Loop Codes

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 F q L for q ∈ {2,3,4,5} and | L | ≤ 7, and also in the group algebras F q G for groups G of order | G | ≤ 12. 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.