Planar additive bases for rectangles

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Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2018
Major/Subject
Mcode
Degree programme
Language
en
Pages
25
Series
JOURNAL OF INTEGER SEQUENCES, Volume 21, issue 9
Abstract
We study a generalization of additive bases into a planar setting. A planar additive basis is a set of non-negative integer pairs whose vector sumset covers a given rectangle. Such bases find applications in active sensor arrays used in, for example, radar and medical imaging. We propose two algorithms for finding the minimal bases of small rectangles: one in the unrestricted case where the basis elements can be anywhere in the rectangle, and another in the restricted case, where the elements are confined to the lower left quadrant. We present numerical results from such searches, including the minimal cardinalities and number of unique solutions for all rectangles up to [0,11]×[0,11] in the unrestricted case, and up to [0,26]×[0,26] in the restricted case. For squares we list the minimal basis cardinalities up to [0,13]×[0,13] in the unrestricted case, and up to [0,46]×[0,46] in the restricted case. Furthermore, we prove asymptotic upper and lower bounds on the minimal basis cardinality for large rectangles.© 2019, University of Waterloo. All rights reserved.
Description
Keywords
Additive basis, Planar basis, Rectangular sumset, Restricted basis
Other note
Citation
Kohonen , J , Rajamäki , R & Koivunen , V 2018 , ' Planar additive bases for rectangles ' , JOURNAL OF INTEGER SEQUENCES , vol. 21 , no. 9 , 18.9.8 . < https://cs.uwaterloo.ca/journals/JIS/VOL21/Rajamaki/raj.pdf >