Planar additive bases for rectangles

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.