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In this paper we address the problem of list decoding of linear codes over an integer residue ring Z q , where q is a power of a prime p. The proposed procedure exploits a particular matrix representation of the linear code over Z p r , called the standard form, and the p -adic expansion of the to-be-decoded vector. In particular, we focus on the erasure channel in which the location of the errors is known. This problem then boils down to solving a system of linear equations with coefficients in Z p r . From the parity-check matrix representations of the code we recursively select certain equations that a codeword must satisfy and have coefficients only in the field p r − 1 Z p r . This yields a step by step procedure obtaining a list of the closest codewords to a given received vector with some of its coordinates erased. We show that such an algorithm actually computes all possible erased coordinates, that is, the provided list is minimal. [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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the properties of two-dimensional (2D) convolutional codes which are obtained from series concatenation of two 2D convolutional codes. For this purpose, we confine ourselves to dealing with finite-support 2D convolutional codes and make use of the Fornasini-Marchesini input-state-output (ISO) model representations. Within these ISO representations, we study when the structural properties of modal reachability and modal observability of the two given ISO representations carry over to the resulting 2D convolutional code. Moreover, we provide necessary conditions for obtaining a systematic concatenated convolutional code. Finally, we present a lower bound on its free distance. [ABSTRACT FROM AUTHOR]

Abstract: Motivated by applications in communications, we consider linear discrete-time systems over the finite ring . We solve the open problem of deriving a theory of row-reduced representations for these systems. We introduce a less restrictive form of representation than the adapted form introduced by Fagnani and Zampieri. We call this form the “composed form”. We define the concept of “p-predictable degree property” and “p-row-reduced”. We demonstrate that these concepts, coupled with the composed form, provide a natural setting that extends several classical results from the field case to the ring case. In particular, the classical rank test on the leading row coefficient matrix is generalized. We show that any annihilator of of a pre-specified degree is uniquely parametrized with finitely many coefficients in terms of a kernel representation in p-row-reduced composed form. The underlying theory is the theory of “reduced p-basis” for submodules of that is developed in this paper. We show how to construct a p-row-reduced kernel representation in composed form by constructing a reduced p-basis for the module . [Copyright &y& Elsevier]

Abstract: Reachability and observability of two-dimensional (2D) discrete state-space models are introduced in two different forms: a local form, which refers to single local states, and a global form, which pertains to the infinite set of local states lying on a separation set [M. Bisiacco, State and output feedback stabilizability of 2D systems, IEEE Trans. Circ. Syst., CAS-32 (1985) 1246–1249; E. Fornasini, G. Marchesini, Global properties and duality in 2-D systems, Syst. Control Lett. 2 (1) (1982) 30–38]. While local reachability and observability can be naturally characterized by resorting to classical state space techniques, their global counterparts are better addressed by means of polynomial techniques. In this paper, reachability and observability are introduced in the context of 2D positive systems and their global versions investigated via a polynomial approach. Necessary and sufficient conditions for the existence of these properties are provided and, in particular, polynomial canonical forms for globally reachable/observable positive systems with scalar inputs/scalar outputs are provided. [Copyright &y& Elsevier]