The Complexity of Local List Decoding

14 years 1 months ago

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We study the complexity of locally list-decoding binary error correcting codes with good parameters (that are polynomially related to information theoretic bounds). We show that computing majority over (1/ ) bits is essentially equivalent to locally list-decoding binary codes from relative distance 1/2 - with list size at most poly(1/ ). That is, a local-decoder for such a code can be used to construct a circuit of roughly the same size and depth that computes majority on (1/ ) bits. On the other hand, there is an explicit locally list-decodable code with these parameters that has a very efficient (in terms of circuit size and depth) local-decoder that uses majority gates of fan-in (1/ ). Using known lower bounds for computing majority by constant depth circuits, our results imply that every constant-depth decoder for such a code must have size almost exponential in 1/ (this extends even to sub-exponential list sizes). This shows that the list-decoding radius of the constant-depth loca...

Dan Gutfreund, Guy N. Rothblum

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