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arXiv:2307.15488 · Quantum Information Processing

MDS, Hermitian almost MDS, and Gilbert–Varshamov quantum codes from generalized monomial-Cartesian codes

Beatriz Barbero-Lucas, Fernando Hernando, Helena Martín-Cruz, Gary McGuire

Published Mar 1, 2024• 2 citations
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Abstract

Abstract We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials inmvariables. When $$m=1$$ m=1 our codes are MDS, and when $$m=2$$ m=2 and our lower bound for the minimum distance is 3, the codes are at least Hermitian almost MDS. For an infinite family of parameters, when $$m=2$$ m=2 we prove that our codes beat the Gilbert–Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.

DOI
10.1007/s11128-024-04297-x

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Abstract We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials inmvariables. When m=1m=1m=1 m=1 our codes are MDS, and when m=2m=2m=2 m=2 and our lower bound for the minimum distance is 3, the codes are at least Hermitian almost MDS. For an infinite family of parameters, when m=2m=2m=2 m=2 we prove that our codes beat the Gilbert–Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.
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Abstract The construction of quantum Maximum Distance Separable (MDS) error-correcting codes is of great significance in practical applications, as it plays a vital role in enhancing the reliability and efficiency of quantum communication and quantum computing. As a result, it remains an active research topic in the field of quantum information science. In this paper, based on generalized Reed-Solomon (GRS) codes and Hermitian construction, we construct two new classes of quantum MDS codes. Our constructed quantum MDS codes are novel in the sense that their parameters differ from all previously known codes. Furthermore, most of our codes exhibit significantly larger minimum distance compared to the existing ones of the same length.
Entanglement-assisted quantum error-correcting codes (EAQECCs) can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism, which greatly promoted the development of quantum coding theory. In this paper, we construct several families of [Formula: see text]-ary entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes of lengths [Formula: see text] with flexible parameters as to the minimum distance [Formula: see text] and the number [Formula: see text] of maximally entangled states. Most of the obtained EAQMDS codes have larger minimum distances than the codes available in the literature.
Hermitian hulls of linear codes are interesting for theoretical and practical reasons alike. In terms of recent application, linear codes whose hulls meet certain conditions have been utilized as ingredients to construct entanglement-assisted quantum error correcting codes. This family of quantum codes is often seen as a generalization of quantum stabilizer codes. Theoretically, compared with the Euclidean setup, the Hermitian case is much harder to deal with. Hermitian hulls of MDS linear codes with low dimensions have been explored, mostly from generalized Reed-Solomon codes. Characterizing Hermitian hulls which themselves are MDS appears to be more involved and has not been extensively studied. This paper introduces some tools to study linear codes whose Hermitian hulls are MDS. Using the tools, we then propose explicit constructions of such codes. We consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of linear MDS codes. We demonstrate that, given the same Hermitian hull dimensions, the codes from our constructions have dimensions which are larger than those in the literature.