Blocker sets, orthogonal arrays and their application to combination locks

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Let A denote a set of order m and let X be a subset of Ak + 1. Then X will be called a blocker (of Ak + 1) if for any element say (a1, a2, ..., ak, ak + 1) of Ak + 1, there is some element (x1, x2, ..., xk, xk + 1) of X such that xi equals ai for at least two i. The smallest size of a blocker set X will be denoted by α (m, k) and the corresponding blocker set will be called a minimal blocker. Honsberger (who credits Schellenberg for the result) essentially proved that α (2 n, 2) equals 2 n2 for all n. Using orthogonal arrays, we obtain precise numbers α (m, k) (and lower bounds in other cases) for a large number of values of both k and m. The case k = 2 that is three coordinate places (and small m) corresponds to the usual combination lock. Supposing that we have a defective combination lock with k + 1 coordinate places that would open if any two coordinates are correct, the numbers α (m, k) obtained here give the smallest number of attempts that will have to be made to ensure that the lock can be opened. It is quite obvious that a trivial upper bound for α (m, k) is m2 since allowing the first two coordinates to take all the possible values in A will certainly obtain a blocker set. The results in this paper essentially prove that α (m, k) is no more than about m2 / k in many cases and that the upper bound cannot be improved. The paper also obtains precise values of α (m, k) whenever suitable orthogonal arrays of strength two (that is, mutually orthogonal Latin squares) exist. © 2007 Elsevier B.V. All rights reserved.

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Discrete Mathematics