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Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes

By Boaz Barak, Zeev Dvir, Avi Wigderson and Amir Yehudayoff


A (q, k, t)-design matrix is an m × n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns intersect in at most t rows. We prove that for m ≥ n, the rank of any (q, k, t)-design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least n − ( ) 2 qtn 2k Using this result we derive the following applications: Impossibility results for 2-query LCCs over large fields. A 2-query locally correctable code (LCC) is an error correcting code in which every codeword coordinate can be recovered, probabilistically, by reading at most two other code positions. Such codes have numerous applications and constructions (with exponential encoding length) ar

Year: 2011
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