TAILIEUCHUNG - Báo cáo toán học: "Honeycomb Arrays"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Honeycomb Arrays. | Honeycomb Arrays Simon R. Blackburn Anastasia Panoui Maura B. Paterson Douglas R. Stinson Submitted Nov 12 2009 Accepted Nov 17 2010 Published Dec 10 2010 Mathematics Subject Classification 05B30 Abstract A honeycomb array is an analogue of a Costas array in the hexagonal grid they were first studied by Golomb and Taylor in 1984. A recent result of Blackburn Etzion Martin and Paterson has shown that in contrast to the situation for Costas arrays there are only finitely many examples of honeycomb arrays though their bound on the maximal size of a honeycomb array is too large to permit an exhaustive search over all possibilities. The present paper contains a theorem that significantly limits the number of possibilities for a honeycomb array in particular the theorem implies that the number of dots in a honeycomb array must be odd . Computer searches for honeycomb arrays are summarised and two new examples of honeycomb arrays with 15 dots are given. 1 Introduction Honeycomb arrays were introduced by Golomb and Taylor 8 in 1984 as a hexagonal analogue of Costas arrays. Examples of honeycomb arrays are given in Figures 7 to 11 below. A honeycomb array is a collection of n dots in a hexagonal array with two properties The hexagonal permutation property There are three natural directions in a hexagonal grid see Figure 1 . Considering rows in each of these three directions the dots occupy n consecutive rows with exactly one dot in each row. The distinct differences property The n n 1 vector differences between pairs of distinct dots are all different. Department of Mathematics Royal Holloway University of London Egham Surrey TW20 0EX United Kingdom. @ Department of Economics Mathematics and Statistics Birkbeck University of London Malet Street London WC1E 7HX United Kingdom. David R. Cheriton School of Computer Science University of Waterloo Waterloo Ontario N2L 3G1 Canada. dstinson@ THE ELECTRONIC JOURNAL OF

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