Local Decoding in Constant Depth

We study the complexity of locally decoding and locally list-decoding error correcting codes with good parameters. We focus on the circuit depth of the decoding algorithm and obtain new efficient constructions and tight lower bounds. our contributions are as follows: 1. We show general transformations from locally decodable and locally list-decodable codes with high-depth, say NC, decoders, into codes with related parameters and constant depth AC decoders. These transformations use the delegation methodology introduced by Goldwasser et al. [STOC08]; the decoding algorithm is made more efficient by shifting some of its work to the encoding algorithm (in an error-robust way). 2. Using these transformations, we obtain explicit families of binary codes that are locally decodable or locally list decodable by constant-depth circuits of size polylogarithmic in the length of the codeword. For example, we build (i) an explicit binary code that is locally decodable in constant depth from constant (relative) distance, and (ii) a locally list-decodable binary code that decodes from relative distance 1/2−ε with list size at most poly(1/ε). This second decoder is constant depth using majority gates of fan-in Θ(1/ε). 3. We show that our positive results are essentially tight. In particular, 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). In fact, we show that 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. 4. As a consequence of our positive result, and using the tight connection between locally listdecodable codes and average-case complexity, we obtain a new, more efficient, worst-case to average-case reduction for languages in EXP.