The complexity of local list decoding

Abstract

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 Θ(1/ε) bits is essentially equivalent to locally list-decoding binary codes from relative distance 1/2 - ε with list size at most poly (1/ε). 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 Θ(1/ε) 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 Θ(1/ε). 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/ε (this extends even to sub-exponential list sizes). This shows that the list-decoding radius of the constant-depth local-list-decoders of Goldwasser et al. [STOC07] is essentially optimal. © 2008 Springer-Verlag Berlin Heidelberg.