Erasures Versus Errors In Local Decoding And Property Testing

We initiate the study of the role of erasures in local decoding and use our understanding to prove a separation between erasure-resilient and tolerant property testing. We first investigate local list-decoding in the presence of erasures. We prove an analog of a famous result of Goldreich and Levin on local list-decodability of the Hadamard code. Specifically, we show that the Hadamard code is locally list-decodable in the presence of a constant fraction of erasures, arbitrarily close to 1, with list sizes and query complexity better than in the Goldreich-Levin theorem. We further study approximate locally erasure list-decodable codes and use them to construct a property that is erasure-resiliently testable with query complexity independent of the input length, n, but requires n omega(1) queries for tolerant testing. We also investigate the general relationship between local decoding in the presence of errors and in the presence of erasures.