On easily computable indecomposable dimension group algebras, and group codes
E. J. García-Claro
TL;DR
This work investigates when a finite group algebra $\mathbb{F}_{q}G$ yields easily computable indecomposable dimension (ECID) algebras and how group codes in $\mathbb{F}_{q}G$ behave in terms of dimension and minimum distance. It develops a framework based on $q$-orbits, splitting fields, and Artin–Wedderburn theory to characterize ECID algebras in both semisimple and non-semisimple settings, and to provide arithmetic tests for idempotent primitivity. It furthermore derives dimension formulas for principal indecomposable modules, identifies finite representation type in the non-semisimple ECID case (with Sylow $p$-subgroups isomorphic to $C_{p}$), and establishes lower bounds for the minimum Hamming distance of group codes arising from ECID algebras. The results enable efficient dimension computations for group codes, practical distance bounds, and concrete illustrations on small groups, advancing both the theory and the construction of codes with desired parameters.
Abstract
An easily computable dimension (or ECD) group code in the group algebra $\mathbb{F}_{q}G$ is an ideal of dimension less than or equal to $p=char(\mathbb{F}_{q})$ that is generated by an idempotent. This paper introduces an easily computable indecomposable dimension (or ECID) group algebra as a finite group algebra for which all group codes generated by primitive idempotents are ECD. Several characterizations are given for these algebras. In addition, some arithmetic conditions to determine whether a group algebra is ECID are presented, in the case it is semisimple. In the non-semisimple case, these algebras have finite representation type where the Sylow $p$-subgroups of the underlying group are simple. The dimension and some lower bounds for the minimum Hamming distance of group codes in these algebras are given together with some arithmetical tests of primitivity of idempotents. Examples illustrating the main results are presented.