Quantum MDS codes from complements of unions of finite-field subsets
Abstract
Let q be an odd prime power. We use complements of unions of subsets of 𝔽q2 as locator sets and establish a sufficient condition under which a generalized Reed--Solomon (GRS) code is Hermitian self-orthogonal. Using cosets of multiplicative subgroups and sets with prescribed trace or norm values, we construct five families of Hermitian self-orthogonal GRS codes over 𝔽q2. The Hermitian construction then yields five corresponding families of q-ary quantum maximum-distance-separable (MDS) codes. Under suitable parameter conditions, these quantum codes have minimum distances greater than q/2+1. By comparing codes of the same length, we give conditions under which our codes have strictly larger minimum distances than those obtainable from several previously known constructions based on trace maps, linear transformations, and cosets of multiplicative subgroups, either directly or via the propagation rule. We further show that such improvements occur for infinitely many values of q.
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BibTeX
@misc{hu2026quantum,
title = {{Quantum MDS codes from complements of unions of finite-field subsets}},
author = {Naihong Hu and Hong Ji},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09943},
eprint = {2609.09943},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09943},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum MDS codes from complements of unions of finite-field subsets
AU - Naihong Hu
AU - Hong Ji
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09943
UR - https://arxiv.org/abs/2609.09943
AB - Let q be an odd prime power. We use complements of unions of subsets of 𝔽_q^2 as locator sets and establish a sufficient condition under which a generalized Reed--Solomon (GRS) code is Hermitian self-orthogonal. Using cosets of multiplicative subgroups and sets with prescribed trace or norm values, we construct five families of Hermitian self-orthogonal GRS codes over 𝔽_q^2. The Hermitian construction then yields five corresponding families of q-ary quantum maximum-distance-separable (MDS) codes. Under suitable parameter conditions, these quantum codes have minimum distances greater than q/2+1. By comparing codes of the same length, we give conditions under which our codes have strictly larger minimum distances than those obtainable from several previously known constructions based on trace maps, linear transformations, and cosets of multiplicative subgroups, either directly or via the propagation rule. We further show that such improvements occur for infinitely many values of q.
ER -