Planar additive bases for rectangles

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dc.contributor Aalto-yliopisto fi
dc.contributor Aalto University en Kohonen, Jukka Rajamäki, Robin Koivunen, Visa 2019-01-14T09:18:33Z 2019-01-14T09:18:33Z 2018
dc.identifier.citation Kohonen , J , Rajamäki , R & Koivunen , V 2018 , ' Planar additive bases for rectangles ' JOURNAL OF INTEGER SEQUENCES , vol. 21 , no. 9 , 18.9.8 . en
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dc.description.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. en
dc.format.extent 25
dc.format.mimetype application/pdf
dc.language.iso en en
dc.relation.ispartofseries JOURNAL OF INTEGER SEQUENCES en
dc.relation.ispartofseries Volume 21, issue 9 en
dc.rights openAccess en
dc.subject.other Discrete Mathematics and Combinatorics en
dc.subject.other 111 Mathematics en
dc.title Planar additive bases for rectangles en
dc.type A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä fi
dc.description.version Peer reviewed en
dc.contributor.department University of Helsinki
dc.contributor.department Department of Signal Processing and Acoustics
dc.subject.keyword Additive basis
dc.subject.keyword Planar basis
dc.subject.keyword Rectangular sumset
dc.subject.keyword Restricted basis
dc.subject.keyword Discrete Mathematics and Combinatorics
dc.subject.keyword 111 Mathematics
dc.identifier.urn URN:NBN:fi:aalto-201901141084
dc.type.version publishedVersion

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