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A pair of planes, both projective or both affine, of the same order and on the same point set are orthogoval if each line of one plane intersects each line of the other plane in at most two points. In this paper we prove new constructions for sets of mutually orthogoval planes, both projective and affine, and review known results that are equivalent to sets of more than two mutually orthogoval planes. We also discuss the connection between sets of mutually orthogoval planes and covering arrays.Mathematics Subject Classifications: 05B25, 05B40, 51E20, 51E21Keywords: Finite geometry, projective planes, affine planes, covering arrays, orthogoval planes

By exploiting symmetries of finite fields, covering perfect hash families provide a succinct representation for covering arrays of index one. For certain parameters, this connection has led to both the best current asymptotic existence results and the best known efficient construction algorithms for covering arrays. The connection generalizes in a straightforward manner to arrays in which every t-way interaction is covered λ > 1 times, i.e., to covering arrays of index more than one. Using this framework, we focus on easily computed, explicit upper bounds on numbers of rows for various parameters with higher index. [ABSTRACT FROM AUTHOR]

Alternative design and analysis methods for screening experiments based on locating arrays are presented. The number of runs in a locating array grows logarithmically based on the number of factors, providing efficient methods for screening complex engineered systems, especially those with large numbers of categorical factors having different numbers of levels. Our analysis method focuses on levels of factors in the identification of important main effects and two-way interactions. We demonstrate the validity of our design and analysis methods on both well-studied and synthetic data sets and investigate both statistical and combinatorial properties of locating arrays that appear to be related to their screening capability. [ABSTRACT FROM AUTHOR]

A known recursive construction for perfect hash families multiplies the number of factors by q, where q is the largest prime power not exceeding the number of symbols. This is generalized to permit multiplication of the number of factors by powers of q; it is also improved upon by restricting the structure of the ingredient designs. For a perfect hash family of strength t, it may be possible to order its rows so that, for each , the first Ni rows form a perfect hash family of strength . This ‘matroshka’ ordering underlies a sometimes substantial reduction in the number of rows generated by the recursive construction. Perfect hash families with useful matroshka orderings are generated using linear orthogonal arrays, and with an effective one-row-at-a-time greedy algorithm.

A perfect hash family PHF(N; k, v, t) is an N × k array on v symbols with v ≥ t, in which in every N × t subarray, at least one row is comprised of distinct symbols. Perfect hash families have a wide range of applications in cryptography, particularly to secure frameproof codes, in database management, and indirectly in software interaction testing. New recursive constructions, new direct constructions, and PHFs found using tabu search are provided here. The first general tables of the best known sizes of PHFs are presented; in the process, the known direct and recursive constructions are surveyed.