On the maximum number of columns for supersaturated designs with s _max ∈{1,3,5}

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 smax=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.