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Filomat 2017 Volume 31, Issue 4, Pages: 885-897
https://doi.org/10.2298/FIL1704885Y
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The homogeneous weight for Rk, related gray map and new binary quasi-cyclic codes

Yildiz Bahattin (Fatih University, Istanbul, Turkey)
Kelebek Ismail Gokhan (Fatih University, Istanbul, Turkey)

Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family Rk, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We find an associated Gray map for the homogeneous weight using first order Reed-Muller codes and we describe some of the general properties of the images of codes over Rk under this Gray map. We then discuss quasi-twisted codes over Rk and their binary images under the homogeneous Gray map. In this way, we find many optimal binary codes which are self-orthogonal and quasi-cyclic. In particular, we find a substantial number of optimal binary codes that are quasi-cyclic of index 8, 16 and 24, nearly all of which are new additions to the database of quasi-cyclic codes kept by Chen.

Keywords: homogeneous weights, cyclic codes, quasi-twisted codes, quasi-cyclic codes, codes over rings