The maximum number of columns in supersaturated designs with smax=2

Cheng and Tang [Biometrika, 88 (2001), pp. 1169–1174] 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 ) ≤ n + 2 for n ≡ 2 (mod 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 ) ≤ 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 ) ≥ n − 1. Hence, we establish B ( n , 2 ) = n + 1 for n = 14 , 18 , 22 , 30 and B ( n , 2 ) ≥ n + 1 for all n ≡ 2 (mod 4) such that n ≤ 270. Our result also implies that B ( n , 2 ) ≥ n + 1 when n + 1 is a prime power and n + 1 ≡ 3 (mod 4). We conjecture that n + 1 = B ( n , 2 ) < B ′ ( n , 2 ) = n + 2 for all n > 10 and n ≡ 2 (mod 4), where B ′ ( n , 2 ) is the maximum number of equiangular lines in R n − 1 with pairwise angle arccos ( 2 ∕ n ).