Codes over local rings of order 16 and binary codes

Abstract

We study codes over the commutative local Frobenius rings of order 16 with maximal ideals of size 8. We define a weight preserving Gray map and study the images of these codes as binary codes. We study self-dual codes and determine when they exist.

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

“…Given the ideal structure of these rings it is possible to write every element of [15,16,25], for a complete explanation).…”

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

“…It is shown in [15] and [16] that if C is a linear code over a Frobenius local ring of order 16 of length n, size 2 k and minimum Lee weight d, then φ(C) is a binary code with length 4n, minimum Hamming distance d and size 2 k .…”

“…Codes over the rings of order 4 and order 8 as well as codes over chain rings have been well studied. Hence, the next case to consider is non-chain local rings of order 16.…”

“…In [25], the local Frobenius non-chain rings of order 16 were classified. Codes over the local Frobenius non-chain rings were studied in [15] and [16] and a Gray map to the binary Hamming space was defined which applies to each of the rings in this family. Cyclic and constacyclic codes over these rings were studied in [12], and in [14] respectively, and relationship between cyclic and constacyclic codes was given.…”