Group codes and their nonassociative generalizations

We give a complete description (with the use of computation) of the best parameters of linear codes that correspond to the left ideals in the loop algebras $\mathbf F_qL$ for $q\in\{2,3,4,5\}$ and $|L|\le 7$, and also in the group algebras $\mathbf F_qG$ for groups $G$ of order $|G|\le12$. We distinguish the linearly optimal codes, the codes satisfying the Varshamov–Hilbert condition as well as those for which the Plotkin bound is attained. The results suggest that the research in codes constructed by using non-associative and non-semisimple non-commutative algebras can open new possibilities and deserves to be developed.
This research was supported by Russian Foundation for Basic Research, grants 02–01–00218, 02–01–00687, and by grants 1910.2003.1 and 2358.2003.9 of President of Russian Federation for supporting the leading scientific schools.
The last two authors thank University of Oviedo for the hospitality.