Locally Testable Codes Require Redundant Testers

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 LTCs 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, until now it has not even been known whether LTCs need to be nontrivially redundant, i.e., need to have one linear dependency among the low-weight codewords in their 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 LTC $\mathcal{C}$. The general theorem can be used, for instance, to provide an arguably simpler proof of the main result of Ben-Sasson, Harsha, and Raskhodnikova [SIAM J. Comput., 35 (2005), pp. 1-21], which says that testing random low density parity check (LDPC) codes requires linear query complexity. Informally, our more general theorem says the following. Take any basis $B$ for the dual code of $\mathcal{C}$ that is composed of words of small support; i.e., every element of $B$ has very few nonzero entries. Then the dual code of $\mathcal{C}$ must contain many words that (i) are not in $B$, (ii) have small support, and, most importantly, (iii) are a linear combination of a constant fraction of $B$.