The Complexity of Local List Decoding

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=²) bits is essentially equivalent to locally list-decoding binary codes from relative distance 1=2 ¡ ² with list size at most poly(1=²). 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=²) bits. On the other hand, there is an explicit locally list-decodable code with these parameters that has a very e±cient (in terms of circuit size and depth) local-decoder that uses majority gates of fan-in £(1=²). 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=² (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.