Optimal, Divisible binary Codes from Gray-Homogeneous Images of Codes over R_{k,m}

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In this work, we find a form for the homogeneous weight over the ring R_{k,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,m} to binary codes. By considering cyclic, constacyclic and quasicyclic codes over R_{k,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 km\geq 4 The results, which have been obtained by computer search are tabulated.


Homogeneous Weight; Frobenius rings; divisible codes; self-orthogonal quasicyclic codes; optimal codes

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