Optimal tristance anticodes in certain graphs

For z1, z2, z3 ∈ Z, the tristance d3(z1, z2, z3) is a generalization of the L1-distance on Z 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 Z with the property that d3(z1, z2, z3) 6 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 Z for all diameters d > 1. We then generalize this result to two related distance models: a different distance structure on Z where d(z1, z2) = 1 if z1, z2 are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when Z is replaced by the hexagonal lattice A2. We also investigate optimal tristance anticodes in Z 3 and optimal quadristance anticodes in Z, and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multidimensional interleaving schemes and to connectivity loci in the game of Go. Research supported by the David and Lucile Packard Foundation and by the National Science Foundation.