On Isometries for Convolutional Codes

TL;DR

This paper investigates the structure of isometries in convolutional codes, establishing a MacWilliams-type theorem for isometries and exploring the limitations of strong isometries and their relation to weight adjacency matrices.

Contribution

It proves a MacWilliams Equivalence Theorem for isometries of convolutional codes, characterizing them via monomial transformations, and discusses the role of weight adjacency matrices.

Findings

Isometries correspond to monomial transformations.

Strong isometries are not fully characterized by monomial transformations.

Weight adjacency matrices provide additional insights into code isometries.

Abstract

In this paper we will discuss isometries and strong isometries for convolutional codes. Isometries are weight-preserving module isomorphisms whereas strong isometries are, in addition, degree-preserving. Special cases of these maps are certain types of monomial transformations. We will show a form of MacWilliams Equivalence Theorem, that is, each isometry between convolutional codes is given by a monomial transformation. Examples show that strong isometries cannot be characterized this way, but special attention paid to the weight adjacency matrices allows for further descriptions. Various distance parameters appearing in the literature on convolutional codes will be discussed as well.