Equivalences among Z_{p^s}-linear Generalized Hadamard Codes

TL;DR

This paper investigates the equivalences among $ ext{Z}_{p^s}$-linear generalized Hadamard codes, establishing new bounds on their number and showing some codes are equivalent, thus advancing classification methods.

Contribution

It proves that certain $ ext{Z}_{p^s}$-linear GH codes of length $p^t$ are equivalent, leading to improved bounds on the count of nonequivalent codes.

Findings

Some $ ext{Z}_{p^s}$-linear GH codes are equivalent for fixed $t$.

New upper bounds for the number of nonequivalent codes are established.

Bounds match known lower bounds up to $t=10$, confirming their tightness.

Abstract

The $\Z_{p^s}$-additive codes of length $n$ are subgroups of $\Z_{p^s}^n$, and can be seen as a generalization of linear codes over $\Z_2$, $\Z_4$, or $\Z_{2^s}$ in general. A $\Z_{p^s}$-linear generalized Hadamard (GH) code is a GH code over $\Z_p$ which is the image of a $\Z_{p^s}$-additive code by a generalized Gray map. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some $\Z_{p^s}$-linear GH codes of length $p^t$ are equivalent, once $t$ is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to $t=10$, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel).