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

Blackburn, S.R. and Etzion, T. and Martin, K.M. and Paterson, Maura B. (2010) Two-dimensional patterns with distinct differences; constructions, bounds, and maximal anticodes. IEEE Transactions on Information Theory 56 (3), pp. 1216-1229. ISSN 0018-9448.

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Official URL: http://dx.doi.org/10.1109/TIT.2009.2039046

Abstract

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.

Item Type: Article
Additional Information: "(c) 2010 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works."
Keyword(s) / Subject(s): Anticodes, Costas arrays, distinct-difference configurations, golomb rectangles, honeycomb arrays
School or Research Centre: Birkbeck Schools and Research Centres > School of Business, Economics & Informatics > Economics, Mathematics and Statistics
Depositing User: Administrator
Date Deposited: 08 Dec 2010 09:37
Last Modified: 17 Apr 2013 12:33
URI: http://eprints.bbk.ac.uk/id/eprint/2906

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