Some new quantum MDS codes derived from generalized Reed-Solomon codes

Xueying Shi; Qin Yue

doi:10.3934/amc.2022069

Article Contents

2022, Volume 16, Issue 4: 947-959. Doi: 10.3934/amc.2022069

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Some new quantum MDS codes derived from generalized Reed-Solomon codes

Xueying Shi^1, and
Qin Yue^2, ,

1.
Department of Mathematics, Taizhou University, Taizhou 225300, China
2.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, China

^*Corresponding author: Qin Yue
^*Corresponding author: Qin Yue

Received: March 2022

Revised: June 2022

Early access: September 2022

Published: November 2022

The work is supported by the National Natural Science Foundation of China (No. 12001396, No. 61772219), the Natural Science Foundation of Jiangsu Province of China (No. BK20200268), the Natural Science Foundation of the Jiangsu Higher Education Institution of China (No. 20KJB110005) and Qing Lan Project of the Jiangsu Higher Education Institutions.

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Abstract

Quantum maximum-distance-separable (MDS) codes are very important in coding theory. In this paper, we construct three classes of new quantum MDS codes with large minimum distance by classical Hermitian self-orthogonal generalized Reed-Solomon codes. Furthermore, these quantum MDS codes have more flexible lengths and larger minimum distance than those of some known quantum MDS codes.

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Mathematics Subject Classification: 81P70.

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Table 1. Quantum MDS Codes

$h_1$	$h_2$	$r_1$	$r_2$	${[[n, k, d]]_q}$	$d$
$7$	$3$	$6$	$1$	${[[\frac{2(q^2-1)}{3}, \frac{2(q^2-1)}{3}-2d+2, d]]_q}$	$2\leq d\leq \frac{5q+9}{7}$
$5$	$7$	$4$	$2$	${[[\frac{4(q^2-1)}{7}, \frac{4(q^2-1)}{7}-2d+2, d]]_q}$	$2\leq d\leq \frac{3q+7}{5}$

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Table 2. Quantum MDS Codes

$h_1$	$h_2$	$r_1$	$r_2$	${[[n, k, d]]_q}$	$d$
$6$	$9$	$2$	$4$	${[[\frac{4(q^2-1)}{9}, \frac{4(q^2-1)}{9}-2d+2, d]]_q}$	$2\leq d\leq \frac{2q+8}{3}$
$5$	$11$	$2·$	$5$	${[[\frac{6(q^2-1)}{11}, \frac{6(q^2-1)}{11}-2d+2, d]]_q}$	$2\leq d\leq \frac{8q+30}{11}$

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Table 3. Quantum MDS Codes

$h_1$	$h_2$	$r_1$	$r_2$	${[[n, k, d]]_q}$	$d$
$12$	$19$	$5$	$15$	${[[\frac{8(q^2-1)}{19}, \frac{8(q^2-1)}{19}-2d+2, d]]_q}$	$2\leq d\leq \frac{3q+5}{4}$ if $\frac{q-1}{12}$ is odd
$16$	$5$	$6$	$4$	${[[\frac{7(q^2-1)}{16}, \frac{7(q^2-1)}{16}-2d+2, d]]_q}$	$2\leq d\leq \frac{11q+21}{16}$ if $\frac{q-1}{16}$ is odd

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