IEEE Transactions on Information Theory
Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
Acquistion sequences in PPM communications
IEEE Transactions on Information Theory
Bounds for the size of radar arrays
IEEE Transactions on Information Theory
An algebraic construction of sonar sequences using M-sequences
SIAM Journal on Algebraic and Discrete Methods
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
On Perfect Codes and Related Concepts
Designs, Codes and Cryptography
Codes and anticodes in the Grassman graph
Journal of Combinatorial Theory Series A
Optimal tristance anticodes in certain graphs
Journal of Combinatorial Theory Series A
Efficient Key Predistribution for Grid-Based Wireless Sensor Networks
ICITS '08 Proceedings of the 3rd international conference on Information Theoretic Security
Golomb rectangles as folded rulers
IEEE Transactions on Information Theory
Improved bounds on maximum size binary radar arrays
IEEE Transactions on Information Theory
Interleaving schemes for multidimensional cluster errors
IEEE Transactions on Information Theory
Genetic search for Golomb arrays
IEEE Transactions on Information Theory
Distinct difference configurations: multihop paths and key predistribution in sensor networks
IEEE Transactions on Information Theory
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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.