Rank and Kernel of $\mathbb{F}_p$-Additive Generalised Hadamard Codes

TL;DR

This paper investigates the properties of $F_p$-additive generalized Hadamard codes, establishing bounds on their rank and kernel dimension, constructing codes for specific parameters, and demonstrating their self-orthogonality and quantum code generation.

Contribution

It provides new bounds on rank and kernel dimension for $F_p$-additive GH codes and constructs codes achieving these bounds, especially for the case $e=2$, also proving their self-orthogonality and quantum code potential.

Findings

Bounds on rank and kernel dimension are established.

Explicit constructions of $F_p$-additive GH codes for certain parameters.

Codes are shown to be self-orthogonal and generate quantum codes.

Abstract

A subset of a vector space $\mathbb{F}_q^n$ is $K$-additive if it is a linear space over the subfield $K\subseteq \mathbb{F}_q$. Let $q=p^e$, $p$ prime, and $e>1$. Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are $\mathbb{F}_p$-additive are established. For specific ranks and dimensions of the kernel within these bounds, $\mathbb{F}_p$-additive GH codes are constructed. Moreover, for the case $e=2$, it is shown that the given bounds are tight and it is possible to construct an $\mathbb{F}_p$-additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.