Top

Designs, Codes and Cryptography

Published in:

16-03-2019

Three-weight codes, triple sum sets, and strongly walk regular graphs

Authors: Minjia Shi, Patrick Solé

Published in: Designs, Codes and Cryptography | Issue 10/2019

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We construct strongly walk-regular graphs as coset graphs of the duals of three-weight codes over \(\mathbb {F}_q.\) The columns of the check matrix of the code form a triple sum set, a natural generalization of partial difference sets. Many infinite families of such graphs are constructed from cyclic codes, Boolean functions, and trace codes over fields and rings. Classification in short code lengths is made for \(q=2,3,4\).

previous article A correction to: on the linear complexity of the Sidelnikov–Lempel–Cohn–Eastman sequences

next article Generalized binary arrays from quasi-orthogonal cocycles

Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Springer, Berlin (1989). https://doi.org/10.1007/978-3-642-74341-2.CrossRefMATH

Brouwer A.E., Haemers W.H.: Spectra of Graphs. Springer, New York (2012). https://doi.org/10.1007/978-1-4614-1939-6.CrossRefMATH

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

Courteau B., Wolfmann J.: On triple sum sets and two or three-weight codes. Discret. Math. 50(2–3), 179–191 (1984).MathSciNetCrossRefMATH

Cohen G.D., Honkala I., Litsyn S., Lobstein A.: Covering Codes. North-Holland, Amsterdam (1997).MATH

Delsarte P.: Weights of linear codes and strongly regular normed spaces. Discret. Math. 3(1), 47–64 (1972).MathSciNetCrossRefMATH

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

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

Ding C., Li C., Li N., Zhou : Three-weight cyclic codes and their weight distributions. Discret. Math. 339(2), 415–427 (2016).MathSciNetCrossRefMATH

10.

Grassl, M.: www.codetables.de

11.

Griera M.: On \(s\)-sum sets and three weight projective codes. Springer Lect. Notes Comput. Sci. 307, 68–76 (1986).MathSciNetCrossRef

12.

Griera M., Rifa J., Hughet L.: On \(s\)-sum sets and projective codes. Springer Lect. Notes Comput. Sci. 229, 135–142 (1986).MathSciNetCrossRefMATH

13.

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

14.

Ma S.L.: A survey of partial differences sets. Des. Codes Cryptogr. 4(4), 221–261 (1994).MathSciNetCrossRefMATH

15.

Magma website http://magma.maths.usyd.edu.au/magma/

16.

Riera C., Solé P., Stanica P.: A complete characterization of plateaued Boolean functions in terms of their Cayley graph. Springer Lect. Notes Comput. Sci. 10831, 1–8 (2018).MathSciNetMATH

17.

Sarwate D.V., Pursley M.B.: Crosscorrelation properties of pseudorandom and related sequences. Proc. IEEE 68(5), 593–619 (1980).CrossRef

18.

Shi M., Rongsheng W., Liu Y., Solé P.: Two and three weight codes over \(\mathbb{F}_{p}+u\mathbb{F}_{p}\). Cryptogr. Commun. 9(5), 637–646 (2017).MathSciNetCrossRefMATH

19.

Shi M., Sepasdar Z., Alahmadi A., Solé P.: On two weight \(\mathbb{Z}_2^k\)-codes. Des. Codes Cryptogr. 86, 1201–1209 (2018).MathSciNetCrossRefMATH

20.

van Dam E.R., Omidi G.R.: Strongly walk-regular graphs. J. Comb. Theory A 120, 803–810 (2013).MathSciNetCrossRefMATH

21.

Yang S., Yao Z.-A.: Complete weight enumerator of a family of three-weight linear codes. Des. Codes Cryptogr. 82(3), 1–12 (2017).MathSciNetCrossRef

Title: Three-weight codes, triple sum sets, and strongly walk regular graphs
Authors: Minjia Shi
Patrick Solé
Publication date: 16-03-2019
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 10/2019
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-019-00628-7

Springer Professional

Three-weight codes, triple sum sets, and strongly walk regular graphs

Abstract

Premium Partner

Springer Professional

Abstract

Please log in to get access to your license.

Other articles of this Issue 10/2019

On the uniqueness of a type of cascade connection representations for NFSRs

On the weight distribution of second order Reed–Muller codes and their relatives

Hadamard matrices, d-linearly independent sets and correlation-immune Boolean functions with minimum Hamming weights

Linear (2, p, p)-AONTs exist for all primes p

Linked systems of symmetric group divisible designs of type II

Theory of supports for linear codes endowed with the sum-rank metric

Premium Partner