Characteristic Generators and Dualization for Tail-Biting Trellises

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 Bahl-Cocke-Jelinek-Raviv (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.