Blocker sets, orthogonal arrays and their application to combination locks

Abstract

Let A denote a set of order m and let X be a subset of A^k+1. Then X will be called a blocker (of A^k+1) if for any element say (a₁,a₂,...,a_k,a_k+1) of A^k+1, there is some element (x₁,x₂,…,x_k,x_k+1) of X such that x_i equals a_i 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(2n,2) equals 2n² 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.