2022/07/06 by Fahim Rahim, Rahim, Fahim, Nicholas J. Cavenagh +1
Decision Sciences · Engineering · #Combinatorics (math.CO) #FOS: Mathematics #Optimal Experimental Design Methods #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2207.02397
openalex publication_date 2022/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The qk (full) factorial design with replication λ is the multi-set consisting of λ occurrences of each element of each q-ary vector of length k; we denote this by λ× [q]k. An m× n row-column factorial design qk of strength t is an arrangement of the elements of λ× [q]k into an m× n array (which we say is of type Ik(m,n,q,t)) such that for each row (column), the set of vectors therein are the rows of an orthogonal array of degree k, size n (respectively, m), q levels and strength t. Such arrays are used in experimental design. In this context, for a row-column factorial design of strength t, all subsets of interactions of size at most t can be estimated without confounding by the row and column blocking factors. In this manuscript, we study row-column factorial designs with strength t≥ 2. Our results for strength t=2 are as follows. For any prime power q and assuming 2≤ M≤ N, we show that there exists an array of type Ik(qM,qN,q,2) if and only if k≤ M+N, k≤ (qM-1)/(q-1) and (k,M,q)≠ (3,2,2). We find necessary and sufficient conditions for the existence of Ik(4m,n,2,2) for small parameters. We also show that Ik+α(2αb,2k,2,2) exists whenever α≥ 2 and 2α+α+1≤ k<2αb-α, assuming there exists a Hadamard matrix of order 4b. For t=3 we focus on the binary case. Assuming M≤ N, there exists an array of type Ik(2M,2N,2,3) if and only if M≥ 5, k≤ M+N and k≤ 2M-1. Most of our constructions use linear algebra, often in application to existing orthogonal arrays and Hadamard matrices.