A quadratic lower bound for three-query linear locally decodable codes over any field
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
High-rate codes with sublinear-time decoding
Proceedings of the forty-third annual ACM symposium on Theory of computing
Proceedings of the forty-third annual ACM symposium on Theory of computing
Three-Query Locally Decodable Codes with Higher Correctness Require Exponential Length
ACM Transactions on Computation Theory (TOCT)
SIAM Journal on Computing
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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.