Optimal quinary negacyclic codes with minimum distance four

Jinmei Fan; Yanhai Zhang

doi:10.3934/amc.2021043

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2021, Volume 0. Doi: 10.3934/amc.2021043

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Optimal quinary negacyclic codes with minimum distance four

Jinmei Fan and
Yanhai Zhang^,

College of Science, Guilin University of Technology, Guilin 541004, China

^* Corresponding author: Yanhai Zhang
^* Corresponding author: Yanhai Zhang

Received: January 31, 2021

Revised: June 30, 2021

Early access: September 2021

This work was supported by Guangxi Natural Science Foundation of China (No. 2018GXNSFBA281019) and the National Natural Science Foundation of China (No. 12061027)

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Abstract

Based on solutions of certain equations over finite yields, a necessary and sufficient condition for the quinary negacyclic codes with parameters $ [\frac{5^m-1}{2},\frac{5^m-1}{2}-2m,4] $ to have generator polynomial $ m_{\alpha^3}(x)m_{\alpha^e}(x) $ is provided. Several classes of new optimal quinary negacyclic codes with the same parameters are constructed by analyzing irreducible factors of certain polynomials over finite fields. Moreover, several classes of new optimal quinary negacyclic codes with these parameters and generator polynomial $ m_{\alpha}(x)m_{\alpha^e}(x) $ are also presented.

Keywords:

Mathematics Subject Classification: Primary: 94B15, 12E12; Secondary: 11T71.

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Table 1. Known optimal quinary negacyclic codes $ \mathcal{N}_5(1,e) $

Type	e	conditions	Reference
1)	$ 5^m-2 $	$ m $ is odd	[28]
2)	$ \frac{5^{k}+1}{2} $	gcd$ (k,2m)=1 $	[28]
3)	$ \frac{2\cdot 5^m-1}{3} $	$ m $ is odd	[28]
4)	$ 5^k+2 $	$ m=2k $, $ k $ is even	[28]
5)	$ \frac{5^{\frac{m+1}{2}}-1}{2}+\frac{5^m-1}{4} $	$ m>1 $ is odd	[28]
6)	$ \frac{5^m-1}{2}-3 $	$ m>1 $ is odd	[28]

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Table 2. Optimal quinary negacyclic codes $ \mathcal{N}_5(1,u) $

Type	u	conditions	Reference
ⅰ)	$ \frac{5^m-1}{2}+\frac{5^t+1}{2} $	$ 1\leq t<m,{\rm{gcd}}(t,m) = 1 $	Section 3.1
ⅱ)	$ 5^{\frac{m+1}{2}}+2 $	$ m \not\equiv 0\left( {{\rm{mod}}\;{\rm{3}}} \right) $	Section 3.2
ⅲ)	$ \frac{5^m-1}{2}+5^s+2 $	$ m = 2s, s \;{\rm{is\;even}} $	Section 3.3

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Table 3. Optimal quinary negacyclic codes $ \mathcal{N}_5(3,u) $

Type	u	conditions	Reference
ⅰ)	$ 5^m-4 $	$ m > 1\; {\rm{is\;odd}} $	Section 4.2
ⅱ)	$ \frac{5^m-1}{2}-1 $	$ m > 1\; {\rm{is\;odd}} $	Section 4.3
ⅲ)	$ \frac{5^m-1}{2}-9 $	$ m > 1\; {\rm{is\;odd}} $	Section 4.3

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Table 4. Weight distributions of $ \mathcal{N}_5(3,121)^{\bot}, \mathcal{N}_5(3,61)^{\bot} \;and\;\mathcal{N}_5(3,53)^{\bot} $

$ \mathcal{N}_5(3,121)^{\bot} $		$ \mathcal{N}_5(3,61)^{\bot} $		$ \mathcal{N}_5(3,53)^{\bot} $
Weight	Frequency	Weight	Frequency	Weight	Frequency
0	1	0	1	0	1
43	744	44	1488	44	1488
46	2232	46	2232	46	2232
47	744	48	744	48	744
49	2976	49	2232	49	2232
50	496	50	3224	50	3224
51	744	51	1488	51	1488
52	2976	52	1736	52	1736
53	744	53	1736	53	1736
54	744	57	744	57	744
48	2232
55	248
56	744

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