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