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.
The homogeneous weight for Rk, related gray map and new binary quasi-cyclic codes