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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 absolutevalues of its m row sums and n column sums. In this paper, we investigatecombinatorial rectangles with minimum discrepancy (0 or 1 for each linedepending on the parity). Specifically, we get explicit formula for the numberof matrices with minimum L^infinity-discrepancy up to 4 rows, and establish theorder of magnitude of the number of such matrices with m rows and n columnswhile m is fixed and n approaches infinity. By considering the number ofcolumn-good matrices with a fixed row-sum vector, we have developed a theory ofdecreasing criterion on based row-sum vectors with majorization relation, whichturns out to be a helpful tool in the proof of our main theorems.

On Combinatorial Rectangles with Minimum $\infty$-Discrepancy