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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

Received  February 2021 Revised  July 2021 Early access September 2021

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

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: Cyclic code, negacyclic code, irreducible polynomial, finite field.
Mathematics Subject Classification: Primary: 94B15, 12E12; Secondary: 11T71.
Citation: Jinmei Fan, Yanhai Zhang. Optimal quinary negacyclic codes with minimum distance four. Advances in Mathematics of Communications, doi: 10.3934/amc.2021043
References:
[1]

E. R. Berlekamp, Negacyclic Codes for the Lee Metric, N. C. Chapel Hill, North Carolina State University, Dept. of Statistics, 1966.

[2]

E. R. Berlekamp, Algebraic Coding Theory, World Scientific, Singapore, 2015. doi: 10.1142/9407.

[3]

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

[4]

Y. Chen, N. Li and X. Zeng, A class of binary cyclic codes with generalized Niho exponents, Finite Fields Appl., 43 (2017), 123-140.  doi: 10.1016/j.ffa.2016.09.005.

[5]

B. Chen, S. Ling and G. Zhang, Application of constacyclic codes to quantum MDS codes, IEEE Trans. Inf. Theory, 61 (2015), 1474-1484.  doi: 10.1109/TIT.2015.2388576.

[6]

R. Chien, Cyclic decoding procedure for the Bose-Chaudhuri-Hocquenghem codes, IEEE Trans. Inf. Theory, 10 (1964), 357-363.  doi: 10.1109/TIT.1964.1053699.

[7]

C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118.

[8]

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.

[9]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theor. Comput. Sci., 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011.

[10]

C. Ding, Y. Yang and X. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inf. Theory, 56 (2010), 3605-3612.  doi: 10.1109/TIT.2010.2048504.

[11]

C. Fan, N. 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.

[12]

J. Fan, Y. Xu, Y. Xia and X. Zeng, Two families of Niho sequences having four-valued cross correlation with $m$-sequences, Science China Mathematics, 60 (2017), 2377-2390.  doi: 10.1007/s11425-016-9061-y.

[13]

J. Fan and Y. Zhang, Optimal quinary cyclic codes with minimum distance four, Chinese J. Electron., 29 (2020), 515-524.  doi: 10.1049/cje.2020.02.011.

[14]

J. Fan, Y. Zhang and X. Shi, Cyclic codes with four weights and sequence families with four-valued correlation functions, Chinese J. Electron., 28 (2019), 288-293.  doi: 10.1049/cje.2018.06.012.

[15]

G. D. Forney, On decoding BCH codes, IEEE Trans. Inf. Theory, 11 (1995), 549-557.  doi: 10.1109/tit.1965.1053825.

[16]

M. Grassl, Bounds on the Minimum Distance of Linear Codes and Quantum Codes, www.codetables.de">arXiv: www.codetables.de, 2007.

[17]

X. Hu, G. Zhang and B. Chen, Construction of new nonbinary quantum codes, Int. J. Theor. Phys., 54 (2015), 92-99.  doi: 10.1007/s10773-014-2204-8.

[18] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[19]

G. Hughes, Constacyclic codes, cocycles and a u+v|u-v construction, IEEE Trans. Inf. Theory, 46 (2000), 674-680.  doi: 10.1109/18.825841.

[20]

X. Kai and S. Zhu, New quantum MDS codes from negacyclic codes, IEEE Trans. Inf. Theory, 59 (2013), 1193-1197.  doi: 10.1109/TIT.2012.2220519.

[21]

A. Krishna and D. V. Sarwate, Pseudocyclic maximum distance separable codes, IEEE Trans. Inf. Theory, 36 (1990), 880-884.  doi: 10.1109/18.53751.

[22]

C. Li, N. Li, T. Helleseth and C. Ding, The weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721.  doi: 10.1109/TIT.2014.2329694.

[23]

N. Li, C. Li, T. Helleseth, C. Ding and X. Tang, Optimal ternary cyclic codes with minimum distance four and five, Finite Fields Appl., 30 (2014), 100-120.  doi: 10.1016/j.ffa.2014.06.001.

[24] R. Lidl and H. Niederreiter, Finite Fields, Encycl. Math. Appl., Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511525926.
[25]

E. Prange, Some Cyclic Error-Correcting Codes with Simple Decoding Algorithms, AFCRC-TN-58–156, Cambridge, Mass, 1985.

[26]

G. Xu, X. Cao and S. Xu, Optimal p-ary cyclic codes with minimum distance four from monomials, Cryptography and Communications, 8 (2016), 541-554.  doi: 10.1007/s12095-015-0159-0.

[27]

X. Zeng, L. Hu, W. Jiang, Q. 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.

[28]

Y. Zhou, X. Kai, S. 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.

show all references

References:
[1]

E. R. Berlekamp, Negacyclic Codes for the Lee Metric, N. C. Chapel Hill, North Carolina State University, Dept. of Statistics, 1966.

[2]

E. R. Berlekamp, Algebraic Coding Theory, World Scientific, Singapore, 2015. doi: 10.1142/9407.

[3]

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

[4]

Y. Chen, N. Li and X. Zeng, A class of binary cyclic codes with generalized Niho exponents, Finite Fields Appl., 43 (2017), 123-140.  doi: 10.1016/j.ffa.2016.09.005.

[5]

B. Chen, S. Ling and G. Zhang, Application of constacyclic codes to quantum MDS codes, IEEE Trans. Inf. Theory, 61 (2015), 1474-1484.  doi: 10.1109/TIT.2015.2388576.

[6]

R. Chien, Cyclic decoding procedure for the Bose-Chaudhuri-Hocquenghem codes, IEEE Trans. Inf. Theory, 10 (1964), 357-363.  doi: 10.1109/TIT.1964.1053699.

[7]

C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118.

[8]

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.

[9]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theor. Comput. Sci., 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011.

[10]

C. Ding, Y. Yang and X. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inf. Theory, 56 (2010), 3605-3612.  doi: 10.1109/TIT.2010.2048504.

[11]

C. Fan, N. 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.

[12]

J. Fan, Y. Xu, Y. Xia and X. Zeng, Two families of Niho sequences having four-valued cross correlation with $m$-sequences, Science China Mathematics, 60 (2017), 2377-2390.  doi: 10.1007/s11425-016-9061-y.

[13]

J. Fan and Y. Zhang, Optimal quinary cyclic codes with minimum distance four, Chinese J. Electron., 29 (2020), 515-524.  doi: 10.1049/cje.2020.02.011.

[14]

J. Fan, Y. Zhang and X. Shi, Cyclic codes with four weights and sequence families with four-valued correlation functions, Chinese J. Electron., 28 (2019), 288-293.  doi: 10.1049/cje.2018.06.012.

[15]

G. D. Forney, On decoding BCH codes, IEEE Trans. Inf. Theory, 11 (1995), 549-557.  doi: 10.1109/tit.1965.1053825.

[16]

M. Grassl, Bounds on the Minimum Distance of Linear Codes and Quantum Codes, www.codetables.de">arXiv: www.codetables.de, 2007.

[17]

X. Hu, G. Zhang and B. Chen, Construction of new nonbinary quantum codes, Int. J. Theor. Phys., 54 (2015), 92-99.  doi: 10.1007/s10773-014-2204-8.

[18] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[19]

G. Hughes, Constacyclic codes, cocycles and a u+v|u-v construction, IEEE Trans. Inf. Theory, 46 (2000), 674-680.  doi: 10.1109/18.825841.

[20]

X. Kai and S. Zhu, New quantum MDS codes from negacyclic codes, IEEE Trans. Inf. Theory, 59 (2013), 1193-1197.  doi: 10.1109/TIT.2012.2220519.

[21]

A. Krishna and D. V. Sarwate, Pseudocyclic maximum distance separable codes, IEEE Trans. Inf. Theory, 36 (1990), 880-884.  doi: 10.1109/18.53751.

[22]

C. Li, N. Li, T. Helleseth and C. Ding, The weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721.  doi: 10.1109/TIT.2014.2329694.

[23]

N. Li, C. Li, T. Helleseth, C. Ding and X. Tang, Optimal ternary cyclic codes with minimum distance four and five, Finite Fields Appl., 30 (2014), 100-120.  doi: 10.1016/j.ffa.2014.06.001.

[24] R. Lidl and H. Niederreiter, Finite Fields, Encycl. Math. Appl., Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511525926.
[25]

E. Prange, Some Cyclic Error-Correcting Codes with Simple Decoding Algorithms, AFCRC-TN-58–156, Cambridge, Mass, 1985.

[26]

G. Xu, X. Cao and S. Xu, Optimal p-ary cyclic codes with minimum distance four from monomials, Cryptography and Communications, 8 (2016), 541-554.  doi: 10.1007/s12095-015-0159-0.

[27]

X. Zeng, L. Hu, W. Jiang, Q. 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.

[28]

Y. Zhou, X. Kai, S. 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.

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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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]
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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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
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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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
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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$ \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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