Characteristic Generators and Dualization for Tail-Biting Trellises

TL;DR

This paper investigates dualization methods for tail-biting trellises based on characteristic generators, demonstrating their relationships and proving a conjecture about dual KV-trellises using linear algebra and BCJR constructions.

Contribution

It introduces a new dualization approach for tail-biting trellises and proves a stronger version of a conjecture relating dual KV-trellises.

Findings

BCJR-dual is a subtrellis of the local dual in general.

For KV-trellises, BCJR-dual and local dual coincide.

Existence of dual characteristic generator sets for code and dual code.

Abstract

This paper focuses on dualizing tail-biting trellises, particularly KV-trellises. These trellises are based on characteristic generators, as introduced by Koetter/Vardy (2003), and may be regarded as a natural generalization of minimal conventional trellises, even though they are not necessarily minimal. Two dualization techniques will be investigated: the local dualization, introduced by Forney (2001) for general normal graphs, and a linear algebra based dualization tailored to the specific class of tail-biting BCJR-trellises, introduced by Nori/Shankar (2006). It turns out that, in general, the BCJR-dual is a subtrellis of the local dual, while for KV-trellises these two coincide. Furthermore, making use of both the BCJR-construction and the local dualization, it will be shown that for each complete set of characteristic generators of a code there exists a complete set of characteristic generators of the dual code such that their resulting KV-trellises are dual to each other if paired suitably. This proves a stronger version of a conjecture formulated by Koetter/Vardy.