We investigate linear and cyclic codes over the ring $F_2[u,v,w]/⟨u^2=v^2,uv=0,w^2=w⟩$. This is a commutative Frobenius non-chain ring, which, to the best of our knowledge, is studied here for the first time in the literature. We define a homogeneous weight on the ring and, with respect to a Gray map induced by this weight, obtain the optimal Reed-Muller code $RM$ $(1,7)$. We analyze the algebraic structure of the ring in detail, determine its ideals, and present code constructions together with their Gray images.