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., $G(d_{1},d_{2}) = G_{2}(d_{2})G_{1}(d_{1})$. 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.