On the maximum number of columns for supersaturated designs with smax∈{1, 3, 5}. Issue 1 (27th May 2020) - Morales, Luis B.
Abstract : Cheng and Tang [Upper bounds on the number of columns in supersaturated designs. Biometrika, 88 (2001), 1169‐1174] provided upper bounds on the maximum number of columns, denoted by B ( n, t ), that can be accommodated in two‐symbol supersaturated designs for a given number, say n, of rows and a maximum correlation in absolute value, say t / n, between any two columns. Recently, Morales et al [On the maximum number of columns in supersaturated designs with s max =2, J. Combin. Des., 27 (2019), 448‐472] proved that B ( n, 2)= n +1 for n =14, 18, 22, 30. However, from the 35 lower bounds for B ( n, t ) provided by Cheng and Tang, only 11 supersaturated designs are known to satisfy these bounds. In this article, by performing a computer search, we show that B (9, 1)=7, B (13, 1)=12, B (17, 1)=15, B (21, 1)=19, B (7, 3)=15, B (9, 3)=12, B (11, 3)= B (13, 3)=15, B (11, 5)=66, and B (15, 3)=17. Likewise, our search produces supersaturated designs that achieve these maximums. Each of these exact values was previously unknown.