Rank and Kernel of Additive Generalized Hadamard Codes

DOI10.1109/TIT.2021.3100433zbMATH Open1487.94182arXiv2001.11609OpenAlexW3185200166MaRDI QIDQ5032569

Steven T. Dougherty, Mercè Villanueva, Josep Rifà

Publication date: 17 February 2022

Published in: IEEE Transactions on Information Theory (Search for Journal in Brave)

Abstract: A subset of a vector space $mathbb{F}_q^n$ is $K$-additive if it is a linear space over the subfield $Ksubseteqmathbb{F}_q$. 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.

Full work available at URL: https://arxiv.org/abs/2001.11609

zbMATH Keywords

bounds on rank and dimensionkernel of additive generalised Hadamard codes

Mathematics Subject Classification ID

Arithmetic codes (94B40)

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