Isometries and Binary Images of Linear Block Codes over $Z_4+uZ_4$ and $Z_8+uZ_8$

TL;DR

This paper develops isometric maps from specific rings to binary spaces to analyze linear block codes, deriving bounds and constructing new codes over rings like $Z_4+uZ_4$ and $Z_8+uZ_8$, with potential generalization to $Z_{2^r}+uZ_{2^r}$.

Contribution

It introduces weight-based isometries for codes over $Z_4+uZ_4$ and $Z_8+uZ_8$, enabling analysis and construction of binary images with bounds on minimum distances.

Findings

Established isometries using Lee and homogeneous weights.

Derived bounds on minimum distances of binary images.

Constructed new codes and their binary images.

Abstract

Let $F_2$ be the binary field and $Z_{2^r}$ the residue class ring of integers modulo $2^r$, where $r$ is a positive integer. For the finite $16$-element commutative local Frobenius non-chain ring $Z_4+uZ_4$, where $u$ is nilpotent of index $2$, two weight functions are considered, namely the Lee weight and the homogeneous weight. With the appropriate application of these weights, isometric maps from $Z_4+uZ_4$ to the binary spaces $F_2^4$ and $F_2^8$, respectively, are established via the composition of other weight-based isometries. The classical Hamming weight is used on the binary space. The resulting isometries are then applied to linear block codes over $Z_4+uZ_4$ whose images are binary codes of predicted length, which may or may not be linear. Certain lower and upper bounds on the minimum distances of the binary images are also derived in terms of the parameters of the $Z_4+uZ_4$ codes. Several new codes and their images are constructed as illustrative examples. An analogous procedure is performed successfully on the ring $Z_8+uZ_8$, where $u^2=0$, which is a commutative local Frobenius non-chain ring of order $64$. It turns out that the method is possible in general for the class of rings $Z_{2^r}+uZ_{2^r}$, where $u^2=0$, for any positive integer $r$, using the generalized Gray map from $Z_{2^r}$ to $F_2^{2^{r-1}}$.