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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.

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]

In this paper, polynomial matrix fraction descriptions (MFDs) are used as a tool for investigating the structure of a (linear) convolutional code and the family of its encoders and syndrome formers. As static feedback and precompensation allow to obtain all minimal encoders (in particular, polynomial encoders and decoupled encoders) of a given code, a simple parametrization of their MFDs is provided. All minimal syndrome formers, by a duality argument, are obtained by resorting to output injection and postcompensation. Decoupled encoders are finally discussed as well as the possibility of representing a convolutional code as a direct sum of smaller ones. [Copyright &y& Elsevier]