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