Tolerant locally testable codes

An error-correcting code is said to be locally testable if it has an efficient spot-checking procedure that can distinguish codewords from strings that are far from every codeword, looking at very few locations of the input in doing so. Locally testable codes (LTCs) have generated a lot of interest over the years, in large part due to their connection to Probabilistically checkable proofs (PCPs). The ability to correct errors that occur during transmission is one of the big advantages of using a code. Hence, from a coding-theoretic angle, local testing is potentially more useful if in addition to accepting codewords, it also accepts strings that are close to a codeword (in contrast, local testers can have arbitrary behavior on such strings, which potentially annuls the benefits of error-correction). This would imply that when the tester accepts, one can follow-up the testing with a (more expensive) decoding procedure to correct the errors and recover the transmitted codeword, while if the tester rejects, we can save the effort of running the more expensive decoding algorithm. In this work, we define such testers, which we call tolerant testers following some recent work in property testing [13]. We revisit some recent constructions of LTCs and show how one can make them locally testable in a tolerant sense. While we do not optimize the parameters, the main message from our work is that there are explicit tolerant LTCs with similar parameters to LTCs.