  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.
