For z1, z2, z3 ∈ ℤn, the tristance d3(z1, z2, z3) is a generalization of the L1-distance on ℤn to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticode Ad of diameter d is a subset of ℤn with the property that d3(z1, z2, z3) ≤ d for all z1, z2, z3 ∈ Ad. 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 ℤ2 for all diameters d ≥ 1. We then generalize this result to two related distance models: a different distance structure on ℤ2 where d(z1, z2) = 1 if z1, z2 are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when ℤ2 is replaced by the hexagonal lattice A2. We also investigate optimal tristance anticodes in ℤ3 and optimal quadristance anticodes in ℤ2., 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.
- Grid graph
- Multidimensional interleaving
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics