Optimal Divisible Binary Codes from Gray-Homogeneous Images of Codes over R-k,R-m

Abstract

In this work, we find a form for the homogeneous weight over the ring R-k,R-m, using the related theoretical results from the literature. We then use the first order Reed-Muller codes to find a distance-preserving map that takes codes over R-k,R-m to binary codes. By considering cyclic, constacyclic and quasicyclic codes over R-k,R-m of different lengths for different values of k and m, we construct a considerable number of optimal binary codes that are divisible with high levels of divisibility. The codes we have obtained are also quasicyclic with high indices and they are all self-orthogonal when k(m) >= 4. The results, which have been obtained by computer search are tabulated.