Efficiently Computable Indecomposable Dimension Group Algebras and Their Group Codes

Core Concepts

This paper introduces the concept of easily computable indecomposable dimension (ECID) group algebras, which are finite group algebras where all group codes generated by primitive idempotents have dimension less than or equal to the characteristic of the underlying field. Several characterizations and conditions are provided for identifying ECID group algebras.

Abstract

The paper introduces the concept of easily computable indecomposable dimension (ECID) group algebras, which are finite group algebras where all group codes generated by primitive idempotents have dimension less than or equal to the characteristic of the underlying field.
The key highlights and insights are:

Several characterizations are provided for ECID group algebras, including conditions related to the size of q-orbits, splitting fields, and the structure of the group algebra.

Necessary and sufficient conditions are given for a group algebra to be a minimal ECD (easily computable dimension) group algebra, where the dimension of all principal indecomposable modules is less than or equal to the characteristic of the field.

For non-semisimple ECID group algebras, it is shown that they have finite representation type and their Sylow p-subgroups are cyclic of order p.

Lower bounds are provided for the dimension and minimum Hamming distance of group codes in ECID group algebras.

Arithmetical tests are presented to determine whether an idempotent in a finite group algebra is not primitive.

Examples are included to illustrate the main results.

Stats

|G| - |H/H'| = 21
max{nj * [Fj : Fq]} = 10
γ = 21

Quotes

"An easily computable dimension (or ECD) group code in the group algebra FqG is an ideal of dimension less than or equal to p = char(Fq) that is generated by an idempotent."
"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."

Key Insights Distilled From

On easily computable indecomposable dimension group algebras, and group codes

by E. J... at arxiv.org 04-10-2024

https://arxiv.org/pdf/2404.05775.pdf

Deeper Inquiries

How can the results in this paper be extended to non-abelian finite groups

The results in this paper can be extended to non-abelian finite groups by considering the structure of the group algebra and the properties of the group codes generated by primitive idempotents. For non-abelian groups, the decomposition of the group algebra into simple components may be more complex, but the concept of easily computable indecomposable dimension (ECID) group algebras can still be applied. The characterization of ECID group algebras and the arithmetic conditions for determining the dimension and minimum distance of group codes can be adapted to non-abelian groups by considering their specific group structures and representation theory.

What are the implications of ECID group algebras for the construction of asymptotically good group codes

ECID group algebras have significant implications for the construction of asymptotically good group codes. By ensuring that all group codes generated by primitive idempotents are easily computable in terms of dimension and minimum distance, ECID group algebras provide a framework for efficiently constructing high-quality group codes. These codes have properties that make them suitable for error correction and data transmission applications, as they offer a balance between information rate and minimum distance, crucial for reliable communication in noisy channels.

Can the techniques developed here be applied to study the representation theory of finite groups in a more general setting

The techniques developed in this paper can be applied to study the representation theory of finite groups in a more general setting by considering the algebraic structures and properties of group algebras. By exploring the characterizations of ECID group algebras and the arithmetic conditions for determining the dimension and minimum distance of group codes, researchers can gain insights into the relationship between group codes and the underlying group structures. This can lead to advancements in understanding the algebraic properties of group representations and their applications in coding theory and communication systems.