MacWilliams Identities for Terminated Convolutional Codes

TL;DR

This paper establishes a MacWilliams identity for the weight distribution generating functions of terminated convolutional codes and their orthogonal codes, showing its usefulness for performance estimation, especially with tail-biting termination.

Contribution

It introduces a MacWilliams identity for the weight distribution of tail-biting terminated convolutional codes and their orthogonal codes, extending previous results.

Findings

MacWilliams identity exists for weight distribution generating functions

Tail-biting termination preserves the identity

Distribution useful for performance estimation

Abstract

Shearer and McEliece [1977] showed that there is no MacWilliams identity for the free distance spectra of orthogonal linear convolutional codes. We show that on the other hand there does exist a MacWilliams identity between the generating functions of the weight distributions per unit time of a linear convolutional code C and its orthogonal code C^\perp, and that this distribution is as useful as the free distance spectrum for estimating code performance. These observations are similar to those made recently by Bocharova, Hug, Johannesson and Kudryashov; however, we focus on terminating by tail-biting rather than by truncation.