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Applicable Algebra in Engineering, Communication and Computing

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15-05-2017 | Original Paper

Linear codes from quadratic forms

Authors: Xiaoni Du, Yunqi Wan

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 6/2017

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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 power q, we present a class of linear codes over finite fields \(F_q\) with quadratic forms via a general construction and then determine the explicit complete weight enumerators of these linear codes. Our construction covers some related ones via quadratic form functions and the linear codes may have applications in cryptography and secret sharing schemes.

previous article An improved method for determining the weight distribution of a family of q-ary cyclic codes

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

Ashikhmin, A., Barg, A., Cohen, G., Huguet, L.: Variations on minimal codewords in linear codes. Appl. Algebra Algebraic Algorithms Error-Correcting Codes 948, 96–105 (1995)CrossRefMATHMathSciNet

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

Calderbank, A.R., Goethals, J.M.: Three-weight codes and association schemes. Philips J. Res. 39, 143–152 (1984)MATHMathSciNet

Calderbank, A.R., Kantor, W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986)CrossRefMATHMathSciNet

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

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

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

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

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)CrossRefMATHMathSciNet

11.

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

12.

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

13.

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

14.

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

15.

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

16.

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

17.

Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite field of odd characteristic. IEEE Trans. Inf. Theory 52, 2018–2032 (2006)CrossRefMATHMathSciNet

18.

Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)CrossRefMATH

19.

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

20.

Klapper, A.: Cross-correlations of geometric sequences in characteristic two. Des. Codes Cryptogr. 3(4), 347–377 (1993)CrossRefMATHMathSciNet

21.

Klapper, A.: Cross-correlations of quadratic form sequences in odd characteristic. Des. Codes Cryptogr. 11(3), 289–305 (1997)CrossRefMATHMathSciNet

22.

Kløve, T.: Codes for Error Detection. World Scientific, Singapore (2007)CrossRefMATH

23.

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)

24.

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

25.

Lidl, R., Niederreiter, H.: Finite Fields Encyclopedia of Mathematics 20. Cambridge University Press, Cambridge (1983)MATH

26.

Tang, C., Li, N., Qi, Y., Zhou, Z., Helleseth, T.: Two-weight and three-weight linear codes from weakly regular bent functions. IEEE Trans. Inf. Theory 62(3), 1166–1176 (2016)CrossRefMATH

27.

Xu, G., Cao, X.: Linear codes with two or three weights from some functions with low Walsh spectrum in odd characteristic. arXiv: 1510.01031 (2015)

28.

Yuan, J., Ding, C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inf. Theory 52, 206–212 (2006)CrossRefMATHMathSciNet

29.

Yang, S., Yao, Z.: Complete weight enumerators of some linear codes. arxiv: 1505.06326v1 (2015)

30.

Yang, S., Yao, Z.: Complete weight enumerators of a family of three-weight linear codes. Des. Codes Cryptogr. doi:10.1007/s10623-016-0191-x

31.

Zhang, D., Fan, C., Peng, D., Tang X.: Complete weight enumerators of some linear codes from quadratic forms. Cryptogr. Commun. doi:10.1007/s12095-016-0190-9

32.

Zhou, Z., Ding, C.: Seven classes of three-weight cyclic codes. IEEE Trans. Commun. 61(10), 4120–4126 (2013)CrossRef

33.

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

34.

Zhou, Z., Li, N., Fan, C., Helleseth, T.: Linear codes with two or three weights from quadratic bent functions. Des. Codes Cryptogr. doi:10.1007/s10623-015-0144-9

Title: Linear codes from quadratic forms
Authors: Xiaoni Du
Yunqi Wan
Publication date: 15-05-2017
Publisher: Springer Berlin Heidelberg
Published in: Applicable Algebra in Engineering, Communication and Computing / Issue 6/2017
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI: https://doi.org/10.1007/s00200-017-0319-x

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