Generic constructions of MDS Euclidean self-dual codes via GRS codes

Ziteng Huang; Weijun Fang; Fang-Wei Fu; Fengting Li

doi:10.3934/amc.2021059

Article Contents

2023, Volume 17, Issue 6: 1453-1467. Doi: 10.3934/amc.2021059

This issue Previous Article New nonexistence results on perfect permutation codes under the hamming metric Next Article Differential spectra of a class of power permutations with Niho exponents

Generic constructions of MDS Euclidean self-dual codes via GRS codes

1.
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
2.
School of Cyber Science and Technology, Shandong University, Qingdao 266237, China
3.
Chern Institute of Mathematics and LPMC, and Tianjin Key Laboratory of Network, and Data Security Technology, Nankai University, Tianjin 300071, China
4.
Ant Group, Beijing 100020, China

^* Corresponding author: Weijun Fang
^* Corresponding author: Weijun Fang

Received: April 2021

Revised: September 2021

Early access: December 2021

Published: December 2023

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Abstract

Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square $ q $, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of $ q $-ary MDS Euclidean self-dual codes of lengths in the form $ s\frac{q-1}{a}+t\frac{q-1}{b} $, where $ s $ and $ t $ range in some interval and $ a, b \,|\, (q -1) $. In particular, for large square $ q $, our constructions take up a proportion of generally more than 34% in all the possible lengths of $ q $-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.

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Mathematics Subject Classification: Primary: 94B05; Secondary: 94B60.

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Table 1. Some known results about MDS self-dual codes with length n

$ q $	$ n $	References
$ q $ even	$ n\leq q $	[12]
$ q $ odd	$ n=q+1 $	[12], [17]
$ q=r^{2} $	$ n\leq r $	[17]
$ q=r^{2} $, \; $ r\equiv3(mod\;4) $	$ n=2tr $, $ t\leq \frac{r-1}{2} $	[17]
$ q\equiv1(mod\;4) $	$ 4^{n}n^{2}\leq q $	[17]
$ q\equiv3(mod\;4) $	$ n\equiv 0 (mod\;4) $ and $ (n-1)\mid (q-1) $	[21]
$ q\equiv1(mod\;4) $	$ (n-1)\mid (q-1) $	[21]
$ q=p^{m}\equiv1(mod\;4) $	$ n=p^{l}+1 $, $ l\leq m $	[6]
$ q=r^{s} $, $ r $ odd, $ s $ even	$ n=2tr^{l} $, $ 0 \leq l\leq s $ and $ 1 \leq t \leq\frac{r-1}{2} $	[6]
$ q=r^{s} $, $ r $ odd, $ s $ even	$ n=(2t+1)r^{l}+1 $, $ 0 \leq l\leq s $ and $ 0 \leq t \leq\frac{r-1}{2} $	[6]
$ q $ odd	$ (n-2) \mid (q-1) $, $ \eta(2-n)=1 $	[22], [6]
$ q $ odd	$ (n-1) \mid (q-1) $, $ \eta(1-n)=1 $	[22]
$ q\equiv1(mod\;4) $	$ n \mid (q-1) $	[22]
$ q=p^{m} $, $ p $ odd	$ n=p^{l}+1 $, $ l\|m $	[22]
$ q=p^{m} $, $ p $ odd	$ n=2p^{l} $, $ l<m $, $ \eta(-1)=1 $	[22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $	$ n=tr $, $ t $ even and $ 2t \mid (r-1) $	[22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $	$ n=tr $, $ t $ even, $ (t-1) \mid (r-1) $ and $ \eta(1-t)=1 $	[22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $	$ n=tr+1 $, $ t $ odd, $ t \mid (r-1) $ and $ \eta(t)=1 $	[22]
$ q=r^{s} $, $ r $ odd, $ s\geq2 $	$ n=tr+1 $, $ t $ odd, $ (t-1) \mid (r-1) $ and $ \eta(t-1)=\eta(-1)=1 $	[22]
$ q=r^{2} $, $ r $ odd	$ n=tr $, $ t $ even, $ 1\leq t\leq r $	[22]
$ q=r^{2} $, $ r $ odd	$ n=tr+1 $, $ t $ odd, $ 1\leq t\leq r $	[22]
$ q=r^{2} $, $ r $ odd	$ n=tm $, $ \frac{q-1}{m} $ even and $ 1 \leq t \leq \frac{r+1}{\gcd(r+1,m)} $	[8]
$ q=r^{2} $, $ r $ odd	$ n=tm+1 $, $ tm $ odd, $ m \mid (q-1) $ and $ 2 \leq t \leq \frac{r+1}{2\gcd(r+1,m)} $	[8]
$ q=r^{2} $, $ r $ odd	$ n=tm+2 $, $ tm $ even, $ m \mid (q-1) $ (except $ t $, $ m $ are even and $ r\equiv1(mod\;4) $), and $ 1 \leq t \leq \frac{r+1}{\gcd(r+1,m)} $	[8]
$ q=r^{2} $, $ r $ odd	$ n=tm $, $ \frac{q-1}{m} $ even, $ 1 \leq t \leq \frac{s(r-1)}{\gcd(s(r-1),m)} $, $ s $ even, $ s \mid m $ and $ \frac{r+1}{s} $ even	[8]
$ q=r^{2} $, $ r $ odd	$ n=tm+2 $, $ \frac{q-1}{m} $ even, $ \frac{r+1}{s} $ even, $ 1 \leq t \leq \frac{s(r-1)}{\gcd(s(r-1), m)} $, $ s $ even and $ s \mid m $	[8]
$ q=r^{2} $, $ r $ odd	$ n=tm $, $ \frac{q-1}{m} $ even, $ 1 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $	[18]
$ q=r^{2} $, $ r $ odd	$ n=tm+1 $, $ tm $ odd, $ m \mid (q-1) $, $ 2 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $	[18]
$ q=r^{2} $, $ r $ odd	$ n=tm+2 $, $ tm $ even, $ m \mid (q-1) $, $ 2 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $	[18]
$ q=r^{2} $, $ r\equiv1(mod\;4) $	$ n=s(r-1)+t(r+1) $, $ s $ even, $ 1 \leq s \leq \frac{r+1}{2} $ and $ 1 \leq t \leq \frac{r-1}{2} $	[9]
$ q=r^{2} $, $ r\equiv3(mod\;4) $	$ n=s(r-1)+t(r+1) $, $ s $ odd, $ 1 \leq s \leq \frac{r+1}{2} $ and $ 1 \leq t \leq \frac{r-1}{2} $	[9]

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Table 2. Proportion of number of possible lengths to $\frac{q}{2}$ ($N$ is the number of possible lengths)

$r$	$q$	$N/(\frac{q}{2}$) of Table 1 (except [9])	$N/(\frac{q}{2})$ of [9]	$N/\frac{q}{2}$ of us	number of new lengths
149	22201	$11.89\%$	$25\%$	$38.61\%$	775
151	22801	$13.16\%$	$25\%$	$34.95\%$	676
157	24649	$10.18\%$	$25\%$	$34.95\%$	758
163	26569	$10.67\%$	$25\%$	$34.28\%$	828
167	27889	$13.90\%$	$25\%$	$34.27\%$	704

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