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.

翻译：群代数$\mathbb{F}_{q}G$中易于计算的维数（ECD）群码是指维数小于或等于$p=char(\mathbb{F}_{q})$且由幂等元生成的理想。本文引入了一种易于计算的不可分解维数（ECID）群代数，即所有由本原幂等元生成的群码均为ECD的有限群代数。给出了这些代数的若干刻画。此外，在半单情形下，提出了判定群代数是否为ECID的一些算术条件。在非半单情形下，这些代数具有有限表示型，且其底层群的Sylow $p$-子群是单群。给出了这些代数中群码的维数及其最小Hamming距离的下界，同时提供了幂等元本原性的算术检验方法。文中还给出了说明主要结果的例子。