### 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 relatively prime. Further, the existence is also guaranteed whenever n=k ^{m} , for some integers k×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) |
---|---|

Pages (from-to) | 143-156 |

Number of pages | 14 |

Journal | Theory of Computing Systems |

Volume | 42 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2008 |

### Fingerprint

### Keywords

- Digital halftoning
- Discrepancy
- Latin square
- Magic square
- Matrix

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Theory of Computing Systems*,

*42*(2), 143-156. https://doi.org/10.1007/s00224-007-9005-x