Multidimensional Systems and Signal Processing, 5, 231-243 (1994) © 1994 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. On 2D Finite Support Convolutional Codes" An Algebraic Approach MARIA ELENA VALCHER AND ETTORE FORNASINI Department of Electronics and Computer Science, University of Padova, 35131 Padova, ltaly Received June 26, 1993; Revised Abstract. Two-dimensional (2D) finite codes are defined as families of compact support sequences indexed in Z x Z and taking values in F n, F a Galois field. Several properties of encoders, decoders and syndrome decod- ers are discussed under different hypotheses on the code structure, and related to the injecfivity and primeness of the corresponding polynomial matrices in two variables. Dual codes are finally introduced as families of parity checks on a given modular code, and related to the standard theory of 2D behaviors. Key Words: convolutional codes, basic codes, 2D behaviors, matrix fraction description, duality 1. Introduction Since the early seventies, the pioneering work of Forney [1, 2] made it quite clear that the theory of discrete-time multidimensional linear systems over a finite field provides a very convenient setting for the analysis of convolutional codes. On the other hand, in the algebraic context many questions concerning convolutional codes proved to have answers that seem quite illuminating and useful for systems and control applications. However, even if both fields exhibit some common research directions and resort to similar mathematical tools, the coding point of view is somewhat different from that of linear systems. Actually, in system theory the interest centers around input-output relations, while in coding theory what is most important is the set of output sequences of the encoder, i.e., the internal struc- ture of the code. Quite recently, the behavioral approach, developed by J.C. WiUems [3] for the analysis of dynamical systems, has been applied to the investigation of 1D and 2D convolutional codes [4-6]. This new framework seems to be quite effective in the 2D case, since it allows us to investigate the internal properties of the code without explicitly referring to the machin- ery which underlies the codeword generation and, in particular, without making any assump- tion on the ordering of two-dimensional data. So, in principle, no artificial notion of causality in Z × Z, and, consequently, no a priori restriction on the supports of the signals are needed. Indeed, the finite-support constraint we shall introduce in a while on two- dimensional codewords does not follow from causality considerations, but corresponds to the fact that most of 2D information sequences encountered in the applications do not infi- nitely extend in Z × Z.