Construction of new quantum codes via Hermitian dual-containing matrix-product codes
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Construction of new quantum codes via Hermitian dual-containing matrix-product codes Meng Cao1 · Jianlian Cui1 Received: 25 September 2019 / Accepted: 29 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In 2001, Blackmore and Norton introduced an important tool called matrix-product codes, which turn out to be very useful to construct new quantum codes of large lengths. To obtain new and good quantum codes, we first give a general approach to construct matrix-product codes being Hermitian dual-containing and then provide the constructions of such codes in the case s|(q 2 − 1), where s is the number of the constituent codes in a matrix-product code. For s|(q +1), we construct such codes with lengths more flexible than the known ones in the literature. For s|(q 2 − 1) and s(q + 1), such codes are constructed in an unusual manner; some of the constituent codes therein are not required to be Hermitian dual-containing. Accordingly, by Hermitian construction, we present two procedures for acquiring quantum codes. Finally, we list some good quantum codes, many of which improve those available in the literature or add new parameters. Keywords Hermitian dual-containing codes · Matrix-product codes · Generalized Reed–Solomon codes · Extended generalized Reed–Solomon codes · Quantum codes
1 Introduction The concept of quantum error-correcting code is one of the most important concepts in quantum computation and quantum communication. It was introduced to deal with the problems of decoherence and quantum noise in quantum information. Since the appearance of the groundbreaking works [4,32,34], the exploration of quantum codes has got great development. Among these studies, the construction of good quantum codes is a significant topic. The famous Hermitian construction
Project supported by the National Natural Science Foundation of China (Grant No. 11271217).
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Jianlian Cui [email protected] Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 0123456789().: V,-vol
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[2] tells us that an [[n, 2k − n, ≥ d]]q quantum code can be generated by a Hermitian dual-containing [n, k, d]q 2 code. Therefore, the construction of q 2 -ary Hermitian dual-containing codes is an effective way to obtain q-ary quantum codes (e.g., see [1,16,21,23,24,26,27,30,37,38]). In 2001, Blackmore and Norton [3] proposed a new class of codes called matrixproduct codes. From then on, some scholars considered several kinds of such codes and obtained many meaningful results related to their minimum distances, duals and decoding algorithms, see [5,6,9,10,17,18,33,35]. In recent years, seeking for the matrix-product codes satisfying Euclidean or Hermitian dual-containing is a new manner to construct good quantum codes. For instance, Galindo et al. [13] obtained some new and good quantum stabilizer codes by the Euclidean dual-containing matrixproduct codes with constituent codes being Reed–Muller [7,22] or hyperbolic [14,31] or affine variety [1
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