Codes over Rk, Gray maps and their binary images

Abstract

We introduce codes over an infinite family of rings and describe two Gray maps to binary codes which are shown to be equivalent. The Lee weights for the elements of these rings are described and related to the Hamming weights of their binary image. We describe automorphisms in the binary image corresponding to multiplication by units in the ring and describe the ideals in the ring, using them to define a type for linear codes. Finally, Reed Muller codes are shown as the image of linear codes over these rings.

A modified bordered construction for self-dual codes from group rings

“…It is shown in [8] that the ring R k is a commutative ring with |R k | = 2 (2 k ) . It is also shown that ∀a ∈ R k a 2 = 1 if a is a unit 0 otherwise.…”

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

“…2 , where w = uv, was given as R 1 in [12]. Notice that this ring is a member of the family of rings R k studied in [17], [18], and [19]. This family of rings has been shown to be very useful in finding good binary self-dual codes as well as good binary QC codes.…”

Codes over local rings of order 16 and binary codes

“…In [7], another Gray map equivalent to φ was defined. It is also possible to define another Gray map on the local rings of order 16 with a maximal ideal of size 8.…”

“…One of the early papers using this was [8] in which binary codes were constructed via a non-linear Gray map from quaternary codes. This work was followed by papers where Gray maps were defined from all of the rings of order 4, see [3], and then to other rings, see [2,7,12]. In general, Gray maps have been a central feature of the study of codes over rings.…”