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

  • Minjia ShiEmail author
  • Patrick Solé


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 \(q=2,3,4\).


Strongly walk-regular graphs Three-weight codes Triple sum sets 

Mathematics Subject Classification

Primary 05 E 30 Secondary 94 B 05 



  1. 1.
    Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Springer, Berlin (1989). Scholar
  2. 2.
    Brouwer A.E., Haemers W.H.: Spectra of Graphs. Springer, New York (2012). Scholar
  3. 3.
    Calderbank R., Kantor W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Courteau B., Wolfmann J.: On triple sum sets and two or three-weight codes. Discret. Math. 50(2–3), 179–191 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cohen G.D., Honkala I., Litsyn S., Lobstein A.: Covering Codes. North-Holland, Amsterdam (1997).zbMATHGoogle Scholar
  6. 6.
    Delsarte P.: Weights of linear codes and strongly regular normed spaces. Discret. Math. 3(1), 47–64 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ding C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 61(6), 3265–3275 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ding K., Ding C.: Binary linear codes with three weight. IEEE Commun. Lett. 18(11), 1879–1882 (2014).CrossRefzbMATHGoogle Scholar
  9. 9.
    Ding C., Li C., Li N., Zhou : Three-weight cyclic codes and their weight distributions. Discret. Math. 339(2), 415–427 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grassl, M.:
  11. 11.
    Griera M.: On \(s\)-sum sets and three weight projective codes. Springer Lect. Notes Comput. Sci. 307, 68–76 (1986).MathSciNetCrossRefGoogle Scholar
  12. 12.
    Griera M., Rifa J., Hughet L.: On \(s\)-sum sets and projective codes. Springer Lect. Notes Comput. Sci. 229, 135–142 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Huffman W.C., Pless V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003).CrossRefzbMATHGoogle Scholar
  14. 14.
    Ma S.L.: A survey of partial differences sets. Des. Codes Cryptogr. 4(4), 221–261 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
  16. 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).MathSciNetzbMATHGoogle Scholar
  17. 17.
    Sarwate D.V., Pursley M.B.: Crosscorrelation properties of pseudorandom and related sequences. Proc. IEEE 68(5), 593–619 (1980).CrossRefGoogle Scholar
  18. 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).MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shi M., Sepasdar Z., Alahmadi A., Solé P.: On two weight \(\mathbb{Z}_2^k\)-codes. Des. Codes Cryptogr. 86, 1201–1209 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    van Dam E.R., Omidi G.R.: Strongly walk-regular graphs. J. Comb. Theory A 120, 803–810 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical Sciences of Anhui UniversityAnhuiPeople’s Republic of China
  2. 2.Aix Marseille Univ, CNRS, Centrale Marseille, I2MMarseilleFrance

Personalised recommendations