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arXiv:2304.08121 · Computational and Applied Mathematics

Entanglement-assisted quantum error-correcting codes from subfield subcodes of projective Reed–Solomon codes

Philippe Giménez, Diego Ruano, Rodrigo San–José

Published Nov 25, 2023• 6 citations
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Abstract

Abstract We study the subfield subcodes of projective Reed–Solomon codes and their duals: we provide bases for these codes and estimate their parameters. With this knowledge, we can construct symmetric and asymmetric entanglement-assisted quantum error-correcting codes, which in many cases have new or better parameters than the ones available in the literature.

DOI
10.1007/s40314-023-02506-4

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Abstract We study the subfield subcodes of projective Reed–Solomon codes and their duals: we provide bases for these codes and estimate their parameters. With this knowledge, we can construct symmetric and asymmetric entanglement-assisted quantum error-correcting codes, which in many cases have new or better parameters than the ones available in the literature.
Long quantum codes using projective Reed–Muller codes are constructed. Projective Reed–Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Reed–Muller codes with flexible amounts of entanglement by considering equivalent codes. Moreover, we also construct quantum codes from subfield subcodes of projective Reed–Muller codes.
Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC by any classical linear code. In this paper, we propose two constructions of generalized Reed–Solomon codes and calculate the dimension of their hulls. With these generalized Reed–Solomon codes, we present two new infinite families of EAQECCs, which are optimal with respect to the Singleton bound for EAQECCs. Notably, the parameters of our EAQECCs are new and flexible.
Abstract Entanglement-assisted quantum error correction codes (EAQEECs) can improve the reliability of transmission, which make use of pre-existing entanglement between two parties. In this paper, we construct two classes of entanglement-assisted quantum maximum distance separable (EAQMDS) codes via generalized Reed-Solomon (GRS) codes. Consequently, the results show that some of these EAQMDS codes have much larger minimum distance than ones available in the literature. Meanwhile, many of these EAQMDS codes are new in the sense that the parameters of these codes are not covered by the previously known ones. (Abstract)
Classical cyclic and constacyclic codes play a fundamental role in coding theory, thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length [Formula: see text] over [Formula: see text]. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric EAQEC (AEAQEC) codes of length [Formula: see text] over [Formula: see text]. We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.
In this paper, we first give the definition of the Euclidean sums of linear codes, and prove that the Euclidean sums of linear codes are Euclidean dual-containing. Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes, and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields. Moreover, these optimal asymmetric quantum error-correcting codes constructed in this paper are different from the ones in the literature.
Entanglement-assisted quantum-error-correcting (EAQEC) codes are quantum codes which use entanglement as a resource. These codes can provide better error correction than the (entanglement unassisted) codes derived from the traditional stabilizer formalism. In this paper, we provide a general method to construct EAQEC codes from cyclic codes. Afterwards, the method is applied to Reed-Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of EAQEC codes. Three families have been created: two families of EAQEC codes are maximal distance separable (MDS), and one is almost MDS or almost near MDS. The comparison of the codes in this paper is mostly based on the quantum Singleton bound.