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arXiv · 1404.3250

On the rank of random matrices over finite fields

Daniel Salmond·Alex Grant·Ian Grivell·Terence Chan
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Abstract

A novel lower bound is introduced for the full rank probability of random finite field matrices, where a number of elements with known location are identically zero, and remaining elements are chosen independently of each other, uniformly over the field. The main ingredient is a result showing that constraining additional elements to be zero cannot result in a higher probability of full rank. The bound then follows by "zeroing" elements to produce a block-diagonal matrix, whose full rank probability can be computed exactly. The bound is shown to be at least as tight and can be strictly tighter than existing bounds.

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Daniel Salmond, Alex Grant, Ian Grivell, Terence Chan. 2014-04-12. On the rank of random matrices over finite fields. https://arxiv.org/abs/1404.3250

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