Abstract
We describe a new connection between the problem of finding rigid matrices, as posed by Valiant (MFCS 1977), and the problem of proving lower bounds for linear locally correctable codes. Our result shows that proving linear lower bounds on locally correctable codes with super-logarithmic query complexity will give new constructions of rigid matrices. The interest in constructing rigid matrices is their connection to circuit lower bounds.
Our results are based on a lemma saying that if the generating matrix of a locally decodable code is 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 any locally decodable code (and in particular Reed Muller codes) is rigid.
This connection gives, on the one hand, a new approach to attack the long-standing open problem of matrix rigidity and, on the other hand, explains the difficulty of advancing our current knowledge on locally correctable codes (in the high-query regime).
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Dvir, Z. On Matrix Rigidity and Locally Self-correctable Codes. comput. complex. 20, 367–388 (2011). https://doi.org/10.1007/s00037-011-0009-1
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DOI: https://doi.org/10.1007/s00037-011-0009-1