On uniform eventowns

Suppose that $n=2m$, $k=2t$ and $n > 10 k^7$. We show that if a family $\mathcal F$ of $k$-subsets of an $n$-set has only even pairwise intersections then $|\mathcal F| \leq \binom{m}{t}$. Moreover, every extremal family has an atomic structure. This result was previously proved by Frankl and Tokushige for $n > n_{FT}(k)$, where $n_{FT}(k)$ is at least exponential. The main technique is Delsarte linear programming.