Reed–Solomon codes and Reed–Muller codes are represented as ideals of the group ring S = QH of an elementary Abelian p-group H over a finite field Q = \( {\mathbb{F}_q} \) of characteristic p. Such representations for these codes are already known. Our technique differs from the previously used method in the following. There, the codes in question were represented as kernels of some homomorphisms; in other words, these were defined by some kind of parity-check relations. Here, we explicitly specify generators for the ideals presenting the codes. In this case Reed–Muller codes are obtained by applying the trace function to some sums of one-dimensional subspaces of Q S in a fixed set of q such subspaces, whose sums also present Reed–Solomon 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.

A code of length n over an alphabet of q > 2 elements is called a full ^-recursive code if it consists of all segments of length π of a recurring sequence that satisfies some fixed (nonlinear in general) recursivity law f(x\ , . . . , jty) of order k < n. Let n(k, q) be the maximal number n such that there exists such a code with distance n — k+ 1 (MDS-code). The condition n(k,q) > n means that the function / together with its n — k — 1 sequential recursive derivatives forms an orthogonal system of fc-quasigroups. We prove that if q £ {2,6, 14, 18,26,42}, then n(2,q) > 4. The proof is reduced to constructing some special pairs of orthogonal Latin squares.