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 \(q=2,3,4\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Springer, Berlin (1989). https://doi.org/10.1007/978-3-642-74341-2.
Brouwer A.E., Haemers W.H.: Spectra of Graphs. Springer, New York (2012). https://doi.org/10.1007/978-1-4614-1939-6.
Calderbank R., Kantor W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986).
Courteau B., Wolfmann J.: On triple sum sets and two or three-weight codes. Discret. Math. 50(2–3), 179–191 (1984).
Cohen G.D., Honkala I., Litsyn S., Lobstein A.: Covering Codes. North-Holland, Amsterdam (1997).
Delsarte P.: Weights of linear codes and strongly regular normed spaces. Discret. Math. 3(1), 47–64 (1972).
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 weight. IEEE Commun. Lett. 18(11), 1879–1882 (2014).
Ding C., Li C., Li N., Zhou : Three-weight cyclic codes and their weight distributions. Discret. Math. 339(2), 415–427 (2016).
Grassl, M.: www.codetables.de
Griera M.: On \(s\)-sum sets and three weight projective codes. Springer Lect. Notes Comput. Sci. 307, 68–76 (1986).
Griera M., Rifa J., Hughet L.: On \(s\)-sum sets and projective codes. Springer Lect. Notes Comput. Sci. 229, 135–142 (1986).
Huffman W.C., Pless V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003).
Ma S.L.: A survey of partial differences sets. Des. Codes Cryptogr. 4(4), 221–261 (1994).
Magma website http://magma.maths.usyd.edu.au/magma/
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).
Sarwate D.V., Pursley M.B.: Crosscorrelation properties of pseudorandom and related sequences. Proc. IEEE 68(5), 593–619 (1980).
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).
Shi M., Sepasdar Z., Alahmadi A., Solé P.: On two weight \(\mathbb{Z}_2^k\)-codes. Des. Codes Cryptogr. 86, 1201–1209 (2018).
van Dam E.R., Omidi G.R.: Strongly walk-regular graphs. J. Comb. Theory A 120, 803–810 (2013).
Yang S., Yao Z.-A.: Complete weight enumerator of a family of three-weight linear codes. Des. Codes Cryptogr. 82(3), 1–12 (2017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. H. Koolen.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research is supported by National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20).
Rights and permissions
About this article
Cite this article
Shi, M., Solé, P. Three-weight codes, triple sum sets, and strongly walk regular graphs. Des. Codes Cryptogr. 87, 2395–2404 (2019). https://doi.org/10.1007/s10623-019-00628-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-019-00628-7