Let n, k, α be integers, n, α>0, p be a prime and q=p α. Consider the complete q-uniform family $$ℱ\left(k,q\right)=\left\{K\subseteq \left[n\right]:\left|K\right|\equiv k\left(modq\right)\right\}$$ We study certain inclusion matrices attached to F(k,q) over the field $${𝔽}_p$$ . We show that if l≤q−1 and 2l≤n then $$ran{k}_{{𝔽}_p}I(ℱ(k,q),\left(\begin{array}{c}\left[n\right]\\ ⩽\ell \end{array}\right))⩽\left(\begin{array}{c}n\\ \ell \end{array}\right)$$ This extends a theorem of Frankl [7] obtained for the case α=1. In the proof we use arguments involving Gröbner bases, standard monomials and reduction. As an application, we solve a problem of Babai and Frankl related to the size of some L-intersecting families modulo q.

We construct new symmetric Hadamard matrices of orders 92, 116, and 172. While the existence of those of order 92 was known since 1978, the orders 116 and 172 are new. Our construction is based on a recent new combinatorial array (GP array) discovered by N. A. Balonin and J. Seberry. For order 116 we used an adaptation of an algorithm for parallel collision search. The adaptation pertains to the modification of some aspects of the algorithm to make it suitable to solve a 3-way matching problem....