Group Codes and the Schreier matrix form

TL;DR

This paper introduces a novel encoder structure for group codes based on the Schreier matrix form, utilizing normal chains and coset representatives, which offers a natural and extendable implementation for group codes.

Contribution

It presents a new encoder design for group codes using Schreier matrix form and coset decompositions, extending classical shift register concepts.

Findings

Encoder uses shortest generator sequences with coset representatives.

The Schreier matrix form provides a natural structure for group code encoding.

Extension to solvable groups with generators from prime cyclic groups.

Abstract

In a group trellis, the sequence of branches that split from the identity path and merge to the identity path form two normal chains. The Schreier refinement theorem can be applied to these two normal chains. The refinement of the two normal chains can be written in the form of a matrix, called the Schreier matrix form, with rows and columns determined by the two normal chains. Based on the Schreier matrix form, we give an encoder structure for a group code which is an estimator. The encoder uses the important idea of shortest length generator sequences previously explained by Forney and Trott. In this encoder the generator sequences are shown to have an additional property: the components of the generators are coset representatives in a chain coset decomposition of the branch group B of the code. Therefore this encoder appears to be a natural form for a group code encoder. The encoder has a register implementation which is somewhat different from the classical shift register structure. This form of the encoder can be extended. We find a composition chain of the branch group B and give an encoder which uses coset representatives in the composition chain of B. When B is solvable, the generators are constructed using coset representatives taken from prime cyclic groups.