Application of Orthogonal Polynomials and Special Matrices to Orthogonal Arrays

  • Nikolai L. ManevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)


Special matrices are explored in many areas of science and technology. Krawtchouk matrix is such a matrix that plays important role in coding theory and theory of orthogonal arrays also called fractional factorial designs in planning of experiments and statistics. In this paper we give explicitly Smith normal forms of Krawtchouk matrix and its extended matrix. Also we propose a computationally effective method for determining the Hamming distance distributions of an orthogonal array with given parameters. The obtained results facilitate the solving of many existence and classification problems in theory of codes and orthogonal arrays.


Orthogonal polynomials Smith normal form Distance distribution Orthogonal arrays Linear programming bound 



This work has been accomplished with the financial support of the Ministry of Education and Science of Bulgaria by the Grant No. D01-221/03.12.2018 for National Centre of High-performance and Distributed Computing—part of the Bulgarian National Roadmap on RIs.


  1. 1.
    Boyvalenkov, P., Kulina, H.: Investigation of binary orthogonal arrays via their distance distributions. Probl. Inf. Transm. 49(4), 320–330 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Boyvalenkov, P., Marinova, T., Stoyanova, M.: Nonexistence of a few binary orthogonal arrays. Discret. Appl. Math. 217(2), 144–150 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Boumova, S., Marinova, T., Stoyanova, M.: On ternary orthogonal arrays. In: Proceedings of 16th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT-XVI), Svetlogorsk, Russia, 2–8 September 2018, pp. 102–105 (2018).
  4. 4.
    Delsarte, P.: Bounds for unrestricted codes by linear programming. Philips Res. Rep. 27, 272–289 (1972)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Delsarte, P.: Four fundamental parameters of a code and their combinatorial significance. Inform. Control 23, 407–438 (1973) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. (10) (1973)Google Scholar
  7. 7.
    Hedayat, A., Sloane, N.J.A., Stufken, J.: Orthogonal Arrays: Theory and Applications. Springer, New York (1999). Scholar
  8. 8.
    Krawtchouk, M.: Sur une généralisation des polynômes d’Hermite. Compt. Rend. 189, 620–622 (1929)Google Scholar
  9. 9.
    Levenshtein, V.I.: Krawtchouk polynomials and universal bounds for codes and designs in hamming spaces. IEEE Trans. Inform. Theory 41(5), 1303–1321 (1995)MathSciNetCrossRefGoogle Scholar
  10. 10.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North Holland, Amsterdam (1977)zbMATHGoogle Scholar
  11. 11.
    Riordan, J.: Combinatorial Identities. Wiley, Hoboken (1968)zbMATHGoogle Scholar
  12. 12.
  13. 13.
  14. 14.
    Szego, G.: Orthogonal Polynomials, vol. 23. AMS Col. Publ., Providence (1939)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsBASSofiaBulgaria

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