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2011 26th Annual IEEE Conference on Computational Complexity
Bounded-Depth Circuits Cannot Sample Good Codes
San Jose, California USA
June 08-June 11
ISBN: 978-0-7695-4411-3
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 on the statistical distance between (i) the output distribution of any small constant-depth (a.k.a. AC0) circuit, and (ii) the uniform distribution over any code that is ``good'', i.e. has constant relative distance and rate. 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 requires an additive logarithmic redundancy, 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.
Index Terms:
Small-depth circuits, sampling distributions, lower bounds
Citation:
Shachar Lovett, Emanuele Viola, "Bounded-Depth Circuits Cannot Sample Good Codes," ccc, pp.243-251, 2011 26th Annual IEEE Conference on Computational Complexity, 2011
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