On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$ Z p r

Napp, Diego; Pinto, Raquel; Toste, Marisa

doi:10.1007/s10623-016-0204-9

Cited by 16 publications

(14 citation statements)

References 27 publications

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“…Note that a block code is a convolutional code with δ = 0. For more general classes of convolutional codes see [4,25] In the sequel, we adopt the notation of McEliece [22, p. 1082] and denote a convolutional code of rate k/n and degree δ an (n, k, δ)-convolutional code.…”

Section: Convolutional Codesmentioning

confidence: 99%

A new rank metric for convolutional codes

Almeida

Napp

2020

Des. Codes Cryptogr.

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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.

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Section: Convolutional Codesmentioning

confidence: 99%

A new rank metric for convolutional codes

Almeida

Napp

2020

Des. Codes Cryptogr.

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“…In this section we analyse two fundamental distance properties, namely, free distance and column distance. Once we recall the definition of free distance [21] and [24], we introduce, for the first time, the concept of column distance of convolutional codes over Z p r . We also derive an upper-bound on these distances which leads to the notion of Maximum Distance Profile convolutional code.…”

Section: Column Distance Of Convolutional Codes Over a Finite Ringmentioning

confidence: 99%

“…Codes achieving such a bound were called Maximal Distance Separable (or MDS). The concrete constructions of MDS convolutional codes over Z p r presented in [24] were restricted to free codes and general constructions were built in [21].…”

mentioning

confidence: 99%

Column Distances of Convolutional Codes Over <inline-formula> <tex-math notation="LaTeX">${\mathbb Z}_{p^r}$ </tex-math> </inline-formula>

Napp

Pinto

Toste

2019

IEEE Trans. Inform. Theory

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Maximum Distance Profile codes over finite non-binary 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. I. INTRODUCTION Massey and Mittelholzer [19] showed that the most appropriate codes for phase modulation are the linear codes over the residue class ring Z M and this class includes the convolutional codes over Z M , where M is a positive integer. Fundamental results of the structural properties of convolutional codes over finite rings can be found, for instance, in [7] and [12]. Fagnani and Zampieri [7] studied the theory of convolutional codes over the ring Z p r in the case when the input sequence space is a free module. The problem of deriving minimal encoders (left prime and row-reduced) was posed by Solé et al. in [26] and solved by Kuijper et al. in [16] and [17] using the concept of minimal p-encoder, which is an extension of the concept of p-basis introduced in [29] to the polynomial context. The search for and design of good convolutional codes over Z p r have been investigated in several works in literature. Unit-memory convolutional codes over Z 4 that give rise to binary trellis codes with high free distances together with several concrete constructions of these codes were reported in [2] and [15]. In [13] two 16-state trellis codes of rate 2/4, again over Z 4 , were found by computer search. Also worth

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“…In [17], a bound on the free distance of convolutional codes over Z p r was developed generalizing the results described in [7]. A construction of nonfree MDS convolutional codes over Z p r is also given in [18] with new upper-bounds on the free distance.…”

Section: Introductionmentioning

confidence: 99%

Feedback equivalence of convolutional codes over finite rings

DeCastro-García

2017

Open Mathematics

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Abstract:The approach to convolutional codes from the linear systems point of view provides us with e ective tools in order to construct convolutional codes with adequate properties that let us use them in many applications. In this work, we have generalized feedback equivalence between families of convolutional codes and linear systems over certain rings, and we show that every locally Brunovsky linear system may be considered as a representation of a code under feedback convolutional equivalence.

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On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$ Z p r

Cited by 16 publications

References 27 publications

A new rank metric for convolutional codes

A new rank metric for convolutional codes

Column Distances of Convolutional Codes Over <inline-formula> <tex-math notation="LaTeX">${\mathbb Z}_{p^r}$ </tex-math> </inline-formula>

Feedback equivalence of convolutional codes over finite rings

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