On the Kernel of $\Z_{2^s}$-Linear Simplex and MacDonald Codes

TL;DR

This paper investigates the kernel of binary images of $ ext{Z}_{2^s}$-additive simplex and MacDonald codes obtained via the Gray map, extending understanding of their structure and relationships.

Contribution

It establishes the kernel of the Gray map images of $ ext{Z}_{2^s}$-additive simplex and MacDonald codes, generalizing previous results to codes over $ ext{Z}_{2^s}$.

Findings

Kernel characterization of $ ext{Z}_{2^s}$-linear simplex codes

Kernel characterization of $ ext{Z}_{2^s}$-linear MacDonald codes

Extension of known results to codes over $ ext{Z}_{2^s}$

Abstract

The $\Z_{2^s}$-additive codes are subgroups of $\Z^n_{2^s}$, and can be seen as a generalization of linear codes over $\Z_2$ and $\Z_4$. A $\Z_{2^s}$-linear code is a binary code which is the Gray map image of a $\Z_{2^s}$-additive code. We consider $\Z_{2^s}$-additive simplex codes of type $\alpha$ and $\beta$, which are a generalization over $\Z_{2^s}$ of the binary simplex codes. These $\Z_{2^s}$-additive simplex codes are related to the $\Z_{2^s}$-additive Hadamard codes. In this paper, we use this relationship to establish the kernel of their binary images, under the Gray map, the $\Z_{2^s}$-linear simplex codes. Similar results can be obtained for the binary Gray map image of $\Z_{2^s}$-additive MacDonald codes.