Composition codes

Abstract

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.