Optimal tristance anticodes in certain graphs

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 A_d of diameter d is a subset of ℤⁿ with the property that d₃(z₁, z₂, z₃) ≤ d for all z₁, z₂, z₃ ∈ 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 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.