New quantum codes from matrix-product codes over small fields
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New quantum codes from matrix-product codes over small fields Hao Song1 · Ruihu Li1
· Yang Liu1 · Guanmin Guo1
Received: 5 July 2019 / Accepted: 10 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we provide methods for constructing Hermitian dual-containing (HDC) matrix-product codes over Fq 2 from some non-singular matrices and a special sequence of HDC codes and determine parameters of obtained matrix-product codes when the input matrix and sequence of HDC codes satisfy some conditions. Then, using some 4 nested HDC BCH codes with lengths n = q a−1 (a = 1 or a = q ± 1), we construct some HDC matrix-product codes with lengths N = 2n or 3n and derive nonbinary quantum codes with length N from these matrix-product codes via Hermitian construction. Four classes of quantum codes over Fq (3 ≤ q ≤ 5) are presented, whose parameters are better than those in the literature. Besides, some of our new quantum codes can exceed the quantum Gilbert-Varshamov (GV) bound. Keywords Matrix-product code · Hermitian dual-containing condition · BCH code · Quantum error-correcting code
This work is supported by the National Natural Science Foundation of China under Grant Nos. 11471011 and 11801564, the Natural Science Foundation of Shaanxi province under Grant No. 2019JM-271, the Natural Science Foundation of Department of Basic Sciences in Air Force Engineering University under Grant No. JK2019105.
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Ruihu Li [email protected] Hao Song [email protected] Yang Liu [email protected] Guanmin Guo [email protected]
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Department of Basic Sciences, Air Force Engineering University, Xi’an 710051, Shaanxi, People’s Republic of China 0123456789().: V,-vol
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1 Introduction As we all know, the theory of quantum error-correcting codes (QECCs) has been extensively studied in the literature [1–14]. Constructing QECCs with good parameters from classical linear codes is by far a hot issue. One of the well-known constructions of QECCs is Hermitian construction (see Theorem 1), which focuses on HDC classical codes. As a special class of linear codes, matrix-product codes, which also can be employed to produce quantum codes, attract coding theorists’ attentions since the introduction by Blackmore and Norton [15]. Matrix-product codes over finite fields can be viewed as a generalization of the Plotkin’s (u|u +v)-construction and the ternary (u +v +w|2u +v|u)-construction, see Refs. [16–20]. It is worth mentioning that Mankean and Jitman gave a necessary condition that matrix-product construction are Euclidean and Hermitian self-orthogonal linear codes [21,22]. In [23], Galindo et al. firstly constructed quantum codes from matrix-product codes via Steane’s enlargement construction. Three classes of Hermitian self-orthogonal matrix-product codes with lengths n = 2(q 2 ± 1) and n = 2q 2 were used to obtain q-ary QECCs, whose minimal distance did not exceed q + 1. By discussing a class of matrices, Liu [24] further generalized those previous results i
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