The dual of convolutional codes over Zpr\mathbb{Z}_{p^r}

An important class of codes widely used in applications is the class of convolutional codes. Most of the literature of convolutional codes is devoted to con- volutional codes over finite fields. The extension of the concept of convolutional codes from finite fields to finite rings have attracted much attention in recent years due to fact that they are the most appropriate codes for phase modulation. However convolutional codes over finite rings are more involved and not fully understood. Many results and features that are well-known for convolutional codes over finite fields have not been fully investigated in the context of finite rings. In this paper we focus in one of these unexplored areas, namely, we investigate the dual codes of convolutional codes over finite rings. In particular we study the p-dimension of the dual code of a convolutional code over a finite ring. This contribution can be considered a generalization and an extension, to the rings case, of the work done by Forney and McEliece on the dimension of the dual code of a convolutional code over a finite field.Comment: submitte

Column Rank Distances of Rank Metric Convolutional Codes

In this paper, we deal with the so-called multi-shot network coding, meaning that the network is used several times (shots) to propagate the information. The framework we present is slightly more general than the one which can be found in the literature. We study and introduce the notion of column rank distance of rank metric convolutional codes for any given rate and finite field. Within this new framework we generalize previous results on column distances of Hamming and rank metric convolutional codes [3, 8]. This contribution can be considered as a continuation follow-up of the work presented in [10]

Noncatastrophic convolutional codes over a finite ring

Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized as polynomial left prime matrices. In this paper, we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in the finite ring Zpr. In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring Zpr. The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated Zpr-modules. We introduce the notion of b-degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on Zp is.The second and third authors were supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciancia e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020. Diego Napp is partially supported by Ministerio de Ciencia e Innovación via the grant with ref. PID2019-108668GB-I00

A new rank metric for convolutional codes

Let F[D] be the polynomial ring with entries in a finite field F. Convolutional codes are submodules of F[D]n that can be described by left prime polynomial matrices. In the last decade there has been a great interest in convolutional codes equipped with a rank metric, called sum rank metric, due to their wide range of applications in reliable linear network coding. However, this metric suits only for delay free networks. In this work we continue this thread of research and introduce a new metric that overcomes this restriction and therefore is suitable to handle more general networks. We study this metric and provide characterizations of the distance properties in terms of the polynomial matrix representations of the convolutional code. Convolutional codes that are optimal with respect to this new metric are investigated and concrete constructions are presented. These codes are the analogs of Maximum Distance Profile convolutional codes in the context of network coding. Moreover, we show that they can be built upon a class of superregular matrices, with entries in an extension field, that preserve their superregularity properties even after multiplication with some matrices with entries in the ground field.publishe

Concatenation of convolutional codes and rank metric codes for multi-shot network coding

In this paper we present a novel coding approach to deal with the transmission of information over a network. In particular we make use of the network several times (multishot)and impose correlation in the information symbols over time. We propose to encode the information via an inner and an outer code, namely, a Hamming metric convolutional code as an outer code and a rank metric code as an inner code. We show how this simple concatenation scheme can exploit the potential of both codes to produce a code that can correct a large number of error patterns

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

Column distances of convolutional codes over Z_p^r

Maximum distance profile codes over finite nonbinary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper, we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of p-encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) maximum distance profile convolutional codes over Zpr is presented.publishe

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