Locally Testable Codes

Error correcting codes are combinatorial objects that allow reliable recovery of information in presence of errors by cleverly augmenting the original information with a certain amount of redundancy. The availability of efficient means of error detection is considered as a fundamental criterion for error correcting codes. Locally testable codes are families of error correcting codes that are highly robust against transmission errors and in addition provide super-efficient (sublinear time) probabilistic algorithms for error detection. In particular, the error detection algorithm probes the received sequence only at a small (or even constant) number of locations. There seems to be a trade-off between the amount of redundancy and the number of probes for the error detection procedure in locally testable codes. Even though currently best constructions allow reduction of redundancy to a nearly linear amount, it is not clear whether this can be further reduced to linear while preserving a constant number of probes. We study the formal notion of locally testable codes and survey several major results in this area. We also investigate closely related concepts, and in particular, polynomial low-degree tests and probabilistically checkable proofs.