Complete enumeration of two-Level orthogonal arrays of strength dd with d+2d+2 constraints

Enumerating nonisomorphic orthogonal arrays is an important, yet very difficult, problem. Although orthogonal arrays with a specified set of parameters have been enumerated in a number of cases, general results are extremely rare. In this paper, we provide a complete solution to enumerating nonisomorphic two-level orthogonal arrays of strength $d$ with $d+2$ constraints for any $d$ and any run size $n=\lambda2^d$. Our results not only give the number of nonisomorphic orthogonal arrays for given $d$ and $n$, but also provide a systematic way of explicitly constructing these arrays. Our approach to the problem is to make use of the recently developed theory of $J$-characteristics for fractional factorial designs. Besides the general theoretical results, the paper presents some results from applications of the theory to orthogonal arrays of strength two, three and four.Comment: Published at http://dx.doi.org/10.1214/009053606000001325 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Divisibility of Trinomials by Maximum Weight Polynomials over F2

Divisibility of trinomials by given polynomials over finite fields has been studied and used to construct orthogonal arrays in recent literature. Dewar et al.\ (Des.\ Codes Cryptogr.\ 45:1-17, 2007) studied the division of trinomials by a given pentanomial over \F_2 to obtain the orthogonal arrays of strength at least 3, and finalized their paper with some open questions. One of these questions is concerned with generalizations to the polynomials with more than five terms. In this paper, we consider the divisibility of trinomials by a given maximum weight polynomial over \F_2 and apply the result to the construction of the orthogonal arrays of strength at least 3.Comment: 10 pages, 1 figur

Generalized resolution for orthogonal arrays

The generalized word length pattern of an orthogonal array allows a ranking of orthogonal arrays in terms of the generalized minimum aberration criterion (Xu and Wu [Ann. Statist. 29 (2001) 1066-1077]). We provide a statistical interpretation for the number of shortest words of an orthogonal array in terms of sums of $R^2$ values (based on orthogonal coding) or sums of squared canonical correlations (based on arbitrary coding). Directly related to these results, we derive two versions of generalized resolution for qualitative factors, both of which are generalizations of the generalized resolution by Deng and Tang [Statist. Sinica 9 (1999) 1071-1082] and Tang and Deng [Ann. Statist. 27 (1999) 1914-1926]. We provide a sufficient condition for one of these to attain its upper bound, and we provide explicit upper bounds for two classes of symmetric designs. Factor-wise generalized resolution values provide useful additional detail.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1205 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal Ramp Schemes and Related Combinatorial Objects

In 1996, Jackson and Martin proved that a strong ideal ramp scheme is equivalent to an orthogonal array. However, there was no good characterization of ideal ramp schemes that are not strong. Here we show the equivalence of ideal ramp schemes to a new variant of orthogonal arrays that we term augmented orthogonal arrays. We give some constructions for these new kinds of arrays, and, as a consequence, we also provide parameter situations where ideal ramp schemes exist but strong ideal ramp schemes do not exist