Planar additive bases for rectangles

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Planar additive bases for rectangles. / Kohonen, Jukka; Rajamäki, Robin; Koivunen, Visa.

In: JOURNAL OF INTEGER SEQUENCES, Vol. 21, No. 9, 18.9.8, 2018.

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@article{0bcc8c49f1554a0d99ee0d733a71e594,
title = "Planar additive bases for rectangles",
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. {\circledC} 2019, University of Waterloo. All rights reserved.",
keywords = "Additive basis, Planar basis, Rectangular sumset, Restricted basis",
author = "Jukka Kohonen and Robin Rajam{\"a}ki and Visa Koivunen",
year = "2018",
language = "English",
volume = "21",
journal = "JOURNAL OF INTEGER SEQUENCES",
issn = "1530-7638",
publisher = "University of Waterloo",
number = "9",

}

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TY - JOUR

T1 - Planar additive bases for rectangles

AU - Kohonen, Jukka

AU - Rajamäki, Robin

AU - Koivunen, Visa

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Additive basis

KW - Planar basis

KW - Rectangular sumset

KW - Restricted basis

UR - http://www.scopus.com/inward/record.url?scp=85059450820&partnerID=8YFLogxK

M3 - Article

VL - 21

JO - JOURNAL OF INTEGER SEQUENCES

JF - JOURNAL OF INTEGER SEQUENCES

SN - 1530-7638

IS - 9

M1 - 18.9.8

ER -

ID: 16404675