|Journal of Integer Sequences, Vol. 21 (2018), Article 18.9.8|
Visa Koivunen and Robin Rajamäki
Department of Signal Processing and Acoustics
P.O. Box 15400
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.
(Concerned with sequences
Received August 15 2018; revised version received December 17 2018. Published in Journal of Integer Sequences, December 17 2018.