A partial $A_{\mathrm{\infty }}$-structure on the cohomology of $C_n\times C_m$

Suppose $k$ is a field of characteristic 2, and $n,m\ge 4$ powers of 2. Then the $A_{\mathrm{\infty }}$-structure of the group cohomology algebras $H^{\ast }(C_n,k)$ and $H^{\ast }(C_m,k)$ are well known. We give results characterizing an $A_{\mathrm{\infty }}$-structure on $H^{\ast }(C_n\times C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.