Locally Testable Codes with constant rate, distance, and locality

TL;DR

This paper constructs locally testable error-correcting codes with constant rate, distance, and locality using a novel two-dimensional complex, advancing the theoretical understanding of LTCs with practical potential.

Contribution

The authors introduce a new two-dimensional complex called a left-right Cayley complex to construct LTCs with constant parameters, solving a major open problem.

Findings

Successfully constructed c^3-LTCs with constant rate, distance, and locality.

Codes are based on functions on squares of a new complex, extending expander code concepts.

The work advances the theoretical foundation of LTCs with practical implications.

Abstract

A locally testable code (LTC) is an error-correcting code that has a property-tester. The tester reads $q$ bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter $q$ is called the locality of the tester. LTCs were initially studied as important components of PCPs, and since then the topic has evolved on its own. High rate LTCs could be useful in practice: before attempting to decode a received word, one can save time by first quickly testing if it is close to the code. An outstanding open question has been whether there exist "$c^3$-LTCs", namely LTCs with *c*onstant rate, *c*onstant distance, and *c*onstant locality. In this work we construct such codes based on a new two-dimensional complex which we call a left-right Cayley complex. This is essentially a graph which, in addition to vertices and edges, also has squares. Our codes can be viewed as a two-dimensional version of (the one-dimensional) expander codes, where the codewords are functions on the squares rather than on the edges.