On isometries for convolutional codes

Heide Gluesing-Luerssen

doi:10.3934/amc.2009.3.179

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.

Highlights

Introduction and Basic NotionsOne of the most famous results in the theory of linear block codes is MacWilliams’ Equivalence Theorem [13, 14]; see [10, Sec. 7.9]
The intrinsic notion of isometry coincides with the extrinsic notion of monomial equivalence and this settles the question of a classification of block codes over fields with respect to their error-corrrecting properties
We will make a step in this direction by studying isometries, that is, weight-preserving F[z]-module isomorphisms between convolutional codes in F[z]n

Summary

Introduction and Basic Notions

One of the most famous results in the theory of linear block codes is MacWilliams’ Equivalence Theorem [13, 14]; see [10, Sec. 7.9]. Even though this result is very useful due to its explicit description of all isometries, it is not quite the answer one is looking for because isometries are too weak of a notion in order to classify convolutional codes in a meaningful way. We will briefly consider weight-preserving F-linear (but not necessarily F[z]-linear) isomorphisms between convolutional codes This much weaker notion of isometry raises plenty of interesting questions, which, again, we have to leave open to future research. We will present some of those parameters below in Definition 2.2

The Weight Adjacency Matrix and Distance Parameters

Isometries and Monomial Equivalence

A MacWilliams Equivalence Theorem for Isometry and zME

Other Notions of Isometries