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Parity check matrices and product representations of squares

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“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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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Assaf Naor

  2. Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0112, USA

    Jacques Verstraëte

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  1. Assaf Naor
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  2. Jacques Verstraëte
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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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