## Abstract

For z_{1}, z_{2}, z_{3} ∈ ℤ^{n}, the tristance d_{3}(z_{1}, z_{2}, z_{3}) is a generalization of the L_{1}-distance on ℤ^{n} to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticode A_{d} of diameter d is a subset of ℤ^{n} with the property that d_{3}(z_{1}, z_{2}, z_{3}) ≤ d for all z_{1}, z_{2}, z_{3} ∈ A_{d}. An anticode is optimal if it has the largest possible cardinality for its diameter d. We determine the cardinality and completely classify the optimal tristance anticodes in ℤ^{2} for all diameters d ≥ 1. We then generalize this result to two related distance models: a different distance structure on ℤ^{2} where d(z_{1}, z_{2}) = 1 if z_{1}, z_{2} are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when ℤ^{2} is replaced by the hexagonal lattice A_{2}. We also investigate optimal tristance anticodes in ℤ^{3} and optimal quadristance anticodes in ℤ^{2}., and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multi-dimensional interleaving schemes and to connectivity loci in the game of Go.

Original language | English |
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Pages (from-to) | 189-224 |

Number of pages | 36 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 113 |

Issue number | 2 |

DOIs | |

State | Published - 1 Feb 2006 |

Externally published | Yes |

## Keywords

- Anticodes
- Grid graph
- L-distance
- Multidimensional interleaving
- Tristance

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics