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Designs, Codes and Cryptography

Erschienen in:

18.02.2016

Complete weight enumerators of a family of three-weight linear codes

verfasst von: Shudi Yang, Zheng-An Yao

Erschienen in: Designs, Codes and Cryptography | Ausgabe 3/2017

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Abstract

Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime p, we present the explicit complete weight enumerator of a family of p-ary linear codes constructed with defining set. The weight enumerator is an immediate result of the complete weight enumerator, which shows that the codes proposed in this paper are three-weight linear codes. Additionally, all nonzero codewords are minimal and thus they are suitable for secret sharing.

Vorheriger Artikel Nonexistence of generalized bent functions from to

Nächster Artikel Sudoku-like arrays, codes and orthogonality

Ashikhmin A., Barg A.: Minimal vectors in linear codes. IEEE Trans. Inf. Theory 44(5), 2010–2017 (1998).

Ashikhmin A., Barg A., Cohen G., Huguet L.: Variations on minimal codewords in linear codes. In: Proceedings of AAECC 1995. Lecture Notes in Computer Science, vol. 948, pp. 96–105. Springer, New York (1995).

Blake I.F., Kith K.: On the complete weight enumerator of Reed–Solomon codes. SIAM J. Discret. Math. 4(2), 164–171 (1991).

Carlet C., Ding C., Yuan J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2102 (2005).

Chu W., Colbourn C.J., Dukes P.: On constant composition codes. Discret. Appl. Math. 154(6), 912–929 (2006).

Ding C.: Optimal constant composition codes from zero-difference balanced functions. IEEE Trans. Inf. Theory 54(12), 5766–5770 (2008).

Ding C.: Codes from Difference Sets. World Scientific, Singapore (2015).

Ding C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 61(6), 3265–3275 (2015).

Ding K., Ding C.: Binary linear codes with three weights. IEEE Commun. Lett. 18(11), 1879–1882 (2014).

10.

Ding K., Ding C.: A class of two-weight and three-weight codes and their applications in secret sharing. IEEE Trans. Inf. Theory 61(11), 5835–5842 (2015).

11.

Ding C., Wang X.: A coding theory construction of new systematic authentication codes. Theor. Comput. Sci. 330(1), 81–99 (2005).

12.

Ding C., Yang J.: Hamming weights in irreducible cyclic codes. Discret. Math. 313(4), 434–446 (2013).

13.

Ding C., Yin J.: A construction of optimal constant composition codes. Des. Codes Cryptogr. 40(2), 157–165 (2006).

14.

Ding C., Helleseth T., Klove T., Wang X.: A generic construction of Cartesian authentication codes. IEEE Trans. Inf. Theory 53(6), 2229–2235 (2007).

15.

Ding C., Liu Y., Ma C., Zeng L.: The weight distributions of the duals of cyclic codes with two zeros. IEEE Trans. Inf. Theory 57(12), 8000–8006 (2011).

16.

Dinh H., Li C., Yue Q.: Recent progress on weight distributions of cyclic codes over finite fields. J. Algebr. Comb. Discret. Struct. Appl. 2(1), 39–63 (2015).

17.

Feng T.: On cyclic codes of length \(2^{2^r}-1\) with two zeros whose dual codes have three weights. Des. Codes Cryptogr. 62(3), 253–258 (2012).

18.

Feng K., Luo J.: Weight distribution of some reducible cyclic codes. Finite Fields Appl. 14(2), 390–409 (2008).

19.

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).

20.

Kith K.: Complete weight enumeration of Reed–Solomon codes. Master’s thesis, Department of Electrical and Computing Engineering, University of Waterloo, Waterloo, Ontario, Canada (1989).

21.

Kuzmin A., Nechaev A.: Complete weight enumerators of generalized Kerdock code and linear recursive codes over Galois ring. In: Workshop on Coding and Cryptography, pp. 333–336 (1999).

22.

Kuzmin A., Nechaev A.: Complete weight enumerators of generalized Kerdock code and related linear codes over Galois ring. Discret. Appl. Math. 111(1), 117–137 (2001).

23.

Li C., Yue Q.: Weight distributions of two classes of cyclic codes with respect to two distinct order elements. IEEE Trans. Inf. Theory 60(1), 296–303 (2014).

24.

Li C., Bae S., Ahn J., Yang S., Yao Z.: Complete weight enumerators of some linear codes and their applications. Des. Codes Cryptogr. (2015). doi:10.1007/s10623-015-0136-9.

25.

Li C., Yue Q., Fu F.: Complete weight enumerators of some cyclic codes. Des. Codes Cryptogr. (2015). doi:10.1007/s10623-015-0091-5.

26.

Lidl R., Niederreiter H.: Finite Fields. Addison-Wesley, Reading, MA (1983).

27.

Luo J., Feng K.: On the weight distributions of two classes of cyclic codes. IEEE Trans. Inf. Theory 54(12), 5332–5344 (2008).

28.

MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing, Amsterdam (1977).

29.

Myerson G.: Period polynomials and Gauss sums for finite fields. Acta Arith. 39(3), 251–264 (1981).

30.

Sharma A., Bakshi G.K.: The weight distribution of some irreducible cyclic codes. Finite Fields Appl. 18(1), 144–159 (2012).

31.

Vega G.: The weight distribution of an extended class of reducible cyclic codes. IEEE Trans. Inf. Theory 58(7), 4862–4869 (2012).

32.

Wang B., Tang C., Qi Y., Yang Y., Xu M.: The weight distributions of cyclic codes and elliptic curves. IEEE Trans. Inf. Theory 58(12), 7253–7259 (2012).

33.

Yang S., Yao Z.: Complete weight enumerators of some linear codes (2015). arXiv:1505.06326.

34.

Yu L., Liu H.: The weight distribution of a family of \(p\)-ary cyclic codes. Des. Codes Cryptogr. (2014). doi:10.1007/s10623-014-0029-3.

35.

Yuan J., Carlet C., Ding C.: The weight distribution of a class of linear codes from perfect nonlinear functions. IEEE Trans. Inf. Theory 52(2), 712–717 (2006).

36.

Zheng D., Wang X., Zeng X., Hu L.: The weight distribution of a family of \(p\)-ary cyclic codes. Des. Codes Cryptogr. 75(2), 263–275 (2015).

37.

Zhou Z., Ding C.: A class of three-weight cyclic codes. Finite Fields Appl. 25, 79–93 (2014).

Titel: Complete weight enumerators of a family of three-weight linear codes
verfasst von: Shudi Yang
Zheng-An Yao
Publikationsdatum: 18.02.2016
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 3/2017
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-016-0191-x

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Complete weight enumerators of a family of three-weight linear codes

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