  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.
