Quantum codes with improved error correction distances found from finite field subsets
Quantum MDS codes from complements of unions of finite-field subsets
Information Theory
Summary
Quantum error-correcting codes help protect information in future quantum computers from mistakes. The authors found new ways to build better quantum codes using special sets of numbers from finite fields. Their method improves the minimum distance of the codes, meaning they can detect and fix more errors than some older codes. These improvements apply to many different quantum code sizes and conditions.
What this means in practice
- •For quantum hardware engineers: Use the new quantum MDS codes to design quantum memory systems with stronger error protection in quantum computers.
- •For classical coding theorists: Incorporate the Hermitian self-orthogonal generalized Reed–Solomon codes with improved parameters into research on classical-to-quantum code constructions.
Authors
Naihong Hu, Hong Ji
Abstract
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 $q$.