On the semi-scalar equivalence of polynomial matrices over a field

Polynomial matrices and of size over a field are semi-scalar equivalent if there exist a nonsingular matrix over and an invertible matrix over such that . The aim of the present report is to present a triangular form of some nonsingular polynomial matrices with respect to semi-scalar equivalence.

Some Further Results on the Reduction of Two-Dimensional Systems

The reduction of two-dimensional systems plays an important role in the theory of systems, which is closely associated with the equivalence of the bivariate polynomial matrices. In this paper, the equivalence problems on several classes of bivariate polynomial matrices are investigated. Some new results on the equivalence of these matrices are obtained. These results are useful for reducing two-dimensional systems.

Noncatastrophic convolutional codes over a finite ring

Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized as polynomial left prime matrices. In this paper, we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in the finite ring [Formula: see text]. In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring [Formula: see text]. The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated [Formula: see text]-modules. We introduce the notion of [Formula: see text]-degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on [Formula: see text] is.