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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.