On Optimal Anticodes over Permutations with the Infinity Norm

Motivated by the set-antiset method for codes over permutations under the infinity norm, we study anticodes under this metric. For half of the parameter range we classify all the optimal anticodes, which is equivalent to finding the maximum permanent of certain $(0,1)$-matrices. For the rest of the cases we show constraints on the structure of optimal anticodes

Two-dimensional patterns with distinct differences; constructions, bounds, and maximal anticodes

A two-dimensional (2-D) grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various 2-D shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid

Optimal tristance anticodes in certain graphs

For z1, z2, z3 ∈ Z n, the tristance d3(z1, z2, z3) is a generalization of the L1-distance on Z n 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 n 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 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(z1, z2) = 1 if z1, z2 are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when Z 2 is replaced by the hexagonal lattice A2. 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 multidimensional interleaving schemes and to connectivity loci in the game of Go