Block components of generalized quaternion group codes

TL;DR

This paper analyzes the structure of codes in the generalized quaternion group algebra over finite fields, providing a decomposition approach and applying it to cyclic group codes.

Contribution

It introduces a method to decompose codes in $ ext{GF}_q[Q_{4n}]$ using Wedderburn decomposition and characterizes their generating idempotents.

Findings

Decomposition of codes into direct sums of components.

Explicit description of code blocks via idempotents.

Application to cyclic group codes.

Abstract

Codes in the generalized quaternion group algebra $\mathbb{F}_q[Q_{4n}]$ are considered. Restricting to char$\mathbb{F}_q \nmid 4n$ the structure of an arbitrary code $C \subseteq \mathbb{F}_q[Q_{4n}]$ is described via the Wedderburn decomposition. Moreover it is known that in this case every code $C \subseteq \mathbb{F}_q[Q_{4n}]$ has a generating idempotent $\lambda \in \mathbb{F}_q[Q_{4n}]$. Given the generating idempotent of a code $C$ we determine the different components in its decomposition $C \cong \bigoplus_{j=1}^{r+s}C_j \oplus \bigoplus_{i=1}^{k+t}C'_{i}.$ Afterwards we apply this result to describe the blocks of codes induced by cyclic group codes.