Parity check matrices and product representations of squares

  title={Parity check matrices and product representations of squares},
  author={Assaf Naor and Jacques Verstra{\"e}te},
Let NF(n, k, r) denote the maximum number of columns in an n-row matrix with entries in a finite field F in which each column has at most r nonzero entries and every k columns are linearly independent over F. We obtain near-optimal upper bounds for NF(n, k, r) in the case k > r. Namely, we show that NF(n, k, r) ¿ n r2 + cr k where c ≈ 4 3 for large k. Our method is based on a novel reduction of the problem to the extremal problem for cycles in graphs, and yields a fast algorithm for finding… CONTINUE READING

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