Abstract
A set of permutations of length v is t-suitable if every element precedes every subset of t − 1 others in at least one permutation. The maximum length of a t-suitable set of N permutations depends heavily on the relation between t and N. Two classical results, due to Dushnik and Spencer, are revisited. Dushnik’s result determines the maximum length when \(t > \sqrt{2N}\). On the other hand, when t is fixed Spencer’s uses a strong connection with binary covering arrays of strength t − 1 to obtain a lower bound on the length that is doubly exponential in t. We explore intermediate values for t, by first considering directed packings and related Golomb rulers, and then by examining binary covering arrays whose number of rows is approximately equal to their number of columns. These in turn are constructed from Hadamard and Paley matrices, for which we present some computational data and questions.
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References
Bennett, F.E., Wei, R., Yin, J., Mahmoodi, A.: Existence of DBIBDs with block size six. Utilitas Math. 43, 205–217 (1993)
Colbourn, C.J.: Covering arrays from cyclotomy. Des. Codes Crypt. 55(2–3), 201–219 (2010)
Colbourn, C.J.: Covering arrays and hash families. In: Information Security and Related Combinatorics, NATO Peace and Information Security, pp. 99–136. IOS Press, Amsterdam (2011)
Colbourn, C.J., Kéri, G.: Binary covering arrays and existentially closed graphs. In: Coding and Cryptology. Lecture Notes in Computer Science, vol. 5557, pp. 22–33. Springer, Berlin (2009)
Colbourn, C.J., Kéri, G., Rivas Soriano, P.P., Schlage-Puchta, J.C.: Covering and radius-covering arrays: constructions and classification. Discret. Appl. Math. 158, 1158–1190 (2010)
Dimitromanolakis, A.: Analysis of the Golomb ruler and the Sidon set problems and determination of large near-optimal Golomb rulers. Technical Report, Technical University of Crete (2002)
Drakakis, K.: A review of the available construction methods for Golomb rulers. Adv. Math. Commun. 3(3), 235–250 (2009)
Dushnik, B.: Concerning a certain set of arrangements. Proc. Am. Math. Soc. 1, 788–796 (1950)
Erdős, P.: On a problem of Sidon in additive number theory. Acta Sci. Math. Szeged 15, 255–259 (1954)
Erdős, P., Turán, P.: On a problem of Sidon in additive number theory, and on some related problems. J. Lond. Math. Soc. 16, 212–215 (1941)
Füredi, Z., Kahn, J.: On the dimensions of ordered sets of bounded degree. Order 3, 15–20 (1986)
Füredi, Z., Hajnal, P., Rödl, V., Trotter, W.T.: Interval orders and shift orders. In: Hajnal, A., Sos, V.T. (eds.) Sets, Graphs, and Numbers, pp. 297–313. North-Holland, New York/Amsterdam (1991)
Graham, R.L., Spencer, J.H.: A constructive solution to a tournament problem. Can. Math. Bull. 14, 45–48 (1971)
Hedayat, A.S., Sloane, N.J.A., Stufken, J.: Orthogonal Arrays. Springer, New York (1999)
Ionin, Y.J., Shrikhande, M.S.: Combinatorics of Symmetric Designs. New Mathematical Monographs, vol. 5. Cambridge University Press, Cambridge (2006)
Johnson, K.A., Entringer, R.: Largest induced subgraphs of the n-cube that contain no 4-cycles. J. Comb. Theory Ser. B 46, 346–355 (1989)
Katona, G.O.H.: Two applications (for search theory and truth functions) of Sperner type theorems. Period. Math. 3, 19–26 (1973)
Kharaghani, H., Tayfeh-Rezaie, B.: On the classification of Hadamard matrices of order 32. J. Comb. Des. 18(5), 328–336 (2010)
Kharaghani, H., Tayfeh-Rezaie, B.: Hadamard matrices of order 32. J. Comb. Des. 21(5), 212–221 (2013)
Kierstead, H.A.: On the order dimension of 1-sets versus k-sets. J. Comb. Theory Ser. A 73, 219–228 (1996)
Kimura, H.: Classification of Hadamard matrices of order 28. Discret. Math. 133(1–3), 171–180 (1994)
Kleitman, D., Spencer, J.: Families of k-independent sets. Discret. Math. 6, 255–262 (1973)
Lawrence, J., Kacker, R.N., Lei, Y., Kuhn, D.R., Forbes, M.: A survey of binary covering arrays. Electron. J. Comb. 18(1), Paper 84, 30 (2011)
Paley, R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933)
Spencer, J.: Minimal scrambling sets of simple orders. Acta Math. Acad. Sci. Hung. 22, 349–353 (1971/1972)
Tang, D.T., Chen, C.L.: Iterative exhaustive pattern generation for logic testing. IBM J. Res. Dev. 28, 212–219 (1984)
Torres-Jimenez, J., Rodriguez-Tello, E.: New upper bounds for binary covering arrays using simulated annealing. Inf. Sci. 185(1), 137–152 (2012)
Yan, J., Zhang, J.: A backtracking search tool for constructing combinatorial test suites. J. Syst. Softw. 81, 1681–1693 (2008)
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Dedicated to Hadi Kharaghani on the occasion on his 70th birthday
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Colbourn, C.J. (2015). Suitable Permutations, Binary Covering Arrays, and Paley Matrices. In: Colbourn, C. (eds) Algebraic Design Theory and Hadamard Matrices. Springer Proceedings in Mathematics & Statistics, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-319-17729-8_3
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DOI: https://doi.org/10.1007/978-3-319-17729-8_3
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