A combinatorial characterization of smooth LTCs and applications

The study of locally testable codes (LTCs) has benefited from a number of nontrivial constructions discovered in recent years. Yet we still lack a good understanding of what makes a linear error correcting code locally testable and as a result we do not know what is the rate-limit of LTCs and whether asymptotically good linear LTCs with constant query complexity exist. In this paper we provide a combinatorial characterization of smooth locally testable codes, which are locally testable codes whose associated tester queries every bit of the tested word with equal probability. Our main contribution is a combinatorial property defined on the Tanner graph associated with the code tester (“well-structured tester”). We show that a family of codes is smoothly locally testable if and only if it has a well-structured tester. As a case study we show that the standard tester for the Hadamard code is “well-structured”, giving an alternative proof of the local testability of the Hadamard code, originally proved by [Blum, Luby, Rubinfeld, STOC 1990]. Additional connections to the works of [Ben-Sasson, Harsha, Raskhodnikova, SICOMP 2005] and of [Lachish, Newman and Shapira, Computational Complexity 2008] are also discussed. The research has received funding from the European Community’s Seventh Framework Programme (FP7/20072013) under grant agreement number 240258. Part of work done while visiting the Computer Science and Artificial Intelligence Laboratory (CSAIL) of MIT, Cambridge, MA, USA. The research has received funding from the European Community’s Seventh Framework Programme (FP7/20072013) under grant agreement number 202405.