[2207.02397] Row-column factorial design with strength greater than or equal to $2$

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overview: A $q^k$ (complete) factorial design with iteration $\lambda$ is a multiset consisting of $\lambda$ occurrences of each element of each $q$-ary vector of length $k$ is.We represent this with $\lambda\times [q]^k$.A $m\times n$ row-column factorial design $q^k$ of strength $t$ is an arrangement of elements in $\lambda \times [q]Convert ^k$ to a $m\times n$ array (say, of type $I_k(m,n,q,t)$), and for each row (column), the set of vectors in it is the next row so that Orthogonal arrays of order $k$, size $n$ (each $m$), $q$ levels, and strength $t$. Such arrays are used in experimental designs. In this context, a row-column factorial design of strength $t$ can estimate all subsets of interactions of size up to $t$ without being confounded by block factors in rows and columns.

In this manuscript, we study row-column factorial designs of strength $t\geq 2$. Here are the results for intensity $t=2$ : Assuming $2\leq M\leq N$ for any prime $q$, $k\leq M+N$, $k\leq (q^M-1)/(q-1)$ and $ (k,M,q)\neq(3,2,2)$. Find a necessary and sufficient condition for $I_{k}(4m,n,2,2)$ to exist for small parameters. $I_{k+\alpha}(2^{\alpha}b,2^k,2,2)$ is $\alpha\geq 2$ and $2^{\alpha}+\alpha+1 \leq k<2 Assume that there exists a Hadamard matrix of order $4b$, ^{\alpha}b-\alpha$.
For $t=3$ we focus on the binary case. Given $M\leq N$, $M\geq 5$, $k\leq M+N$ and $k\leq 2^{M-1}$. Most of our construction uses linear algebra and is often applied to existing orthogonal arrays and Hadamard matrices.