doi: 10.3934/amc.2021033
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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

Received  December 2020 Revised  June 2021 Early access August 2021

Fund Project: 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).

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.

Citation: Yan Liu, Xiwang Cao, Wei Lu. Two classes of new optimal ternary cyclic codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021033
References:
[1]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.  Google Scholar

[2]

R. S. Coulter, The number of rational points of a class of Artin-Schreier curves, Finite Fields Appl., 8 (2002), 397-413.  doi: 10.1006/ffta.2001.0348.  Google Scholar

[3]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  doi: 10.1109/tit.1975.1055435.  Google Scholar

[4]

C. Ding and T. Helleseth, Optimal ternary cyclic codes from monomials, IEEE Trans. Inf. Theory, 59 (2013), 5898-5904.  doi: 10.1109/TIT.2013.2260795.  Google Scholar

[5]

C. Ding and S. Ling, A $q$-polynomial approach to cyclic codes, Finite Fields Appl., 20 (2013), 1-14.  doi: 10.1016/j.ffa.2012.12.005.  Google Scholar

[6]

C. FanN. Li and Z. Zhou, A class of optimal ternary cyclic codes and their duals, Finite Fields Appl., 37 (2016), 193-202.  doi: 10.1016/j.ffa.2015.10.004.  Google Scholar

[7]

W. FangJ. Wen and F. Fu, A $q$-polynomial approach to constacyclic codes, Finite Fields Appl., 47 (2017), 161-182.  doi: 10.1016/j.ffa.2017.06.009.  Google Scholar

[8]

K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409.  doi: 10.1016/j.ffa.2007.03.003.  Google Scholar

[9]

T. Feng, On cyclic codes of length $2^{2^{r}}-1$ with two zeros whose dual codes have three weights, Des. Codes Crypogr., 62 (2012), 253-258.  doi: 10.1007/s10623-011-9514-0.  Google Scholar

[10]

D. Han and H. Yan, On an open problem about a class of optimal ternary cyclic codes, Finite Fields Appl., 59 (2019), 335-343.  doi: 10.1016/j.ffa.2019.07.002.  Google Scholar

[11] W. Huffman and V. Pless, Foundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[12]

C. Li and Q. Yue, Weight distributions of two classes of cyclic codes with respect to two distinct order elements, IEEE Trans. Inf. Theory, 60 (2014), 296-303.  doi: 10.1109/TIT.2013.2287211.  Google Scholar

[13]

N. LiC. LiT. HellesethC. Ding and X. Tang, Optimal ternary cyclic codes with minimun distance four and five, Finite Fields Appl., 30 (2014), 100-120.  doi: 10.1016/j.ffa.2014.06.001.  Google Scholar

[14]

N. Li, Z. Zhou and T. Helleseth, On a conjecture about a class of optimal ternary cyclic codes, Signal Design and its Applications in Communications (IWSDA), seventh Intertional Workshop on Signal, (2015). doi: 10.1109/IWSDA.2015.7458415.  Google Scholar

[15]

D. LiaoX. KaiS. Zhu and P. Li, A class of optimal cyclic codes with two zeros, IEEE Commun. Lett., 23 (2019), 1293-1296.  doi: 10.1109/LCOMM.2019.2921330.  Google Scholar

[16]

Y. LiuH. Yan and C. Liu, A class of six-weight cyclic codes and their weight distribution, Des. Codes Crypogr., 77 (2015), 1-9.  doi: 10.1007/s10623-014-9984-y.  Google Scholar

[17]

Y. LiuX. Cao and W. Lu, On some conjectures about optimal ternary cyclic codes, Des. Codes Crypogr., 88 (2020), 297-309.  doi: 10.1007/s10623-019-00679-w.  Google Scholar

[18]

J. Luo and K. Feng, Cyclic codes and sequences form generalized Coulter-Matthews function, IEEE Trans. Inf. Theory, 54 (2008), 5345-5353.  doi: 10.1109/TIT.2008.2006394.  Google Scholar

[19]

C. MaL. ZengY. LiuD. Feng and C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 397-402.  doi: 10.1109/TIT.2010.2090272.  Google Scholar

[20]

M. ShiS. Yang and S. Zhu, Good p-ary quasicyclic codes from cyclic codes over $\mathbb{F}_{p}+v\mathbb{F}_{p}$, J. Syst. Sci. Complex., 25 (2012), 375-384.  doi: 10.1007/s11424-012-0076-7.  Google Scholar

[21]

M. Shi and Y. Zhang, Quasi-twisted codes with constacyclic constituent codes, Finite Fields Appl., 39 (2016), 159-178.  doi: 10.1016/j.ffa.2016.01.010.  Google Scholar

[22]

L. Wang and G. Wu, Several classes of optimal ternary cyclic codes with minimal distance four, Finite Fields Appl., 40 (2016), 126-137.  doi: 10.1016/j.ffa.2016.03.007.  Google Scholar

[23]

M. Xiong and N. Li, Optimal cyclic codes with generalized Niho-type zeros and the weight distribution, IEEE Trans. Inf. Theory, 61 (2015), 4914-4922.  doi: 10.1109/TIT.2015.2451657.  Google Scholar

[24]

G. XuX. Cao and S. Xu, Optimal $p$-ary cyclic codes with minimum distance four from monomials, Cryptogr. Commun., 8 (2016), 541-554.  doi: 10.1007/s12095-015-0159-0.  Google Scholar

[25]

H. YanZ. Zhou and X. Du, A family of optimal ternary cyclic codes from the Niho-type exponent, Finite Fields Appl., 54 (2018), 101-112.  doi: 10.1016/j.ffa.2018.08.004.  Google Scholar

[26]

S. YangZ. Yao and C. Zhao, The weight distributions of two classes of $p$-ary cyclic codes with few weights, Finite Fields Appl., 44 (2017), 76-91.  doi: 10.1016/j.ffa.2016.11.004.  Google Scholar

[27]

X. ZengL. HuW. JiangQ. Yue and X. Cao, The weight distribution of a class of $p$-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.  doi: 10.1016/j.ffa.2009.12.001.  Google Scholar

[28]

Z. Zha and L. Hu, New classes of optimal ternary cyclic codes with minimum distance four, Finite Fields Appl., 64 (2020), 1016710. doi: 10.1016/j.ffa.2020.101671.  Google Scholar

[29]

Y. ZhouX. KaiS. Zhu and J. Li, On the minimum distance of negacyclic codes with two zeros, Finite Fields Appl., 55 (2019), 134-150.  doi: 10.1016/j.ffa.2018.09.006.  Google Scholar

[30]

Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93.  doi: 10.1016/j.ffa.2013.08.005.  Google Scholar

show all references

References:
[1]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.  Google Scholar

[2]

R. S. Coulter, The number of rational points of a class of Artin-Schreier curves, Finite Fields Appl., 8 (2002), 397-413.  doi: 10.1006/ffta.2001.0348.  Google Scholar

[3]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  doi: 10.1109/tit.1975.1055435.  Google Scholar

[4]

C. Ding and T. Helleseth, Optimal ternary cyclic codes from monomials, IEEE Trans. Inf. Theory, 59 (2013), 5898-5904.  doi: 10.1109/TIT.2013.2260795.  Google Scholar

[5]

C. Ding and S. Ling, A $q$-polynomial approach to cyclic codes, Finite Fields Appl., 20 (2013), 1-14.  doi: 10.1016/j.ffa.2012.12.005.  Google Scholar

[6]

C. FanN. Li and Z. Zhou, A class of optimal ternary cyclic codes and their duals, Finite Fields Appl., 37 (2016), 193-202.  doi: 10.1016/j.ffa.2015.10.004.  Google Scholar

[7]

W. FangJ. Wen and F. Fu, A $q$-polynomial approach to constacyclic codes, Finite Fields Appl., 47 (2017), 161-182.  doi: 10.1016/j.ffa.2017.06.009.  Google Scholar

[8]

K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409.  doi: 10.1016/j.ffa.2007.03.003.  Google Scholar

[9]

T. Feng, On cyclic codes of length $2^{2^{r}}-1$ with two zeros whose dual codes have three weights, Des. Codes Crypogr., 62 (2012), 253-258.  doi: 10.1007/s10623-011-9514-0.  Google Scholar

[10]

D. Han and H. Yan, On an open problem about a class of optimal ternary cyclic codes, Finite Fields Appl., 59 (2019), 335-343.  doi: 10.1016/j.ffa.2019.07.002.  Google Scholar

[11] W. Huffman and V. Pless, Foundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[12]

C. Li and Q. Yue, Weight distributions of two classes of cyclic codes with respect to two distinct order elements, IEEE Trans. Inf. Theory, 60 (2014), 296-303.  doi: 10.1109/TIT.2013.2287211.  Google Scholar

[13]

N. LiC. LiT. HellesethC. Ding and X. Tang, Optimal ternary cyclic codes with minimun distance four and five, Finite Fields Appl., 30 (2014), 100-120.  doi: 10.1016/j.ffa.2014.06.001.  Google Scholar

[14]

N. Li, Z. Zhou and T. Helleseth, On a conjecture about a class of optimal ternary cyclic codes, Signal Design and its Applications in Communications (IWSDA), seventh Intertional Workshop on Signal, (2015). doi: 10.1109/IWSDA.2015.7458415.  Google Scholar

[15]

D. LiaoX. KaiS. Zhu and P. Li, A class of optimal cyclic codes with two zeros, IEEE Commun. Lett., 23 (2019), 1293-1296.  doi: 10.1109/LCOMM.2019.2921330.  Google Scholar

[16]

Y. LiuH. Yan and C. Liu, A class of six-weight cyclic codes and their weight distribution, Des. Codes Crypogr., 77 (2015), 1-9.  doi: 10.1007/s10623-014-9984-y.  Google Scholar

[17]

Y. LiuX. Cao and W. Lu, On some conjectures about optimal ternary cyclic codes, Des. Codes Crypogr., 88 (2020), 297-309.  doi: 10.1007/s10623-019-00679-w.  Google Scholar

[18]

J. Luo and K. Feng, Cyclic codes and sequences form generalized Coulter-Matthews function, IEEE Trans. Inf. Theory, 54 (2008), 5345-5353.  doi: 10.1109/TIT.2008.2006394.  Google Scholar

[19]

C. MaL. ZengY. LiuD. Feng and C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 397-402.  doi: 10.1109/TIT.2010.2090272.  Google Scholar

[20]

M. ShiS. Yang and S. Zhu, Good p-ary quasicyclic codes from cyclic codes over $\mathbb{F}_{p}+v\mathbb{F}_{p}$, J. Syst. Sci. Complex., 25 (2012), 375-384.  doi: 10.1007/s11424-012-0076-7.  Google Scholar

[21]

M. Shi and Y. Zhang, Quasi-twisted codes with constacyclic constituent codes, Finite Fields Appl., 39 (2016), 159-178.  doi: 10.1016/j.ffa.2016.01.010.  Google Scholar

[22]

L. Wang and G. Wu, Several classes of optimal ternary cyclic codes with minimal distance four, Finite Fields Appl., 40 (2016), 126-137.  doi: 10.1016/j.ffa.2016.03.007.  Google Scholar

[23]

M. Xiong and N. Li, Optimal cyclic codes with generalized Niho-type zeros and the weight distribution, IEEE Trans. Inf. Theory, 61 (2015), 4914-4922.  doi: 10.1109/TIT.2015.2451657.  Google Scholar

[24]

G. XuX. Cao and S. Xu, Optimal $p$-ary cyclic codes with minimum distance four from monomials, Cryptogr. Commun., 8 (2016), 541-554.  doi: 10.1007/s12095-015-0159-0.  Google Scholar

[25]

H. YanZ. Zhou and X. Du, A family of optimal ternary cyclic codes from the Niho-type exponent, Finite Fields Appl., 54 (2018), 101-112.  doi: 10.1016/j.ffa.2018.08.004.  Google Scholar

[26]

S. YangZ. Yao and C. Zhao, The weight distributions of two classes of $p$-ary cyclic codes with few weights, Finite Fields Appl., 44 (2017), 76-91.  doi: 10.1016/j.ffa.2016.11.004.  Google Scholar

[27]

X. ZengL. HuW. JiangQ. Yue and X. Cao, The weight distribution of a class of $p$-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.  doi: 10.1016/j.ffa.2009.12.001.  Google Scholar

[28]

Z. Zha and L. Hu, New classes of optimal ternary cyclic codes with minimum distance four, Finite Fields Appl., 64 (2020), 1016710. doi: 10.1016/j.ffa.2020.101671.  Google Scholar

[29]

Y. ZhouX. KaiS. Zhu and J. Li, On the minimum distance of negacyclic codes with two zeros, Finite Fields Appl., 55 (2019), 134-150.  doi: 10.1016/j.ffa.2018.09.006.  Google Scholar

[30]

Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93.  doi: 10.1016/j.ffa.2013.08.005.  Google Scholar

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}}) $
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}}) $
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}}) $
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}}) $
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) $
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) $
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) $
$ 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) $
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) $
$ 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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