# Characteristic Matrices and Trellis Reduction for Tail-Biting Convolutional Codes

## TL;DR

This paper explores the properties of characteristic matrices for tail-biting convolutional codes and demonstrates how cyclic transformations and polynomial matrix reductions can lead to trellis complexity reduction.

## Contribution

It introduces a cyclic structure-based analysis of characteristic matrices and proposes a method for trellis reduction using polynomial matrix transformations and partial cyclic shifts.

## Key findings

- Characteristic matrices have a cyclic structure related to the code's generator matrix.
- Trellis reduction can be achieved through polynomial matrix reduction and cyclic shifts.
- Partial cyclic shifts of code sequences facilitate trellis complexity reduction.

## Abstract

Basic properties of a characteristic matrix for a tail-biting convolutional code are investigated. A tail-biting convolutional code can be regarded as a linear block code. Since the corresponding scalar generator matrix Gt has a kind of cyclic structure, an associated characteristic matrix also has a cyclic structure, from which basic properties of a characteristic matrix are obtained. Next, using the derived results, we discuss the possibility of trellis reduction for a given tail-biting convolutional code. There are cases where we can find a scalar generator matrix Gs equivalent to Gt based on a characteristic matrix. In this case, if the polynomial generator matrix corresponding to Gs has been reduced, or can be reduced by using appropriate transformations, then trellis reduction for the original tail-biting convolutional code is realized. In many cases, the polynomial generator matrix corresponding to Gs has a monomial factor in some column and is reduced by dividing the column by the factor. Note that this transformation corresponds to cyclically shifting the associated code subsequence (a tail-biting path is regarded as a code sequence) to the left. Thus if we allow partial cyclic shifts of a tail-biting path, then trellis reduction is accomplished.