On ${\rm STD}_6[18,3]$'s and ${\rm STD}_7[21,3]$'s Admitting a Semiregular Automorphism Group of Order 9

Abstract

We characterize symmetric transversal designs ${\rm STD}_{\lambda}[k,u]$'s which have a semiregular automorphism group $G$ on both points and blocks containing an elation group of order $u$ using the group ring ${\bf Z}[G]$. Let $n_\lambda$ be the number of nonisomorphic ${\rm STD}_{\lambda}[3\lambda,3]$'s. It is known that $n_1=1,\ n_2=1,\ n_3=4, n_4=1$, and $n_5=0$. We classify ${\rm STD}_6[18,3]$'s and ${\rm STD}_7[21,3]$'s which have a semiregular noncyclic automorphism group of order 9 on both points and blocks containing an elation of order 3 using this characterization. The former case yields exactly twenty nonisomorphic ${\rm STD}_6[18,3]$'s and the latter case yields exactly three nonisomorphic ${\rm STD}_7[21,3]$'s. These yield $n_6\geq20$ and $n_7\geq 5$, because B. Brock and A. Murray constructed two other ${\rm STD}_7[21,3]$'s in 1991. We used a computer for our research.

Orderly generation of Butson Hadamard matrices

“…In this section we briefly report on our computational results regarding the BH(21, 3) matrices. The classification of BH (18,3) matrices was reported earlier in [15] and independently in [28], while several examples of BH (21,3) matrices were reported in [2].…”

“…The case BH (18,3) was reported in [15] and independently in [28]. Several cases of BH(21, 3) were found by Brock and Murray as reported in [2] along with additional examples. There are no BH(15, 3) matrices [15], [17,Theorem 6.65], [28,Theorem 3.2.2].…”

“…Complete classification is available up to n ≤ 32, see [17,Section 7.5], [22]. The number of BH(36, 2) matrices is at least 1.8 × 10 7 [34], while according to [26] the number of BH (40,2) matrices is at least 3.66 × 10 11 . q = 3: Complete classification is available up to n ≤ 21, see Section 4.3.…”

“…In the following we introduce the type of a BH(n, 4) matrix, a concept which is invariant up to monomial equivalence. A similar idea was used during the classification of BH (32,2) matrices [22]. Definition 4.1.…”