Abstract
In this paper, we give the complete weight enumerators of two classes of linear codes over the finite field \(\mathbb {F}_{p}\), where p is a prime. These linear codes are the torsion codes of MacDonald codes over the finite non-chain ring \(\mathbb {F}_{p}+v\mathbb {F}_{p}\), where v 2 = v. We also employ these linear codes to construct systematic authentication codes with new parameters.
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AL-Ashker, M., Isleem, I.: Simplex Linear Codes Over the Ring f 2 + v f 2. An-Najah Univ J. Res. (N. Sc.) 22, 25–42 (2008)
Bae, S., Li, C., Yue, Q.: On the complete weight enumerator of some reducible cyclic codes. Discret. Math. 338, 2275–2287 (2015)
Blake, I. F., Kith, K.: On the complete weight enumerator of Reed-Solomon codes. SIAM J. Discret. Math. 4(2), 164–171 (1991)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symb. Comp. 24(3), 235–265 (1997)
Cengellenmis, Y., Al-Ashker, M.M.: MacDonald codes over the ring \(\mathbb {F}_{3}+v\mathbb {F}_{3}\). IUG Journal of Natural and Engineering Studies 20(1), 103–112 (2012)
Cengellenmis, Y., Al-Ashker, M.M.: Simplex code of type beta over \(\mathbb {F}_{3}+v\mathbb {F}_{3}\). Proceedings of the Jangjeon Mathematical Society 14(3), 277–284 (2011)
Dertli, A., Cengellenmis, Y.: MacDonald codes over the ring \(\mathbb {F}_{2}+v\mathbb {F}_{2}\). Int. J. Algebra. 5(20), 985–991 (2011)
Ding, C., Wang, X.: A coding theory construction of new systematic authentication codes. Theor. Comput. Sci. 330(1), 81–99 (2005)
Ding, C., Helleseth, T., Kløve, T., Wang, X.: A general construction of authentication codes. IEEE Trans. Inf. Theory 53(6), 2229–2235 (2007)
Ding, C., Yin, J.: Algebraic constructions of constant composition codes. IEEE Trans. Inf. Theory 51(4), 1585–1589 (2005)
Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 52(5), 2018–2032 (2006)
Kith, K.: Complete weight enumeration of Reed-Solomon codes. Masters Thesis, Department of Electrical and Computing Engineering, University of Waterloo, Waterloo, ON, Canada (1989)
Kuzmin, A.S., Nechaev, A.A.: Complete weight enumerators of generalized Kerdock code and linear recursive codes over Galois rings. In: Proceedings of the WCC99 Workshop on Coding and Cryptography, pp. 332-336, Paris, France, 11-14 January (1999)
Kuzmin, A. S., Nechaev, A. A.: Complete weight enumerators of generalized Kerdock code and related linear codes over Galois rings. Discret. Appl. Math. 111, 117–137 (2001)
Li, C., Yue, Q., Fu, F.-W.: Complete weight enumerators of some cyclic codes. Des. Codes Cryptogr. preprint (2015)
Li, C., Bae, S., Ahn, J., Yang, S., Yao, Z.: Complete weight enumerators of linear codes and their applications. Des. Codes Cryptogr. preprint (2015)
Luo, J., Helleseth, T.: Constant composition codes as subcodes of cyclic codes. IEEE Trans. Inf. Theory 57(11), 7482–7488 (2011)
MacWilliams, F. J., Sloane, N. J. A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
MacWilliams, F.J., Mallows, C.L., Sloane N.J.A.: Generalizations of Gleasons theorem on weight enumerators of self-dual codes. IEEE Trans. Inf. Theory 18(6), 794–805 (1972)
MacDonald, J.E.: Design methods for maximum minimun-distance error-coorcting codes. IBM J. Res. Dev. 4(1), 43–57 (1960)
Rees, R.S., Stinson, D.R.: Combinatorial Characterizations of Authentication Codes II. Des. Codes Cryptogr. 7(3), 239–259 (1996)
Simmons, G.J.: Authentication Theory/Coding Theory. In: Advances in CryptologyCRYPTO84. Lecture Notes in Computer Science, vol. 196, pp 411–431. Springer, Berlin (1984)
Wang, X., Gao, J., Fu, F. -W.: Secret sharing schemes from linear codes over \(\mathbb {F}_{p}+v\mathbb {F}_{p}\). Int. J. Found. Comput. Sci to appear (2015)
Xiang, C., Liu, H.: The complete weight enumerator of a class of linear codes. arXiv:hep-th/1505.06502v2 (2015)
Yang, S., Yao, Z.: Complete weight enumerators of some linear codes. arXiv:1505.06326v2 (2015)
Yang, S., Yao, Z.: The complete weight enumerator of several cyclic codes. arXiv:1505.05575v2 (2015)
Yang, S., Yao, Z.: Complete weight enumerator of a family of linear Codes from Cyclotomy. arXiv:1507.05732v1 (2015)
Yang, S., Yao, Z.: Complete weight enumerators of a family of three-weight linear codes. Des. Codes Cryptogr (2016)
Acknowledgments
Part of this work was done when J. Gao was visiting the Chern Institute of Mathematics, Nankai University, Tianjin, China. J. Gao would like to thank the institution for the kind hospitality. This research is supported by the National Key Basic Research Program of China (973 Program Grant No. 2013CB834204), the National Natural Science Foundation of China (Nos. 61571243, 61171082 and 11526045), the Doctoral Research Foundation of Shandong University of Technology (No. 4041/415059) and the Innovation Research Program of Postgraduate Teaching of Shandong University of Technology (No. 4052/115017).
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Wang, X., Gao, J. & Fu, FW. Complete weight enumerators of two classes of linear codes. Cryptogr. Commun. 9, 545–562 (2017). https://doi.org/10.1007/s12095-016-0198-1
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DOI: https://doi.org/10.1007/s12095-016-0198-1