We continue here the research on (quasi)group codes over (quasi)group rings. We give some constructions of [n,n-3,3]q-codes over Fq for n=2q and n=3q. These codes are linearly optimal, i.e. have maximal dimension among linear codes having a given length and distance. Although codes with such parameters are known, our main results state that we can construct such codes as (left) group codes. In the paper we use a construction of Reed–Solomon codes as ideals of the group ring FqG where G is an elementary abelian group of order q.
том 16 ВЫПУСК 1 * 2004 УДК 519.7 Групповые коды и их неассоциативные обобщения © 2004 г.С. Гонсалес, Е. Коусело, В. Т. Марков, А
In [2,4] the notion of a recursive code was introduced and some constructions of recursive MDS codes were proposed. The main result was that for any q ∉ {2, 6} (except possibly ∈ {14, 18, 26, 42}) there exists a recursive MDS-code in an alphabet of q elements of length 4 and combinatorial dimension 2(i.e. a recursive {4, 2, 3}q-code). One of the constructions we used there was that of pseudogeometries; it enabled us to show that for any q > 126 (except possibly q = 164) there exists a recursive [4,2,3]q-code that contains all the “constants”. One part of the present note is the further application of the pseudogeometry construction which shows that for any q > 164 (resp. q > 26644) there exists a recursive [7,2,6]q-code (resp. [13,2,12]q-code) containing “constants”. Another result presented here is a negative one: we show that there is no nontrivial pseudogeometry consisting of 14, 18, 26 or 42 points with no lines of order 2, 3, 4 or 6, so the pseudogeometry construction cannot be applied for settling the question mentioned in the above. In both cases the usage of computer is essential.
Let q = pt, where p is a prime. Then for any natural m, except the case m = 3 and q ≥ 8 is even, there exists a linear over the space Zpt m-dimensional recursive MDS-code of length q + 1 (q + 2 if q = m + 1 is even) which (in according to well-known conjecture of Bush, Blokhuis, Bruen & Thas) is believed to be the maximum of lengths of linear over the field GF(q) m-dimensional MDS-codes.
том 10 ВЫПУСК 2 * 1998 УДК 519.7 Рекурсивные МДР-коды и рекурсивно дифференцируемые квазигруппы © 1998 г
