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 $\Theta(1/\eps)$ bits is essentially equivalent to locally list-decoding binary codes from relative distance $1/2-\eps$ with list size at most $\poly(1/\eps)$. 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 $\Theta(1/\eps)$ 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 $\Theta(1/\eps)$.

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 $1/\eps$ (this extends even to sub-exponential list sizes). This shows that the list-decoding radius of the constant-depth local-list-decoders of Goldwasser {\em et al.} [STOC07] is essentially optimal.

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 $\Theta(1/\eps)$ bits is essentially equivalent to locally list-decoding binary codes from relative distance $1/2-\eps$ with list size $\poly(1/\eps)$. 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 $\Theta(1/\eps)$ 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 $\Theta(1/\eps)$.

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 $1/\eps$. This shows that the list-decoding radius of the constant-depth local-list-decoders of Goldwasser {\em et al.} [STOC07] is essentially optimal.

Using the tight connection between locally-list-decodable codes and hardness amplification, we obtain similar limitations on the complexity of uniform (and even somewhat non-uniform) fully-black-box worst-case to average-case reductions. Very recently, Shaltiel and Viola [ECCC07] obtained similar limitations for completely non-uniform fully-black-box worst-case to average-case reductions, but only for the special case that the reduction is {\em non-adaptive}.