The maximum number of columns in supersaturated designs with

Abstract

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 ) goodbreakinfix≤ n goodbreakinfix+ 2 for n goodbreakinfix≡ 2 (mod 4) and n goodbreakinfix> 6. However, the only known SSD satisfying this upper bound is when n goodbreakinfix= 10. By utilizing a computer search, we prove that B ( n , 2 ) goodbreakinfix≤ n goodbreakinfix+ 1 for n goodbreakinfix= 18 goodbreakinfix, 22 goodbreakinfix, 30, and B ( 14 , 2 ) goodbreakinfix= 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 goodbreakinfix+ 2 columns, for n goodbreakinfix= 14 goodbreakinfix, 18 goodbreakinfix, 22, and 30. Finally, we show that a skew‐type Hadamard matrix of order n can be used to construct an SSD with n goodbreakinfix− 2 rows and n goodbreakinfix− 1 columns that proves B ( n − 2 , 2 ) goodbreakinfix≥ n goodbreakinfix− 1. Hence, we establish B ( n , 2 ) goodbreakinfix= n goodbreakinfix+ 1 for n goodbreakinfix= 14 goodbreakinfix, 18 goodbreakinfix, 22 goodbreakinfix, 30 and B ( n , 2 ) goodbreakinfix≥ n goodbreakinfix+ 1 for all n goodbreakinfix≡ 2 (mod 4) such that n goodbreakinfix≤ 270. Our result also implies that B ( n , 2 ) goodbreakinfix≥ n goodbreakinfix+ 1 when n goodbreakinfix+ 1 is a prime power and n goodbreakinfix+ 1 goodbreakinfix≡ 3 (mod 4). We conjecture that n goodbreakinfix+ 1 goodbreakinfix= B ( n , 2 ) goodbreakinfix< B ′ ( n , 2 ) goodbreakinfix= n goodbreakinfix+ 2 for all n goodbreakinfix> 10 and n goodbreakinfix≡ 2 (mod 4), where B ′ ( n , 2 ) is the maximum number of equiangular lines in R n − 1 with pairwise angle arccos ( 2 ∕ n ).

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

“…To reduce the size of the search tree for finding a maximum clique, we used an isomorph rejection scheme. Our method was able to find the previously unknown exact values of B(n, t) for n = 9, 13, 17, 21 when t = 1, for n = 7, 9, 11, 13, 15 when t = 3 and for (n, t) = (11,5). These values show that there are substantial gaps between the exact values and the upper bounds given in Reference 4.…”

“…However, for the other cases, it was necessary to resort to the complete method (ie, r ≥ 1). Table 2 provides the cardinalities of ( )'s for (n, t) = (11,5), (13,3), (15,3), (17, 1), (21, 1). The CPU times for determining B (13,3), B (11,5), B (15,3), B(17, 1), and B(21, 1) were about 21 minutes, 4.6 days, 193 days, 8 seconds, and 140 seconds, respectively.…”

“…In this case, the Gram matrix D T D is block ‐circulant with k 2 blocks of ( n × n ) matrices, where n is the number of rows in D . A possible future research direction is to extend our investigation by combining the ideas in References 5, 15, 16, and this article.…”

“…They also showed that B ( n ,2)≥ n +1 if there exists an n × n Hadamard matrix of skew type. However, up until now, only 11 exact values were known for B ( n , t ) with n ≤26 and t ≤6; see References 4 and 5. In this article, by using a correspondence between SSDs and certain resolvable pairwise designs (RPDs) and an exhaustive computer search, we obtained the previously unknown values of B ( n , t ) for n =9,13,17,21 when t =1, for n =7,9,11,13,15 when t =3, and for ( n , t )=(11,5).…”

“…One such bound was obtained by utilizing error-correcting codes and another bound was obtained by using lower bounds on E(s 2 ). In particular, they tabulated upper bounds on B(n, t) for n ≤ 24, t ≤ 6 for even n, and t ≤ 5 for odd n. Recently, Morales et al 5 proved that B(n, 2) = n + 1 for n = 14, 18, 22, 30 by using a computer search. They also showed that B(n, 2) ≥ n + 1 if there exists an n × n Hadamard matrix of skew type.…”