一类最优7-元循环码
A Class of Optimal 7-Ary Cyclic Codes
DOI: 10.12677/pm.2026.164110, PDF,   
作者: 张 鲜：内蒙古师范大学数学科学学院，内蒙古 呼和浩特；黄月梅*：内蒙古师范大学数学科学学院，内蒙古 呼和浩特；内蒙古自治区应用数学中心，内蒙古 呼和浩特；无穷维哈密尔顿系统及其算法应用教育部重点实验室，内蒙古 呼和浩特
关键词: 有限域；完美非线性函数；最小距离；球填充界；循环码；Finite Field； Perfect Nonlinear Function； Minimum Distance； Sphere Packing Bound； Cyclic Codes
摘要: 基于其丰富的代数结构和高效的实现，有限域上的循环码成为编码理论中的一个热门课题。虽然二元和三元循环码已被广泛研究，但诸如 F 7 等更大域上的码字研究还相对较少。根据已有判定七元循环码最优性的方法，构造出参数为 [ 7 m −1, 7 m −2m−2,4 ] 的最优7-元循环码。对比现有构造，本文的参数选取条件给出了新的不相交的码族，丰富了更高阶循环码的最优构造理论。
Abstract: Due to the rich algebraic structures and efficient implementations, cyclic codes over finite fields have become a popular topic in coding theory. While binary and ternary cyclic codes have been extensively studied, research on codes over larger fields, such as F 7 , remains relatively limited. Using a verified method for determining the optimality of 7-ary cyclic codes, we constructs a new 7-ary cyclic codes with parameters [ 7 m −1, 7 m −2m−2,4 ] . Compared with existing constructions, the parameter selection conditions in this paper yield a new disjoint family of codes, thereby enriching the theory of optimal constructions for higher-order cyclic codes.
文章引用：张鲜, 黄月梅. 一类最优7-元循环码[J]. 理论数学, 2026, 16(4): 252-261. https://doi.org/10.12677/pm.2026.164110

参考文献

[1] Charpin, P., Tietäväinen, A. and Zinoviev, V. (1999) On the Minimum Distances of Non-Binary Cyclic Codes. Designs, Codes and Cryptography, 17, 81-85. [Google Scholar] [CrossRef] 
[2] Charpin, P., Tietvinen, A. and Zinoviev, V. (1997) On Binary Cyclic Codes with Minimum Distance d=3. Problems Inf. Transmiss, 33, 3-14.
[3] Lidl, R. and Niederreiter, H. (1977) Finite Fields. Cambridge University Press.
[4] Macwilliams, F. and Sloane, N. (1977) The Theory of Error-Correcting Codes. Elsevier.
[5] Ireland, K. and Rosen, M. (1990) A Classical Introduction to Modern Number Theory. Springer-Verlag.
[6] El Rouayheb, S.Y., Georghiades, C.N., Soljanin, E. and Sprintson, A. (2007) Bounds on Codes Based on Graph Theory. 2007 IEEE International Symposium on Information Theory, Nice, 24-29 June 2007, 1876-1879. [Google Scholar] [CrossRef] 
[7] Carlet, C., Ding, C.S. and Yuan, J. (2005) Linear Codes from Highly Nonlinear Functions and Their Secret Sharing Schemes. IEEE Transactions on Information Theory, 51, 2089-2102.
[8] Ding, C.S. and Helleseth, T. (2013) Optimal Ternary Cyclic Codes from Monomials. IEEE Transactions on Information Theory, 59, 5898-5904. [Google Scholar] [CrossRef] 
[9] Liu, Q., Dong, X.B. and Lian, Z.Z. (2025) Several Classes of Optimal Ternary Cyclic Codes with Two Zeros. Cryptography and Communications, 18, 251-278. [Google Scholar] [CrossRef] 
[10] Wu, G.F., You, Z.H., Zha, Z.B. and Zhang, Y.Q. (2024) Several New Classes of Optimal Ternary Cyclic Codes with Two or Three Zeros. Designs, Codes and Cryptography, 93, 769-786. [Google Scholar] [CrossRef] 
[11] Han, D.C. and Yan, H.D. (2019) On an Open Problem about a Class of Optimal Ternary Cyclic Codes. Finite Fields and Their Applications, 59, 335-343. [Google Scholar] [CrossRef] 
[12] Li, L.Q., Zhu, S.X. and Liu, L. (2019) Three Classes of Optimal Ternary Cyclic Codes and the Weight Distributions of Their Duals. Chinese Journal of Electronics, 28, 674-681. [Google Scholar] [CrossRef] 
[13] Li, N., Li, C.L., Helleseth, T., Ding, C.S. and Tang X.H. (2014) Optimal Ternary Cyclic Codes with Minimum Distance Four and Five. Finite Fields and Their Applications, 30, 100-120. [Google Scholar] [CrossRef] 
[14] Li, N., Zhou, Z.C. and Helleseth, T. (2015) On a Conjecture about a Class of Optimal Ternary Cyclic Codes. 2015 7th International Workshop on Signal Design and Its Applications in Communications (IWSDA), Bengaluru, 14-18 September 2015, 62-65. [Google Scholar] [CrossRef] 
[15] Liu, Q. and Liu, X.M. (2022) On Some Conjectures about Optimal Ternary Cyclic Codes. Applicable Algebra in Engineering, Communication and Computing, 33, 419-436. [Google Scholar] [CrossRef] 
[16] Liu, Y., Cao, X.W. and Lu, W. (2020) On Some Conjectures about Optimal Ternary Cyclic Codes. Designs, Codes and Cryptography, 88, 297-309. [Google Scholar] [CrossRef] 
[17] Liu, Y. and Cao, X.W. (2023) Optimal P-Ary Cyclic Codes with Two Zeros. Applicable Algebra in Engineering, Communication and Computing, 34, 129-138. [Google Scholar] [CrossRef] 
[18] Wang, D.D. and Cao, X.W. (2022) A Family of Optimal Ternary Cyclic Codes with Minimum Distance Five and Their Duals. Cryptography and Communications, 14, 1-13. [Google Scholar] [CrossRef] 
[19] Wang, L.S. and Wu, G.F. (2016) Several Classes of Optimal Ternary Cyclic Codes with Minimal Distance Four. Finite Fields and Their Applications, 40, 126-137. [Google Scholar] [CrossRef] 
[20] Yan, H.D., Zhou, Z.C. and Du, X.N. (2018) A Family of Optimal Ternary Cyclic Codes from the Niho-Type Exponent. Finite Fields and Their Applications, 54, 101-112. [Google Scholar] [CrossRef] 
[21] Zha, Z.B. and Hu, L. (2020) New Classes of Optimal Ternary Cyclic Codes with Minimum Distance Four. Finite Fields and Their Applications, 64, Article 101671. [Google Scholar] [CrossRef] 
[22] Zha, Z.B., Hu, L., Liu, Y. and Cao, X.W. (2021) Further Results on Optimal Ternary Cyclic Codes. Finite Fields and Their Applications, 75, Article 101898. [Google Scholar] [CrossRef] 
[23] Zhao, H., Luo, R. and Sun, T.J. (2022) Two Families of Optimal Ternary Cyclic Codes with Minimal Distance Four. Finite Fields and Their Applications, 79, Article 101995. [Google Scholar] [CrossRef] 
[24] Xu, G.K., Cao, X.W. and Xu, S.D. (2016) Optimal P-Ary Cyclic Codes with Minimum Distance Four from Monomials. Cryptography and Communications, 8, 541-554. [Google Scholar] [CrossRef] 
[25] Zhou, Y.J., Kai, X.S., Zhu, S.X. and Li, J. (2019) On the Minimum Distance of Negacyclic Codes with Two Zeros. Finite Fields and Their Applications, 55, 134-150. [Google Scholar] [CrossRef] 
[26] Wu, T.T., Zhu, S.X, Liu, L. and Li, L.Q. (2024) Optimal Quinary Cyclic Codes with Three Zeros. Cryptography and Communications, 16, 801-823. [Google Scholar] [CrossRef] 
[27] Liao, D.C., Kai, X.S., Zhu, S.X. and Li, P. (2019) A Class of Optimal Cyclic Codes with Two Zeros. IEEE Communications Letters, 23, 1293-1296. [Google Scholar] [CrossRef] 
[28] Liu, Y. and Cao, X.W. (2020) Four Classes of Optimal Quinary Cyclic Codes. IEEE Communications Letters, 24, 1387-1390. [Google Scholar] [CrossRef] 
[29] Liu, Y., Cao, X.W. and Zha, Z.B. (2023) More Classes of Optimal Quinary Cyclic Codes of Form . Applicable Algebra in Engineering, Communication and Computing, 36, 327-339. [Google Scholar] [CrossRef] 
[30] Liu, Q., Huang, J.H., Zheng, D.B., Jiang, R and Zhang L.P. (2025) Several Classes of Optimal Quinary Cyclic Codes with Minimum Distance Four. Cryptography and Communications, 17, 1521-1542. [Google Scholar] [CrossRef] 
[31] Liu, Y. and Cao, X.W. (2025) Three New Classes of Optimal Quinary Cyclic Codes with Minimum Distance Four. Applicable Algebra in Engineering, Communication and Computing, 36, 493-501. [Google Scholar] [CrossRef] 
[32] Wu, T.T., Liu, L. and Li, L.Q. (2025) Several Classes of Optimal Cyclic Codes with Three Zeros. Applicable Algebra in Engineering, Communication and Computing, 36, 743-767. [Google Scholar] [CrossRef] 

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