Matrix rigidity of random Toeplitz matrices

@article{Goldreich2015MatrixRO,
  title={Matrix rigidity of random Toeplitz matrices},
  author={Oded Goldreich and Avishay Tal},
  journal={computational complexity},
  year={2015},
  volume={27},
  pages={305-350}
}
A matrix A is said to have rigidity s for rank r if A differs from any matrix of rank r on more than s entries. We prove that random n-by-n Toeplitz matrices over $${\mathbb{F}_{2}}$$ F2 (i.e., matrices of the form $${A_{i,j} = a_{i-j}}$$ Ai,j=ai-j for random bits $${a_{-(n-1)}, \ldots, a_{n-1}}$$ a-(n-1),…,an-1 ) have rigidity $${\Omega(n^3/(r^2\log n))}$$ Ω(n3/(r2logn)) for rank $${r \ge \sqrt{n}}$$ r≥n , with high probability. This improves, for $${r = o(n/\log n \log\log n)}$$ r=o(n… CONTINUE READING

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