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We prove that random n-by-n Toeplitz (alternatively Hankel) matrices over F 2 have rigidity Ω( n 3 r 2 log n ) for rank r ≥ √ n, with high probability. This improves, for r = o(n/ log n log log n), over the Ω( n 2 r · log( n r )) bound that is known for many explicit matrices. Our result implies that the explicit trilinear [n] × [n] × [2n] function defined by F (x, y, z) =doi:10.1007/s00037-016-0144-9 fatcat:jwuajua4nnajlojbcchrucuzhe