The C^3 problem: error-correcting codes with a constant rate,   constant distance, and constant locality

Alex Lubotzky, Hebrew University and IAS

Online Talk 

An error-correcting code is locally testable (LTC)  if there is a random tester that reads only a small 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 are 2-dimensional versions of the expander codes constructed by Sipser and Spielman (1996). The main result and lecture will be self-contained. But we hope also to explain how the seminal work Howard Garland ( 1972) on the cohomology of quotients of the Bruhat-Tits buildings of p-adic Lie group has led to this construction ( even though, it is not used at the end). 

Based on joint work with I. Dinur, S. Evra, R. Livne, and S. Mozes