The minimum number of rows in covering arrays (equivalently, surjective codes) and radius-covering arrays (equivalently, surjective codes with a radius) has been determined precisely only in special cases. In this paper, explicit constructions for numerous best known covering arrays (upper bounds) are found by a combination of combinatorial and computational methods. For radius-covering arrays, explicit constructions from covering codes are developed. Lower bounds are improved upon using connections to orthogonal arrays, partition matrices, and covering codes, and in specific cases by computation. Consequently for some parameter sets the minimum size of a covering array is determined precisely. For certain of these, a complete classification of all inequivalent covering arrays is determined, again using computational techniques. Existence tables for up to 10 columns, up to 8 symbols, and all possible strengths are presented to report the best current lower and upper bounds, and classifications of inequivalent arrays.

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Let g,h in S_n be chosen at random. Using character estimates we show that in various aspects the elements gh^i behave like independent random variables. As application we show that almost surely the Cayley graph determined by g and h has diameter O(n^3 log n), and the directed Cayley graph has almost surely diameter O(n^4 log n). Further we desribe an algorithm for the black-box-recognition of the symmetric group, and show that for each element g moving a positive proportion of all points, the number of cycles of a random element h and of gh are nearly independent

Let Γ be a free product of finitely many finite-and infinite-cyclic groups. For a subgroup Δ of finite index given by its coset representation we compute its isomorphism type, i.e., its decomposition as a free product of finite-and infinite-cyclic groups. We determine the set of isomorphism types realized by finite-index subgroups, the asymptotics of the subgroup numbers with prescribed isomorphism types, and the distribution of the isomorphism types among subgroups of fixed index. Apart from group-theoretic arguments, the proofs of the present paper make use of asymptotic, combinatorial, and probabilistic ideas and techniques. Crown

We determine asymptotically the maximal order of log d(d(n)), where d(n) is the number of positive divisors of n. This solves a problem first put forth by Ramanujan in 1915.