A note on $k$-wise oddtown problems

For integers $2\le t\le k$, we consider a collection of $k$ set families ${\mathcal{A}}_j:1\le j\le k$ where $$\mathcal{A}$j = { A{j,i} \subseteq [n] : 1 \leq i \leq m }$and$|A_{1, i_1} \cap \cdots \cap A_{k,i_k}|$ is even if and only if at least $t$ of the $i_j$ are distinct. In this paper, we prove that $m=O(n^{1\mathrm{/}\lfloor k\mathrm{/}2\rfloor })$ when $t=k$ and $m=O(n^{1\mathrm{/}(t-1)})$ when $2t-2\le k$ and prove that both of these bounds are best possible. Specializing to the case where $$\mathcal{A}$ = $\mathcal{A}$_1 = \cdots = $\mathcal{A}$_k$, we recover a variation of the classical oddtown problem.