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arXiv:0906.2509 · Physical Review A

Construction of [[n,n−4,3]]q quantum codes for odd prime power q

Ruihu Li, Zongben Xu

Published Nov 16, 2010• 47 citations
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Abstract

For each odd prime power $q$, let $4\ensuremath{\leqslant}n\ensuremath{\leqslant}{q}^{2}+1$. Hermitian self-orthogonal $[n,2,n\ensuremath{-}1]$ codes over ${\mathbf{F}}_{{q}^{2}}$ with dual distance three are constructed by using finite field theory. Hence, $[[n,n\ensuremath{-}4,3]]{}_{q}$ quantum maximal-distance-separable (MDS) codes for $4\ensuremath{\leqslant}n\ensuremath{\leqslant}{q}^{2}+1$ are obtained.

DOI
10.1103/physreva.82.052316

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Let qqq be an odd prime power. We use complements of unions of subsets of Fq2\mathbb F_{q^2}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\mathbb F_{q^2}Fq2​. The Hermitian construction then yields five corresponding families of qqq-ary quantum maximum-distance-separable (MDS) codes. Under suitable parameter conditions, these quantum codes have minimum distances greater than q/2+1q/2+1q/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 qqq.
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Quantum maximum-distance-separable (MDS for short) codes are an important class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed-Solomon (GRS for short) codes, we construct five new classes of qqq-ary quantum MDS codes with minimum distance larger than q/2+1q/2+1q/2+1. Furthermore, the parameters of our quantum MDS code cannot be obtained from the previous constructions.
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