Abstract
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.
Highlights
The notion of primeness plays a central role in the polynomial matrix approach to several areas of pure and applied mathematics, such as systems and control theory or coding theory
We provide a characterization of zero left prime polynomial matrices from which it follows that when a convolutional code admit a left zero prime generator matrix, i.e., a noncatastrophic encoder, the code can be described by means of a parity-check polynomial matrix
A polynomial matrix A(d) ∈ Zpr [d]k×n is left zero-prime if the ideal generated by all the k-th order minors of A(d) is Zpr [d]
Summary
The notion of primeness plays a central role in the polynomial matrix approach to several areas of pure and applied mathematics, such as systems and control theory or coding theory. In this work we define convolutional codes as finitely generated free Zpr [d]-modules of Zpr [d]n, where Zpr [d] is the polynomial ring with coefficients in Zpr and study noncatastrophicity in this setting [13, 18, 19, 25]. We show that a convolutional code over Zp is optimal with respect to the column distance if and only if its projection is In each section we briefly provide some preliminaries: in Section 2 we recall known results of primeness of polynomial matrices over finite fields and in Section 3 the definitions of convolutional codes, free distance and column distances are presented
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