Linear tail-biting trellises: Characteristic generators and the BCJR-construction

TL;DR

This paper explores the construction and properties of tail-biting trellises for linear codes, focusing on characteristic generators, their influence on trellis structures, and proving a duality conjecture for minimal trellises.

Contribution

It generalizes the concept of characteristic generators, links KV-trellises to BCJR-trellises, and proves a duality conjecture for minimal trellises.

Findings

KV-trellises are always non-mergeable.

Each KV-trellis is a BCJR-trellis.

The duality conjecture is proved for minimal trellises.

Abstract

We investigate the constructions of tail-biting trellises for linear block codes introduced by Koetter/Vardy (2003) and Nori/Shankar (2006). For a given code we will define the sets of characteristic generators more generally than by Koetter/Vardy and we will investigate how the choice of characteristic generators affects the set of resulting product trellises, called KV-trellises. Furthermore, we will show that each KV-trellis is a BCJR-trellis, defined in a slightly stronger sense than by Nori/Shankar, and that the latter are always non-mergeable. Finally, we will address a duality conjecture of Koetter/Vardy by making use of a dualization technique of BCJR-trellises and prove the conjecture for minimal trellises.