Bounded-depth circuits cannot sample good codes

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(1) on the statistical distance between (i) the output distribution of any small constant-depth (a.k.a. AC0 ) 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 (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 (log n), if each bit of the codeword can be retrieved by a small AC0 circuit; (2) for some choice of the underlying combinatorial designs, the output distribution of Nisan's pseudorandom generator against AC0 circuits of depth d cannot be sampled by small AC0 circuits of depth less than d.

• Type: Journal
• Year: 2010
• Where: ECCC
• Authors: Shachar Lovett, Emanuele Viola
• Keywords: Classical Circuit-lower-bound Problems, ECCC 2010, Lower Bound, Output Distribution