An error-correcting code is locally testable (LTC) if there is a random tester that reads only a constant number of bits of a given word and decides whether the word is in the code, or at least close to it. A long-standing problem asks if there exists such a code that also satisfies the golden standards of coding theory: constant rate and constant distance. Unlike the classical situation in coding theory, random codes are not LTC, so this problem is a challenge of a new kind.

We construct such codes based on what we call (Ramanujan) Left/Right Cayley square complexes. These 2-dimensional objects seem to be of independent interest.

The lecture will be self-contained. Based on joint work with I. Dinur, S. Evra, R. Livne, and S. Mozes.