Algebraic Construction of Tail-Biting Trellises for Linear Block Codes

TL;DR

This paper introduces an algebraic method for constructing tail-biting trellises for linear block codes, demonstrating their isomorphism to existing trellis types and analyzing their complexity.

Contribution

It presents a novel algebraic construction approach for tail-biting trellises using state space expressions and homomorphism theorem, extending previous methods.

Findings

Constructed trellises are isomorphic to KV and BCJR trellises.

The method generalizes minimal-span generator matrices.

Analyzed the complexity of the new trellises.

Abstract

In this paper, we present an algebraic construction of tail-biting trellises. The proposed method is based on the state space expressions, i.e., the state space is the image of the set of information sequences under the associated state matrix. Then combining with the homomorphism theorem, an algebraic trellis construction is obtained. We show that a tail-biting trellis constructed using the proposed method is isomorphic to the associated Koetter-Vardy (KV) trellis and tail-biting Bahl-Cocke-Jelinek-Raviv (BCJR) trellis. We also evaluate the complexity of the obtained tail-biting trellises. On the other hand, a matrix consisting of linearly independent rows of the characteristic matrix is regarded as a generalization of minimal-span generator matrices. Then we show that a KV trellis is constructed based on an extended minimal-span generator matrix. It is shown that this construction is a natural extension of the method proposed by McEliece (1996).