有限域子集并集补集的量子MDS码
Quantum MDS codes from complements of unions of finite-field subsets
- East China Normal University（华东师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文利用有限域子集并集补集构造Hermitian自正交GRS码，进而得到五族量子MDS码，其最小距离可超过q/2+1，并证明在无穷多个q值下优于已知构造。
AI中文摘要：
设$q$为奇素数幂。我们使用$\mathbb F_{q^2}$的子集并集的补集作为定位集，并建立了一个广义Reed-Solomon（GRS）码为Hermitian自正交的充分条件。利用乘法子群的陪集以及具有指定迹或范数值的集合，我们在$\mathbb F_{q^2}$上构造了五族Hermitian自正交GRS码。Hermitian构造随后产生了五族相应的$q$元量子最大距离可分（MDS）码。在适当的参数条件下，这些量子码的最小距离大于$q/2+1$。通过比较相同长度的码，我们给出了我们的码的最小距离严格大于从几个先前已知的基于迹映射、线性变换和乘法子群陪集的构造（直接或通过传播规则）所获得的码的条件。我们进一步证明，对于无穷多个$q$值，这种改进都会发生。
英文摘要：
Let $q$ be an odd prime power. We use complements of unions of subsets of $\mathbb F_{q^2}$ 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 $\mathbb F_{q^2}$. 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 $q$.
