Article Contents

2023, Volume 17, Issue 4: 979-993. Doi: 10.3934/amc.2021033

This issue Previous Article On the number of distinct k-decks: Enumeration and bounds Next Article New type i binary [72, 36, 12] self-dual codes from composite matrices and R₁ lifts

Two classes of new optimal ternary cyclic codes

1.
College of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, 224003, China
2.
School of Mathematical Sciences, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
3.
Key Laboratory of Mathematical Modeling and High Performance Computing of Air Vehicles, (NUAA), MIIT, Nanjing, 210016, China
4.
School of Mathematics, Southeast University, Nanjing, 210016, China

^* Corresponding author: Xiwang Cao
^* Corresponding author: Xiwang Cao

Received: December 2020

Revised: June 2021

Early access: August 2021

Published: August 2023

The first author is supported by the National Natural Science Foundation of China (No. 12001475) and the Natural Science Foundation of Jiangsu Province (No. BK20201059). The second author is supported by the National Natural Science Foundation of China (No. 12171241). The third author is supported by the National Natural Science Foundation of China (No. 11801070).

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Abstract

Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting subject of study in recent years. The construction of optimal cyclic codes over finite fields is important as they have maximal minimum distance once the length and dimension are given. In this paper, we present two classes of new optimal ternary cyclic codes $ \mathcal{C}_{(2,v)} $ by using monomials $ x^2 $ and $ x^v $ for some suitable $ v $ and explain the novelty of the codes. Furthermore, the weight distribution of $ \mathcal{C}_{(2,v)}^{\perp} $ for $ v = \frac{3^{m}-1}{2}+2(3^{k}+1) $ is determined.

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Mathematics Subject Classification: 94B15, 11T71.

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Table 1. Value distribution of $ T(a,b) $

Value	Frequency
0	$ (3^{m}-1)(3^{m}-2\cdot3^{m-1}+1) $
$ 2\cdot3^{m} $	$ 1 $
$ 3^{\frac{m+1}{2}} $	$ (3^{m}-1)(3^{m-1}+3^{\frac{m-1}{2}}) $
$ - 3^{\frac{m+1}{2}} $	$ (3^{m}-1)(3^{m-1}-3^{\frac{m-1}{2}}) $

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Table 2. Weight Distribution of $ \mathcal{C}_{(2,v)}^{\perp} $

Weight	Frequency
$ 0 $	1
$ 3^{m}-3^{m-1}-3^{\frac{m-1}{2}} $	$ (3^{m}-1)(3^{m-1}+3^{\frac{m-1}{2}}) $
$ 2\cdot 3^{m-1} $	$ (3^{m}-1)(3^{m}-2\cdot3^{m-1}+1) $
$ 3^{m}-3^{m-1}+ 3^{\frac{m-1}{2}} $	$ (3^{m}-1)(3^{m-1}-3^{\frac{m-1}{2}}) $

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Table 3. Known optimal ternary cyclic codes $ C_{1,e} $

Conditions	Case
$ m $ is odd, $ e $ is even, $ e(3^{s}-1) \equiv 3^{t}-1 \pmod{ 3^{m}-1} $, $ 1\leq s,t\leq m-1 $, $ \gcd(m,s)=\gcd(m,t)=1 $, $ \gcd(3^{m}-1,e-1)=1 $.[22]	$ 1) $
$ e(3^{s}+1) \equiv 3^{t}+1 \pmod {3^{m}-1} $, $ 0\leq s,t\leq m-1 $, $ \gcd(3^{m}-1,e-1)= $ $ \gcd(3^{m}-1,3^{t}-e)=1 $, $ m $ is either odd or even with $ \gcd(m,t)=1 $ and $ \frac{m}{\gcd(m,s)} $ is odd.[22]	$ 2) $
$ e \equiv \frac{3^{m}-1}{2}+3^{s}+1 \pmod {3^{m}-1} $, $ m $ is even, $ m/\gcd(m,s) $ is odd.[22]	$ 3) $
$ e \equiv \frac{3^{m}-1}{2}+3^{s}-1 \pmod {3^{m}-1} $, $ m $ is even, $ \gcd(m,s)= $ $ \gcd(3^{s}-2,3^{m}-1)=1 $, $ s=1,3,5,7,9 $.[22]	$ 4) $
$ e = 3^{s}+5 $, $ m\equiv 0 \pmod 4 $ and $ s=\frac{m}{2} $ or $ m\equiv 2 \pmod 4 $ and $ s=\frac{m+2}{2} $.[10]	$ 5) $
$ m $ is odd and $ 1\leq s <m $, $ e = \frac{3^{s}+7}{2} $ if s is even, $ e =\frac{3^{m}-1}{2}+ \frac{3^{s}+7}{2} $ if s is odd.[28]	$ 6) $
$ e = \frac{3^{\frac{m+1}{2}}+5}{2} $ if $ m \equiv 1 \pmod 4 $, $ e =\frac{3^{m}-1}{2}+ \frac{3^{\frac{m+1}{2}}+5}{2} $ if $ m \equiv 3 \pmod 4 $. [28]	$ 7) $
$ m $ is odd, $ e $ is even, $ e(3^{s}+1) \equiv \frac{3^{m}+1}{2} \pmod{ 3^{m}-1} $, $ 0\leq s \leq m-1 $.[28]	$ 8) $
$ 3 \nmid m $, $ 5e \equiv 2 \pmod{ 3^{m}-1} $.[28]	$ 9) $
$ 5 \nmid m $, $ 7e \equiv 2 \pmod{ 3^{m}-1} $, $ \gcd(m,6)=1 $ or $ m\equiv 3 \pmod{ 6} $.[28]	$ 10) $
$ m >2 $, $ 3 \nmid m $ and $ 5 \nmid m $, $ 5e \equiv 4 \pmod{ 3^{m}-1} $.[28]	$ 11) $

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Table 4. Known optimal ternary cyclic codes $ C_{1,e} $

$ e $	Conditions	Case
2	$ m \geq 2 $[1]	$ 12) $
$ \frac{3^{s}+1}{2} $	$ s $ is odd, $ m \geq 2 $, $ \gcd(m,s)=1 $.[1]	$ 13) $
$ 3^{s}+1 $	$ m \geq 2 $, $ m/\gcd(m,s) $ is odd.[1]	$ 14) $
$ 3^{m-1}-1, \frac{3^{m}+1}{4}+\frac{3^{m}-1}{2} $	$ m \geq 3 $, $ m $ is odd.[4]	$ 15) $
$ 3^{\frac{m+1}{2}}-1 $	$ m $ is odd.[4]	$ 16) $
$ \frac{3^{m}-3}{2} $	$ m \geq 5 $, $ m $ is odd.[4]	$ 17) $
$ (3^{\frac{m+1}{4}}-1)(3^{\frac{m+1}{2}}+1) $	$ m \equiv 3 \pmod 4 $.[4]	$ 18) $
$ \frac{3^{(m+1)/2}-1}{2}+\frac{3^{m}-1}{2}, \frac{3^{m+1}-1}{8}+\frac{3^{m}-1}{2} $	$ m \equiv 1 \pmod 4 $.[4]	$ 19) $
$ \frac{3^{(m+1)/2}-1}{2}, \frac{3^{m+1}-1}{8} $	$ m \equiv 3 \pmod 4 $.[4]	$ 20) $
$ \frac{3^{s}-1}{2} $	$ m $ is odd, $ s $ is even, $ \gcd(m,s)=\gcd(m,s-1)=1 $.[4]	$ 21) $
$ 3^{s}-1 $	$ \gcd(m,s)=\gcd(3^{m}-1,3^{s}-2)=1 $.[4]	$ 22) $
$ \frac{3^{m-1}}{2}-2, \frac{3^{m-1}}{2}+10 $	$ m \equiv 2 \pmod{4} $.[13]	$ 23) $
$ \frac{3^{m-1}}{2}-5, \frac{3^{m-1}}{2}+7 $	$ m $ is odd.[13]	$ 24) $
$ 2(3^{m-1}-1), 5(3^{m-1}-1), 16 $	$ m $ is odd, $ 3\nmid m $.[13]	$ 25) $
$ 2(3^{s}+1) $	$ m $ is odd.[14]	$ 26) $

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Table 5. Known optimal ternary cyclic codes $ C_{u,v'} $

$ u $	$ v' $	Conditions	Case
$ \frac{3^{m}+1}{2} $	$ \frac{3^{s}+1}{2} $	$ m $ is odd, $ s $ is even and $ \gcd(m,s)=1 $.[30]	$ 27) $
	$ 2 \cdot 3^{\frac{m-1}{2}}+1 $	$ m $ is odd.[6]	$ 28) $
	$ 3^{s}+2 $	$ m $ is odd, such that $ 9\nmid m $ and $ 4s \equiv1 \pmod m $.[25]	$ 29) $

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