### Abstract

A semimagic square of order n is an n × n matrix containing the integers 0,... ,n^{2} -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 = k^{m}, 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.

Original language | English (US) |
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Pages (from-to) | 89-100 |

Number of pages | 12 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 3341 |

DOIs | |

State | Published - 2004 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*3341*, 89-100. https://doi.org/10.1007/978-3-540-30551-4_10