On Matrix Rigidity and Locally Self-correctable Codes

Abstract

e describe a new approach for the problem of finding {\rm rigid} matrices, as posed by Valiant \cite{Val77}, by connecting it to the, seemingly unrelated, problem of proving lower bounds for linear locally self-correctable codes. This approach, if successful, could lead to a non-natural property (in the sense of Razborov and Rudich \cite{RR97}) implying super-linear lower bounds for linear functions in the model of logarithmic-depth arithmetic circuits. Our results are based on a lemma saying that, if the generating matrix of a locally decodable code is {\bf not} rigid, then it defines a locally self-correctable code with rate close to one. Thus, showing that such codes cannot exist will prove that the generating matrix of {\em any} locally decodable code (and in particular Reed Muller codes) is rigid.