Two new classes of Hermitian self-orthogonal non-GRS MDS codes and their applications

Gaojun Luo; Xiwang Cao; Martianus Frederic Ezerman; San Ling

doi:10.3934/amc.2022070

Article Contents

2022, Volume 16, Issue 4: 921-933. Doi: 10.3934/amc.2022070

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Two new classes of Hermitian self-orthogonal non-GRS MDS codes and their applications

1.
School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link 637371, Singapore
2.
Department of Mathematics and Key Laboratory of Mathematical Modeling and High Performance Computing of Air Vehicles, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

^* Corresponding author: Gaojun Luo
^* Corresponding author: Gaojun Luo

Dedicated to Professor Cunsheng Ding on his 60^th Birthday

Received: March 2022

Revised: July 2022

Early access: September 2022

Published: November 2022

Nanyang Technological University Grant Number 04INS000047C230GRT01 supports the research carried out by G. Luo, M. F. Ezerman, and S. Ling. National Natural Science Foundation of China (Grant No. 11971175) partially supports S. Ling. National Natural Science Foundation of China (Grant No. 12171241) provides funding for X. Cao and G. Luo

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Abstract

Hermitian self-orthogonal codes form an essential ingredient in the construction of quantum stabilizer codes. In this paper, we present two new classes of Hermitian self-orthogonal MDS codes from twisted generalized Reed-Solomon codes. The constructed codes are monomially inequivalent to generalized Reed-Solomon codes. Two new classes of Hermitian LCD MDS codes can then be derived. Finally, based on the constructed Hermitian self-orthogonal MDS codes, we present two classes of quantum MDS codes with new parameters.

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Mathematics Subject Classification: Primary: 94B27, 81P73; Secondary: 14G50.

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Table 1. Parameters of known quantum MDS codes of length $ s(q+1) $ or $ s(q-1) $. Here $ s $ can be a rational number as long as the code length remains an integer

No.	Length	Distance	Constraints	Reference(s)
$ 1 $	$ \frac{q^2-1}{2} $	$ 3\leq d\leq q $	$ q $ is odd	[27]
$ 2 $	$ \ell(q+1) $	$ 2\leq d\leq \frac{q+1}{2}+\ell $	$ \ell $ is odd and $ \ell\mid(q-1) $
$ 3 $	$ 2\ell(q+1) $	$ 2\leq d\leq \frac{q+1}{2}+2\ell $	$ \ell $ is odd, $ q\equiv 1\pmod{4} $
			and $ \ell\mid(q-1) $
$ 4 $	$ \frac{q^2-1}{r} $	$ 2\leq d\leq \frac{q+1}{2}+\frac{q+1}{2r}-1 $	$ r $ is odd	[39]
$ 5 $	$ \frac{q^2-1}{r} $	$ 2\leq d\leq \frac{q+1}{2}+\frac{q+1}{2}-1 $	$ r $ is even	[27,39]
$ 6 $	$ \frac{q^2-1}{r} $	$ 2\leq d\leq \frac{(q+1)(r+1)}{2r}-2\ell $	$ r\mid(q+1) $ and $ r\in\{3, 5, 7\} $	[9]
$ 7 $	$ 2\ell(q+1) $	$ 2\leq d\leq 6\ell+1 $	$ 8\mid(q+1) $ and $ \ell\mid(q+1) $
$ 8 $	$ 3(q-1) $	$ 2\leq d\leq \frac{q+5}{2} $	$ q $ is odd and $ 9\mid(q+1) $
$ 9 $	$ \frac{r(q^2-1)}{2\ell+1} $	$ d\leq \frac{q(\ell+1)(2\ell+1)}{2r}-\frac{\ell}{2\ell+1} $	$ 1\leq r\leq 2\ell+1 $, $ \gcd(r, q)=1 $	[24]
			and $ q\equiv -1\pmod{2\ell+1} $
$ 10 $	$ (2\ell+2) \left(\frac{q^2-1}{r}\right) $	$ 2\leq d\leq \frac{(q+1)(r+2\ell+3)}{2r}-1 $	$ 0\leq \ell\leq \frac{r-3}{2} $, $ r $ is odd	[38]
			and $ q=rm-1, m>1 $
$ 11 $	$ (2\ell+2) \left(\frac{q^2-1}{r}\right) $	$ 2\leq d\leq \frac{(q+1)(r+2\ell+2)}{2r}-1 $	$ 0\leq \ell\leq \frac{r-3}{2} $, $ r $ is even
			and $ q=rm-1, m>1 $
$ 12 $	$ (\ell+1) \left(\frac{q^2-1}{r}\right) $	$ 2\leq d\leq \frac{(q+1)(\ell+1)}{r} $	$ 0\leq \ell\leq \frac{r-3}{2} $ and $ q=rm+1 $
$ 13 $	$ \ell(q-1) $	$ 2\leq d\leq \left\lfloor\frac{\ell q-1}{q+1}\right\rfloor+1 $	$ 1\leq \ell\leq q $	[19]
$ 14 $	$ \ell(q+1) $	$ 2\leq d\leq \ell $	$ 1\leq \ell\leq q-1 $
$ 15 $	$ (2\ell+1) \left(\frac{q^2-1}{2r}\right) $	$ 2\leq d\leq (r+\ell) \left(\frac{q+1}{2r}\right)-1 $	$ 2r\mid(q+1) $ and $ 1\leq \ell\leq r-1 $	[12]
$ 16 $	$ (t+1) \left(\frac{q+1}{r}\right) $	$ 4\leq d\leq\left\lfloor\frac{t(q+1)-r}{r(q+1)}\right\rfloor+1 $	$ r\mid(q+1) $ and $ r<t\leq q-2 $	Theorem 4.2
$ 17 $	$ (t+1) \left(\frac{q-1}{h}\right) $	$ 4\leq d\leq\left\lfloor\frac{tq-h-1}{h(q+1)}\right\rfloor+1 $	$ h\mid(q-1) $, $ h+1\leq t\leq q-h $	Theorem 4.3
			and $ h\mid(q-h-t) $

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Table 2. Parameters of known non-GRS MDS codes which are Hermitian linear complementary dual over $ {\mathbb F}_{q^2} $, with $ \upsilon_p(N) $ denoting the largest power of $ p $ that divides $ N $

No.	Length	Dimension	Constraints	Reference
$ 1 $	$ 2k $	$ k $	$ q $ is odd, $ {\mathbb F}_s $ is a subfield of $ {\mathbb F}_{q^2} $, $ k\mid(s-1) $,	[40]
			and $ 2<k<(s-1)/2 $
$ 2 $	$ k+3 $	$ k $	$ q $ is odd, $ k\mid(q-1) $, and $ \gcd(k+1, q)=1 $
$ 3 $	$ 2k+2 $	$ k $	$ q $ is odd, $ k\mid(q-1) $, and there exists an odd
			prime $ p $ such that $ \upsilon_p(k)<\upsilon_p(q-1) $
$ 4 $	$ (t+1) \frac{q+1}{r} $	$ k $	$ r \mid (q+1) $, $ 1< r< t \leq q-2 $, and $ 2< k \leq \left \lfloor\frac{t}{r} - \frac{1}{q+1}\right\rfloor $	Corollary 1
$ 5 $	$ (t+1) \frac{q-1}{h} $	$ k $	$ h \mid \gcd(q-1, q-h-t) $, $ 2< h+1\leq t\leq q-h $,	Corollary 2
			and $ 2< k\leq \left\lfloor\frac{t \, q-h-1}{h(q+1)}\right\rfloor $

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