February  2009, 3(1): 35-52. doi: 10.3934/amc.2009.3.35

Common distance vectors between Costas arrays

1. 

School of Electrical, Electronic & Mechanical Engineering, University College Dublin, Belfield, Dublin 4

2. 

School of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland

Received  September 2008 Revised  January 2009 Published  January 2009

We investigate the distance vectors contained in individual Costas arrays and in pairs of Costas arrays, and prove some rigorous results in the case of the algebraically constructed arrays. Overall, it appears that the set with the property that every Costas array has a distance vector in this set, or that every pair of Costas arrays with a common vector have a common vector in this set, is in both cases surprisingly small. Further, we study Costas arrays with the additional property that they represent configurations of non-attacking kings or queens: in the former case, we demonstrate that such arrays are either sporadic or produced by a sub-method of the Lempel construction; in the latter case, partially answering a question asked by S. Golomb 26 years ago, we prove that (non-trivial) such arrays can only be sporadic and conjecture they do not exist at all.
Citation: Konstantinos Drakakis, Roderick Gow, Scott Rickard. Common distance vectors between Costas arrays. Advances in Mathematics of Communications, 2009, 3 (1) : 35-52. doi: 10.3934/amc.2009.3.35
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