--We show that CC[m] circuits cannot compute MOD_q in sub-linear size when m, q \ge 1 are co-prime integers. No non-trivial lower bounds were known before on the size of CC[m] circuits of constant depth for computing MOD_q. On the other hand, our results show circuits of type MAJ \circ CC_{o(n)} \left[ m \right] need exponential size to compute MOD_q. Using Bourgain's recent breakthrough result on estimates of exponential sums, we extend our bound to the case where small fan-in AND gates are allowed at the bottom of such circuits i.e. circuits of type MAJ \circ CC_{o(n)} \left[ m \right] \circ AND_{ \in \log n,} where \in > 0 is a sufficiently small constant.

--CC[m] circuits of constant depth need superlinear number of wires to compute both the AND and MOD_q functions. To prove this, we show that any circuit computing such functions has a certain connectivity property that is similar to that of superconcentration. We show a superlinear lower bound on the number of edges of such graphs extending results on superconcentrators.