Optimal tristance anticodes in certain graphs

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.