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In this work, we analyze the problem of catastrophicity of encoders of convolutional codes over the Laurent series with coefficients in ℤ p r , ℤ p r ((d)). Kuijper and Pinto proved in [M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Autom. Control55(11) (2009) 4890–4897] that, contrary to what happens for codes over ((d)) , where is a field, when dealing with ℤ p r ((d)) there are convolutional codes that do not admit non-catastrophic encoders. Nevertheless it was conjectured that any catastrophic convolutional code admits another type of non-catastrophic encoder called p -encoder. In this paper we solve this conjecture for a class of (2 , 1) convolutional codes over ℤ p 2 and show that, in fact, these codes always admit a non-catastrophic p-encoder. We also describe a constructive procedure that allows us to obtain a non-catastrophic p -encoder. [ABSTRACT FROM AUTHOR]

Maximum distance separable convolutional codes are characterized by the property that the free distance reaches the generalized Singleton bound, which makes them optimal for error correction. However, the existing constructions of such codes are available over fields of large size. In this paper, we present the unique construction of MDS convolutional codes of rate 1 / 2 and degree 5 over the field 1 1 . [ABSTRACT FROM AUTHOR]

Maximum-distance separable (MDS) convolutional codes are characterized by the property that their free distance reaches the generalized Singleton bound. In this paper, new criteria to construct MDS convolutional codes are presented. These codes also possess optimal first (reverse) column distances. The new criteria allow to relate the construction of MDS convolutional codes to those of reverse superregular Toeplitz matrices. Moreover, using the new criteria as well as the help of computer search, examples for MDS convolutional codes over small finite fields are given. [ABSTRACT FROM AUTHOR]

It is well known that convolutional codes are linear systems when they are defined over a finite field. A fundamental issue in the implementation of convolutional codes is to obtain a minimal state representation of the code. Compared with the literature on one-dimensional (1D) time-invariant convolutional codes, there exist relatively few results on the realization problem for time-varying 1D convolutional codes and even fewer if the convolutional codes are two-dimensional (2D). In this paper we consider 2D periodic convolutional codes and address the minimal state space realization problem for this class of codes. This is, in general, a highly nontrivial problem. Here, we focus on separable Roesser models and show that in this case it is possible to derive, under weak conditions, concrete formulas for obtaining a 2D Roesser state space representation. Moreover, we study minimality and present necessary conditions for these representations to be minimal. Our results immediately lead to constructive algorithms to build these representations.

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 ℤ p r . In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring ℤ p r . The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated ℤ p r -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 ℤ p is. [ABSTRACT FROM AUTHOR]

Maximum distance separable convolutional codes are the codes that present best performance in error correction among all convolutional codes with certain rate and degree. In this paper, we show that taking the constant matrix coefficients of a polynomial matrix as submatrices of a superregular matrix, we obtain a column reduced generator matrix of an MDS convolutional code with a certain rate and a certain degree. We then present two novel constructions that fulfill these conditions by considering two types of superregular matrices. [ABSTRACT FROM AUTHOR]

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 ${\mathbb Z}_{p^{r}}$ is presented. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) block codes and MDS 1D convolutional codes are the most robust codes for error correction within the class of block codes of a fixed rate and 1D convolutional codes of a certain rate and degree, respectively. In this paper, we generalize this concept to the class of 2D convolutional codes. For that, we introduce a natural bound on the distance of a 2D convolutional code of rate k/n and degree \delta , which generalizes the Singleton bound for block codes and the generalized Singleton bound for 1D convolutional codes. Then, we prove the existence of 2D convolutional codes of rate k/n and degree \delta if k {\nmid } \delta , or n \geq k (({(({\delta }/{k}) + 1)(({\delta }/{k}) + 2)})/{2})$ if $k \mid \delta $ , by presenting a concrete constructive procedure. [ABSTRACT FROM AUTHOR]

A convolutional code C over Z p r [ D ] is a Z p r [ D ] -submodule of Z p r n [ D ] where Z p r [ D ] stands for the ring of polynomials with coefficients in Z p r . In this paper, we study the list decoding problem of these codes when the transmission is performed over an erasure channel, that is, we study how much information one can recover from a codeword w ∈ C when some of its coefficients have been erased. We do that using the p -adic expansion of w and particular representations of the parity-check polynomial matrix of the code. From these matrix polynomial representations we recursively select certain equations that w must satisfy and have only coefficients in the field p r − 1 Z p r . We exploit the natural block Toeplitz structure of the sliding parity-check matrix to derive a step by step methodology to obtain a list of possible codewords for a given corrupted codeword w , that is, a list with the closest codewords to w. [ABSTRACT FROM AUTHOR]