Better lower bounds than 3n for non-boolean functions?
According to the paper A $5n − o(n)$ Lower Bound on the Circuit Size over $U_2$ of a Linear Boolean Function by  Kulikov, Melanich, and Mihajlin, when $m=o(n)$ there are no lower bounds known better than $3n - o(n)$. It also outlines a method for obtaining functions for which a $4n - o(n)$ lower bound holds, when $m=n$, based on a result of Lamagne and Savage.