Local Irreducibility of Tail-Biting Trellises

TL;DR

This paper explores the properties of tail-biting trellises for linear codes, focusing on their irreducibility, duality, and reduction methods, with implications for code analysis and trellis simplification.

Contribution

It introduces a new characterization of irreducibility for tail-biting trellises using observability and controllability, and presents a constructive reduction procedure.

Findings

Fragment observability and controllability are equivalent to irreducibility for certain trellis segments.

A reduction procedure for reducible trellises is developed.

Conditions for dual trellis product factorization into elementary trellises are characterized.

Abstract

This paper investigates tail-biting trellis realizations for linear block codes. Intrinsic trellis properties are used to characterize irreducibility on given intervals of the time axis. It proves beneficial to always consider the trellis and its dual simultaneously. A major role is played by trellis properties that amount to observability and controllability for fragments of the trellis of various lengths. For fragments of length less than the minimum span length of the code it is shown that fragment observability and fragment controllability are equivalent to irreducibility. For reducible trellises, a constructive reduction procedure is presented. The considerations also lead to a characterization for when the dual of a trellis allows a product factorization into elementary ("atomic") trellises.