On Combinatorial Rectangles with Minimum $\infty$-Discrepancy

Chunwei Song; Bowen Yao

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arXiv · 1909.05648

On Combinatorial Rectangles with Minimum $\infty$-Discrepancy

Chunwei Song·Bowen Yao

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Abstract

A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial rectangles with minimum discrepancy (0 or 1 for each line depending on the parity). Specifically, we get explicit formula for the number of matrices with minimum L^infinity-discrepancy up to 4 rows, and establish the order of magnitude of the number of such matrices with m rows and n columns while m is fixed and n approaches infinity. By considering the number of column-good matrices with a fixed row-sum vector, we have developed a theory of decreasing criterion on based row-sum vectors with majorization relation, which turns out to be a helpful tool in the proof of our main theorems.

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Chunwei Song, Bowen Yao. 2019-09-12. On Combinatorial Rectangles with Minimum $\infty$-Discrepancy. https://arxiv.org/abs/1909.05648

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