  Using theoretical results about the homogeneous weights for Frobenius rings,
we describe the homogeneous weight for the ring family $R_k$, 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 $R_k$ under this Gray map. We then discuss quasitwisted
codes over $R_k$ and their binary images under the homogeneous Gray map. In
this way, we find many optimal binary codes which are self-orthogonal and
quasicyclic. In particular, we find a substantial number of optimal binary
codes that are quasicyclic of index 8, 16 and 24, nearly all of which are new
additions to the database of quasicyclic codes kept by Chen.
