On Skew Convolutional and Trellis Codes

TL;DR

This paper introduces two novel classes of skew codes, skew convolutional and skew trellis codes, which are defined over skew fields and can be decoded using standard algorithms, expanding the coding theory landscape.

Contribution

The paper proposes the definitions and properties of skew convolutional and skew trellis codes, including their representations and decoding methods, which are new contributions to coding theory.

Findings

Skew convolutional codes can be represented as periodic time-varying convolutional codes.

Skew trellis codes are generally nonlinear over the base field.

Both classes have associated code trellises and can be decoded with Viterbi or BCJR algorithms.

Abstract

Two new classes of skew codes over a finite field $\F$ are proposed, called skew convolutional codes and skew trellis codes. These two classes are defined by, respectively, left or right sub-modules over the skew fields of fractions of skew polynomials over $\F$. The skew convolutional codes can be represented as periodic time-varying ordinary convolutional codes. The skew trellis codes are in general nonlinear over $\F$. Every code from both classes has a code trellis and can be decoded by Viterbi or BCJR algorithms.