Parity check matrices and product representations of squares

“It is certainly odd to have an instruction in an algorithm asking you to play with some numbers to find a subset with product a square … Why should we expect to find such a subsequence, and, if it exists, how can we find it efficiently?”

Carl Pomerance [31].

Abstract

Let \( N_{\mathbb{F}} \)(n,k,r) denote the maximum number of columns in an n-row matrix with entries in a finite field \( \mathbb{F} \) in which each column has at most r nonzero entries and every k columns are linearly independent over \( \mathbb{F} \). We obtain near-optimal upper bounds for \( N_{\mathbb{F}} \)(n,k,r) in the case k > r. Namely, we show that \( N_\mathbb{F} (n,k,r) \ll n^{\frac{r} {2} + \frac{{cr}} {k}} \) where \( c \approx \frac{4} {3} \) 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 short linear dependencies. We present additional applications of this method to a problem on hypergraphs and a problem in combinatorial number theory.

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Correspondence to Assaf Naor.

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Naor, A., Verstraëte, J. Parity check matrices and product representations of squares. Combinatorica 28, 163–185 (2008). https://doi.org/10.1007/s00493-008-2195-2

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Mathematics Subject Classification (2000)

  • 05Cxx
  • 05Dxx
  • 94B65