Locally testable codes are error-correcting codes that admit very efficient codeword tests. Specifically, using

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography.

Locally testable codes (LTCs) are error-correcting codes for which membership, in the code, of a given word can be tested by examining it in very few locations. Most known constructions of locally testable codes are linear codes, and give error-correcting codes whose duals have (superlinearly) many small weight codewords. Examining this feature appears to be one of the promising approaches to proving limitation results for (i.e., upper bounds on the rate of) LTCs. Unfortunately till now it was not even known if LTCs need to be non-trivially redundant, i.e., need to have one linear dependency among the low-weight codewords in its dual. In this paper we give the first lower bound of this form, by showing that every positive rate constant query strong LTC must have linearly many redundant low-weight codewords in its dual. We actually prove the stronger claim that the actual test itself must use a linear number of redundant dual codewords (beyond the minimum number of basis elements required to characterize the code); in other words, nonredundant (in fact, low redundancy) local testing is impossible. Our main theorem is a special case of a more general theorem that applies to any tester for an arbitrary linear locally testable code C. The general theorem can be used, for instance, to provide an arguably

We initiate the study of the tradeoff between the length of a probabilistically checkable proof of proximity (PCPP) and the maximal soundness that can be guaranteed by a 3-query verifier with oracle access to the proof. Our main observation is that a verifier limited to querying a short proof cannot obtain the same soundness as that obtained by a verifier querying a long proof. Moreover, we quantify the soundness deficiency as a function of the proof-length and show that any verifier obtaining “best possible” soundness must query an exponentially long proof. In terms of techniques, we focus on the special class of inspective verifiers that read at most 2 proof-bits per invocation. For such verifiers we prove exponential length-soundness tradeoffs that are later on used to imply our main results for the case of general (i.e., not necessarily inspective) verifiers. To prove the exponential tradeoff for inspective verifiers we show a connection between PCPP proof length and property-testing query complexity, that may be of independent interest. The connection is that any linear property that can be verified with proofs of length ℓ by linear inspective verifiers must be testable with query complexity ≈ log ℓ.

1.1 Deterministic proofs vs. proofs “beyond any reasonable doubt ”........ 1–1 1.2 Statement of (two variants of) the PCP Theorem............... 1–3 1.3 A (brief and incomplete) history of the PCP Theorem............. 1–3