum Distance Profile
(MDP) the codes that attain the maximum possible column distances.
Furthermore, we show the existence of MDP codes. We note however that the
MDP codes presented here are not completely general as their parameters
need to satisfy certain conditions.
Finally, we study the dual code of a convolutional code defined in (( )) r p
′ D .
Dual codes of convolutional codes over finite fields have been thoroughly
investigated as it is reflected in the large body of literature on this topic. They
are relevant as they provide value information on the weight distribution of the
code and therefore fit in the scope of this thesis. Another important reason for
the study of dual codes is that they can be very useful for the development of
decoding algorithms of convolutional codes over the erasure channel. In this
thesis some fundamental properties have been analyzed. In particular, we
show that convolutional codes defined in (( )) r p
′ D admit a parity-check matrix.
Moreover, we provide a constructive method to explicitly compute an encoder
of the dual code
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