SOME ASPECTS OF THE ALGEBRAIC STRUCTURE OF 2D CONVOLUTIONAL CODES

In recent times the 2D system analysis has constituted a stimulating research topic, both for the theoretical interest and the applications in two-dimensional digital signal processing ([1], [2]). As well known, 2D discrete filtering usually resorts to input/output ARMA rep-resentations or to transfer matrices whose entries are rational functions in two vari-ables. In the last few years an innovative approach to the 2D system representation has been proposed by J.C.Willems and P.Rocha ([3], [4]). They have introduced a mathematical framework for analyzing the trajectories of dynamical systems on the discrete plane and for representing their evolution laws, that seems suitable for studying arbitrary two-dimensional data structures. The convolutional codes theory has largely made use of polynomial matrix alge-bra and some other tools borrowed from Linear Systems Analysis, thus supplying further results whose interest largely exceeds the original field of research. Quite recently, G.D.Forney ([5]) has applied Willems viewpoint to the investigation of the properties of convolutional codes over groups, specified through the set of all their possible trajectories (i.e. the behaviour). On the same streamline, this paper aims to give a preliminary report on 2D convolutional codes, by providing some new results and pointing out open problems and directions of further investigation. The first part of the paper is devoted to the study of some properties a good encoder must be endowed with. A natural requirement is the external controllability (see Willems, [5]), that is the possibility of generating code sequences whose restric-tion to a finite region of the plane entails no loss of freedom on the values assumed far away from that region. A controllable code C can be represented as the image of a 2D polynomial matrix C = ImG(σ1, σ2) where σ1 and σ2 are the shift operators along the coordinate axes. A further requirement for a good encoder is to produce code sequences which, if received incorrupted, allow an exact reconstruction of the generating information sequence. If we look at the polynomial encoder as a linear input/output map on 2D sequence spaces, the above property entails the injectivity of the map. It can b