MRD Rank Metric Convolutional Codes

So far, in the area of Random Linear Network Coding, attention has been given to the so-called one-shot network coding, meaning that the network is used just once to propagate the information. In contrast, one can use the network more than once to spread redundancy over different shots. In this paper, we propose rank metric convolutional codes for this purpose. The framework we present is slightly more general than the one which can be found in the literature. We introduce a rank distance, which is suitable for convolutional codes, and derive a new Singleton-like upper bound. Codes achieving this bound are called Maximum Rank Distance (MRD) convolutional codes. Finally, we prove that this bound is optimal by showing a concrete construction of a family of MRD convolutional codes

Systematic maximum sum rank codes

In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors.publishe

Códigos convolucionais para codificação em rede com múltiplos envios

In this thesis, we aim to provide a general overview of the area of multi-shot codes for network coding. We will review the approaches and results proposed so far and present slightly more general definitions of rank metric block and convolutional codes that allows a wider set of rates than the definitions of rank metric codes that exist in the literature. We also present, within this new framework, the notion of column rank distance of a rank metric convolutional code. We investigate it properties and derive an upper-bound that allows us to extend the notions of Maximum Distance Profile and Strongly-Maximum Distance Separable convolutional codes to some rank metric codes analogues. We focused on the development of channel encoders as a mechanism that allows the recovery of the data lost during the transmission. We also concentrate on the construction of novel classes of MRD convolutional codes. In particular we aim at extending the constructions presented by Napp, Pinto, Rosenthal and Vettori, in order to increase the degree of the code and consequently it error correction capability. As alternative to rank metric convolutional codes, we present a novel scheme by concatenation of a Hamming metric convolutional code (as outer code) and a rank metric block code (as a inner code). The proposed concatenated code is defined over the base finite field instead of over several extension finite fields and pretend to reduce the complexity of encoding and decoding process and moreover use the more general definition of rank metric code in order to be more natural.Nesta tese, pretendemos mostrar uma visão geral da área de códigos multishot na codificação em redes. Para o efeito, iremos rever as abordagens e resultados propostos até agora e apresentar definições um pouco mais gerais de códigos a blocos e códigos convolucionais que permitem uma ampliação das definições de códigos de métrica rank que já existem na literatura. Também apresentamos, dentro desta nova estrutura, a noção de distância de coluna de um código convolucional de métrica rank. Investigamos as suas propriedades e derivamos um limite superior para o valor da mesma, que nos permite estender as noções de MDP e Strongly MDS para os códigos de métrica rank. Iremos também focar-nos no desenvolvimento de codificadores de canal como mecanismo que permite uma melhor recuperação dos dados perdidos durante o processo de transmissão. Também nos concentramos na construção de novas classes de códigos convolucionais MRD. Em particular, pretendemos estender as construções apresentadas por Napp, Pinto, Rosenthal e Vettori, com o intuito de incrementar o grau do código e, consequentemente, melhorar a sua capacidade corretora. Como alternativa aos códigos convolucionais de métrica rank, apresentamos um novo esquema usando concatenação de um código convolucional de métrica Hamming (como código externo) e um código a bloco de métrica rank (como um código interno). O código concatenado proposto é definido sobre o corpo finito base, com o intuito de reduzir a complexidade do processo de codificação e decodificação e, além disso, usa a definição mais geral de código de métrica rank, tornando o processo mais natural.Programa Doutoral em Matemátic

Convolutional Codes in Rank Metric with Application to Random Network Coding

Random network coding recently attracts attention as a technique to disseminate information in a network. This paper considers a non-coherent multi-shot network, where the unknown and time-variant network is used several times. In order to create dependencies between the different shots, particular convolutional codes in rank metric are used. These codes are so-called (partial) unit memory ((P)UM) codes, i.e., convolutional codes with memory one. First, distance measures for convolutional codes in rank metric are shown and two constructions of (P)UM codes in rank metric based on the generator matrices of maximum rank distance codes are presented. Second, an efficient error-erasure decoding algorithm for these codes is presented. Its guaranteed decoding radius is derived and its complexity is bounded. Finally, it is shown how to apply these codes for error correction in random linear and affine network coding.Comment: presented in part at Netcod 2012, submitted to IEEE Transactions on Information Theor

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

On rank metric convolutional codes and concatenated codes

In the recent history of the theory of network coding the multi-shot network coding has been prove as a good alternative for the classical one-shot network theory which is managed by using block codes. To perform communications in this multi-shot context we have, among others, rank-metric convolutional codes and concatenated codes (using a convolutional code as an outer code and a rank-metric code as inner code). In this work we analyse their performance over the rank deficiency channel (described by Gilbert-Elliot channel model) in terms of the correction capabilities and the complexity of the two decoding schemes.publishe