Nonexistence of \((17, 108, 3)\) ternary orthogonal array. (English) Zbl 1474.05031

Boumova, Silvia; Marinova, Tanya; Ramaj, Tedis; Stoyanova, Maya

God. Sofiĭ. Univ., Fak. Mat. Inform. 106, 117-126 (2019).

Summary: We develop a combinatorial method for computing and reducing of the possibilities of distance distributions of ternary orthogonal array (TOA) of given parameters \((n, M, \tau)\). Using relations between distance distributions of arrays under consideration and their relatives we prove certain constraints on the distance distributions of TOAs. This allows us to collect rules for removing distance distributions as infeasible. The main result is nonexistence of \((17,108,34)\) TOA. Our approach allows substantial reduction of the number of feasible distance distributions for known arrays. This could be helpful for other investigations over the classification of the ternary orthogonal arrays.

MSC:

05B15	Orthogonal arrays, Latin squares, Room squares
94B25	Combinatorial codes

Keywords:

distance distributions; Hamming space; Krawtchouk polynomials; orthogonal arrays

References:

[1]	Abramowitz, M., Stegun, I. A.:Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. New York, Dover, 1964. · Zbl 0171.38503
[2]	Boyvalenkov, P., Kulina, H.: Investigation of binary orthogonal arrays via their distance distributions.Probl. Inf. Transm.,49(4), 2013, 320-330. · Zbl 1308.05027
[3]	Boyvalenkov, P., Kulina, H., Marinova, T., Stoyanova, M.: Nonexistence of binary orthogonal arrays via their distance distributions.Probl. Inf. Transm.,51(4), 2015, 326-334. · Zbl 1347.05020
[4]	Boyvalenkov, P., Marinova, T., Stoyanova, M.: Nonexistence of a few binary orthogonal arrays.Discrete Appl. Math.,217(2), 2017, 144-150. · Zbl 1358.05041
[5]	D. A. Bulutoglu, D. A., Margot, F.: Classification of orthogonal arrays by integer programming.J. Statist. Plann. Inference,138, 2008, 654-666. · Zbl 1139.62041
[6]	Delsarte, P.: An Algebraic Approach to the Association Schemes in Coding Theory. Philips Res. Rep. Suppl.,10, 1973. · Zbl 1075.05606
[7]	Delsarte, P., Levenshtein, V. I.: Association schemes and coding theory.IEEE Trans. Inform. Theory,44, 1998, 2477-2504. · Zbl 0946.05086
[8]	Hedayat, A., Sloane, N. J. A., Stufken, J.:Orthogonal Arrays: Theory and Applications. Springer-Verlag, New York, 1999. · Zbl 0935.05001
[9]	Levenshtein, V. I.: Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces.IEEE Trans. Infor. Theory,41, 1995, 1303-1321. · Zbl 0836.94025
[10]	Levenshtein, V. I.: Universal bounds for codes and designs. In:Handbook of Coding Theory. (V. S. Pless and W. C. Huffman, Eds.),Ch. 6, Elsevier, Amsterdam, 1998, 499-648. · Zbl 0983.94056
[11]	http://neilsloane.com/oadir/index.html
[12]	Schoen, E.D., Eendebak, P. T., Nguyen, M. V. M.: Complete enumeration of purelevel and mixed-level orthogonal arrays.J. Combin. Des.,18, 2009, 123-140. · Zbl 1287.05017
[13]	Seiden, E., Zemach, R.: On orthogonal arrays.Ann. Math. Statist.,37, 1996, 1355- 1370. · Zbl 0147.19002
[14]	Szeg˝o, G.:Orthogonal Polynomials. American Mathematical Society Colloquium Publications 23, AMS, Providence, RI, 1939.
[15]	https://store.