The Taub Faculty of Computer Science Events and Talks

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) {\em 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 {\em one} linear dependency among its low-weight codewords. In this paper we give the first lower bound of this form, by showing that every positive rate constant query LTC must have linearly many redundant low-weight codewords in its dual. We actually prove the stronger claim that the {\em actual test itself} must use $\Omega(n)$ redundant dual codewords (beyond the minimum number of basis elements required to characterize the code); in other words, non-redundant (in fact, low redundancy) local testing is impossible.

Joint work with Eli Ben-Sasson, Venkatesan Guruswami, Tali Kaufman and Madhu Sudan.