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Using theoretical results about the homogeneous weights for Frobenius rings,we describe the homogeneous weight for the ring family $R_k$, a recentlyintroduced family of Frobenius rings which have been used extensively in codingtheory. We find an associated Gray map for the homogeneous weight using firstorder Reed-Muller codes and we describe some of the general properties of theimages of codes over $R_k$ under this Gray map. We then discuss quasitwistedcodes over $R_k$ and their binary images under the homogeneous Gray map. Inthis way, we find many optimal binary codes which are self-orthogonal andquasicyclic. In particular, we find a substantial number of optimal binarycodes that are quasicyclic of index 8, 16 and 24, nearly all of which are newadditions to the database of quasicyclic codes kept by Chen.

The homogeneous weight for $R_k$, related Gray map and new binary quasicyclic codes