Morales, Luis B.; Bulutoglu, Dursun A.; Arasu, K. T. The maximum number of columns in supersaturated designs with \(s_{\max}=2\). (English) Zbl 1429.05025 J. Comb. Des. 27, No. 7, 448-472 (2019). Summary: C.-S. Cheng and B. Tang [Biometrika 88, No. 4, 1169–1174 (2001; Zbl 0986.62064)] derived an upper bound on the maximum number of columns \(B(n,t)\) that can be accommodated in a two-symbol supersaturated design (SSD) for a given number of rows (\(n\)) and a maximum in absolute value correlation between any two columns (\(t/n\)). In particular, they proved that \(B(n,2)\le n+2\) for \(n\equiv 2 \pmod 4\) and \(n>6\). However, the only known SSD satisfying this upper bound is when \(n=10\). By utilizing a computer search, we prove that \(B(n,2)\le n+1\) for \(n=18,22,30\), and \(B(14,2)=15\). These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the \(E(s^2)\) lower bound for SSDs with \(n\) rows and \(n+2\) columns, for \(n=14,18,22\), and \(30\). Finally, we show that a skew-type Hadamard matrix of order \(n\) can be used to construct an SSD with \(n-2\) rows and \(n-1\) columns that proves \(B(n-2,2)\ge n-1\). Hence, we establish \(B(n,2)=n+1\) for \(n=18,22,30\) and \(B(n,2)\ge n+1\) for all \(n\equiv2 \pmod 4\) such that \(n\le270\). Our result also implies that \(B(n,2)\ge n+1\) when \(n+1\) is a prime power and \(n+1\equiv 3 \pmod 4\). We conjecture that \(n+1=B(n,2) < B^\prime(n,2)=n+2\) for all \(n>10\) and \(n\equiv 2 \pmod 4\), where \(B^\prime(n,2)\) is the maximum number of equiangular lines in \(\mathbb{R}^{n-1}\) with pairwise angle \(\arccos(2/n)\). Cited in 1 Document MSC: 05B15 Orthogonal arrays, Latin squares, Room squares 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 62K99 Design of statistical experiments Keywords:backtrack search; isomorph rejection; maximum clique; parallel class intersection matrix; resolvable incomplete block designs; skew-type Hadamard matrix Citations:Zbl 0986.62064 PDFBibTeX XMLCite \textit{L. B. Morales} et al., J. Comb. Des. 27, No. 7, 448--472 (2019; Zbl 1429.05025) Full Text: DOI