In this paper, two-dimensional convolutional codes constituted by sequences in (F n ) Z 2 where F is a finite field, are considered. In particular, we restrict to codes with rate 1 n and we investigate the problem of minimal dimension for realizations of such codes by separable Roesser models. The encoders which allow to obtain such minimal realizations, called R-minimal encoders, are characterized.

Abstract. In this work the minimality of state-space realizations of an input/output operator (encoder) and of the corresponding output behavior (code) are analyzed. Moreover, a procedure to obtain a minimal realization of a convolutional code starting from a minimal realization of an encoder of the code is provided.

In this paper we consider a special class of 2D convolutional codes (composition codes) with encoders G(d 1 , d 2 ) that can be decomposed as the product of two 1D encoders, i.e.,are prime we provide constructions of syndrome formers of the code, directly from G 1 (d 1 ) and G 2 (d 2 ). Moreover we investigate the minimality of 2D state-space realization by means of a separable Roesser model of syndrome formers of composition codes, where G 2 (d 2 ) is a quasi-systematic encoder.

In this paper we introduce a special class of 2D convolutional codes, called composition codes, which admit encoders G(d 1 , d 2) that can be decomposed as the product of two 1D encoders, i.e., G(d 1 , d 2) = G 2 (d 2)G 1 (d 1). Taking into account this decomposition, we obtain syndrome formers of the code directly from G 1 (d 1) and G 2 (d 2), in case G 1 (d 1) and G 2 (d 2) are right prime. Moreover we consider 2D state-space realizations by means of a separable Roesser model of the encoders and syndrome formers of a composition code and we investigate the minimality of such realizations. In particular, we obtain minimal realizations for composition codes which admit an encoder G(d 1 , d 2) = G 2 (d 2)G 1 (d 1) with G 2 (d 2) a systematic 1D encoder. Finally, we investigate the minimality of 2D separable Roesser state-space realizations for syndrome formers of these codes.