Abstract. We study a variant of the classical circuit-lower-bound problems: proving lower bounds for sampling distributions given random bits. We prove a lower bound of 1 − 1/n Ω(1) on the statistical distance between (i) the output distribution of any small constant-depth (a.k.a. AC 0 ) circuit f : {0, 1} poly(n) → {0, 1} n , and (ii) the uniform distribution over any code C ⊆ {0, 1} n that is "good", i.e. has relative distance and rate both Ω(1). This seems to be the first lower bound of this kind. We give two simple applications of this result: (1) any data structure for storing codewords of a good code C ⊆ {0, 1} n requires redundancy Ω(log n), if each bit of the codeword can be retrieved by a small AC 0 circuit; (2) for some choice of the underlying combinatorial designs, the output distribution of Nisan's pseudorandom generator against AC 0 circuits of depth d cannot be sampled by small AC 0 circuits of depth less than d.