The Oddtown problem modulo a composite number

A family of subsets $\mathcal{A}$ of an $n$-element set is called an $\mathrm{\ell }$-Oddtown if the sizes of all sets are not divisible by $\mathrm{\ell }$, but the sizes of pairwise intersections are divisible by $\mathrm{\ell }$. Berlekamp and Graver showed that when is a $\mathrm{\ell }$ is a prime, the maximum size of an $\mathrm{\ell }$-Oddtown is $n$. For composite moduli with $\omega$ distinct prime factors, the argument of Szegedy gives an upper bound of $\omega n-\omega {\log }_{2}n$ on the size of an $\mathrm{\ell }$-Oddtown. We improve this to $\omega n-(2\omega +\epsilon ){\log }_{2}n$ for most $\mathrm{\ell }$ and $n$ using a combination of linear algebraic and Fourier-analytic arguments.