Row-column factorial designs with strength at least 2

The qk (full) factorial design with replication λ is the multi-set consisting of occurrences each element q-ary vector length k; we denote this by λ×[q]k. An m×n row-column strength t an arrangement elements λ×[q]k into array (which say type Ik(m,n,q,t)) such that for row (column), set vectors therein are rows orthogonal degree k, size n (respectively, m), q levels and t. Such arrays used in experimental design. In context, a t, all subsets interactions at most can be estimated without confounding column blocking factors. manuscript study designs t≥2. Our results t=2 as follows. For any prime power assuming 2≤M≤N, show there exists Ik(qM,qN,q,2) if only k≤M+N, k≤(qM−1)/(q−1) (k,M,q)≠(3,2,2). We find necessary sufficient conditions existence Ik(4m,n,2,2) small parameters. also Ik+α(2αb,2k,2,2) whenever α≥2 2α+α+1≤k<2αb−α, Hadamard matrix order 4b. t=3 focus on binary case. Assuming M≤N, Ik(2M,2N,2,3) M≥5, k≤M+N k≤2M−1. Most our constructions use linear algebra, often application to existing matrices.