Journal of Combinatorial Theory Series B
Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
Anticodes for the Grassman and bilinear forms graphs
Designs, Codes and Cryptography
The complete nontrivial-intersection theorem for systems of finite sets
Journal of Combinatorial Theory Series A
Regular Article: The Diametric Theorem in Hamming Spaces驴Optimal Anticodes
Advances in Applied Mathematics
Handbook of Coding Theory
Codes and anticodes in the Grassman graph
Journal of Combinatorial Theory Series A
On the Optimality of Coloring with a Lattice
SIAM Journal on Discrete Mathematics
Optimal 2-dimensional 3-dispersion lattices
AAECC'03 Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes
Interleaving schemes for multidimensional cluster errors
IEEE Transactions on Information Theory
Two-dimensional interleaving schemes with repetitions: constructions and bounds
IEEE Transactions on Information Theory
Two-dimensional patterns with distinct differences: constructions, bounds, and maximal anticodes
IEEE Transactions on Information Theory
Optimal permutation anticodes with the infinity norm via permanents of (0,1)-matrices
Journal of Combinatorial Theory Series A
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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.