Codes over a family of local Frobenius rings, Gray maps and self-dual codes

Skew constacyclic codes over the local Frobenius non-chain rings of order 16

“…The Gray map used in [15] and [16] follows from the general Gray map given in [13] and is defined as follows:…”

“…Specifically, any ring can be viewed as the direct product of local rings via the Chinese Remainder Theorem (see [11] for a complete description of this). Additionally, alphabets of size 2 k for some k have been increasingly important due to the fact that many have a natural Gray map to the binary Hamming space, see [13] for a complete description. Codes over the rings of order 4 and order 8 as well as codes over chain rings have been well studied.…”

“…Central to this study, we define non-trivial automorphisms on each of the rings. We use the Gray map, established in [13], to study the binary images of codes over these rings and we find numerous optimal binary codes. Additionally, we use a Gray map to the quaternary space to find a number of new linear codes over Z 4 .…”