Abstract
Orthogonal arrays play an important role in statistics and experimental design. Like other combinatorial constructions, the most important and studied problems are questions about their existence and classification. An essential step to solving such problems is determination of Hamming distance distributions of an orthogonal array with given parameters. In this paper we propose an algorithm for computing possible distance distributions of an orthogonal array with arbitrary parameters with respect to any vector of the space. The possible distance distributions are all nonnegative integer solutions of special linear systems with integer coefficients. The proposed algorithm reduces the problem to checking signs of only t + 1 coordinates of vectors of a subset of integer solutions of the system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hedayat, A.S., Sloane, N.J.A., and Stufken, J., Orthogonal Arrays: Theory and Applications, New York: Springer, 1999.
A Library of Orthogonal Arrays (online tables; maintained by Sloane N.J.A.), http://neilsloane.com/oadir/index.html.
A Library of Distance Distributions of Ternary Orthogonal Arrays (online tables), https://store.fmi.uni-sofia.bg/fmi/algebra/stoyanova/toa.html.
Boyvalenkov, P. and Kulina, H., Investigation of Binary Orthogonal Arrays via Their Distance Distributions, Probl. Peredachi Inf., 2013, vol. 49, no. 4, pp. 28–40 [Probl. Inf. Transm. (Engl. Transl.), 2013, vol. 49, no. 4, pp. 322–332].
Boyvalenkov, P., Marinova, T., and Stoyanova, M., Nonexistence of a Few Binary Orthogonal Arrays, Discrete Appl. Math., 2017, vol. 217, Part 2, pp. 144–150.
Boumova, S., Marinova, T., and Stoyanova, M., On Ternary Orthogonal Arrays, in Proc. 16th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT-XVI), Svetlogorsk, Russia, Sept. 2–8, 2018, pp. 102–105. Available at https://www.dropbox.com/s/h7u89lh8vyirww9/Proceedings%20final.pdf?dl=0.
Levenshtein, V.I., Krawtchouk Polynomials and Universal Bounds for Codes and Designs in Hamming Spaces, IEEE Trans. Inform. Theory, 1995, vol. 41, no. 5, pp. 1303–1321.
Krawtchouk, M., Sur une généralisation des polynômes d’Hermite, C. R. Acad. Sci. Paris, 1929, vol. 189, no. 17, pp. 620–622.
Szegő, G., Orthogonal Polynomials, New York: Amer. Math. Soc., 1939.
MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz’, 1979.
Delsarte P. Bounds for Unrestricted Codes, by Linear Programming // Philips Res. Rep. 1972, vol. 27, pp. 272–289.
Delsarte, P., Four Fundamental Parameters of a Code and Their Combinatorial Significance, Inform. Control, 1973, vol. 23, no. 5, pp. 407–438.
Delsarte, P., An Algebraic Approach to the Association Schemes of Coding Theory, Philips Res. Rep. Suppl., 1973, no. 10.
Bose, R.C. and Bush, K.A., Orthogonal Arrays of Strength Two and Three, Ann. Math. Statist., 1952, vol. 23, no. 4, pp. 508–524.
Evangelaras, H., Koukouvinos, C., and Lappas, E., 18-Run Nonisomorphic Three Level Orthogonal Arrays, Metrika, 2007, vol. 66, no. 1, pp. 31–37.
Riordan, J., Combinatorial Identities, New York: Wiley, 1968.
Funding
This work has been partially supported by the Ministry of Education and Science of Bulgaria under Grant no. D01-271/16.12.2019 “National Centre for High-Performance and Distributed Computing.”
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2020, published in Problemy Peredachi Informatsii, 2020, Vol. 56, No. 1, pp. 51–62.
Rights and permissions
About this article
Cite this article
Manev, N.L. On Distance Distributions of Orthogonal Arrays. Probl Inf Transm 56, 45–55 (2020). https://doi.org/10.1134/S0032946020010056
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0032946020010056