Summary: 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. $\mathrm{AC}^0$) circuit $f:\{0, 1\}^{\mathrm{poly}(n)} \rightarrow \{0,1\}^n$, and (ii) the uniform distribution over any code $\mathcal C \subseteq \{0, 1\}^n$ that is “good", that is, 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 $\mathcal C\subseteq \{0, 1\}^n$ requires redundancy $Ω(\log n)$, if each bit of the codeword can be retrieved by a small $\mathrm{AC}^0$ circuit; and (2) for some choice of the underlying combinatorial designs, the output distribution of Nisan’s pseudorandom generator against $\mathrm{AC}^0$ circuits of depth $d$ cannot be sampled by small $\mathrm{AC}^0$ circuits of depth less than $d$.