Minimal Realizations of Syndrome Formers of a Special Class of 2D Codes

Abstract

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., G(d(1), d(2)) = G(2)(d2)G(1)(d(1)). In case that G(1)(d(1)) and G(2)(d(2)) 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.