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A semimagic square of order n is an n × n matrix containing the integers 0, ..., n2–1 arranged in such a way that each row and column add up to the same value We generalize this notion to that of a zero k×k-discrepancy matrix by replacing the requirement that the sum of each row and each column be the same by that of requiring that the sum of the entries in each k × k square contiguous submatrix be the same We show that such matrices exist if k and n are both even, and do not if k and n are are relatively prime Further, the existence is also guaranteed whenever n = km, for some integers k,m ≥ 2 We present a space-efficient algorithm for constructing such a matrix. Another class that we call constant-gap matrices arises in this construction We give a characterization of such matrices. An application to digital halftoning is also mentioned.