Abstract

The existence of a class regular symmetric transversal design is equivalent to a generalized Hadamard matrix of order over . Let be the number of nonisomorphic 's. It is known that , , , , , , and . In this paper, it is shown that .

1. Introduction

A symmetric transversal design (STD) is an incidence structure satisfying the following three conditions, where , and . (i)Each block contains exactly points.(ii)The point set is partitioned into point sets of equal size such that any two distinct points are incident with exactly blocks or no block according as they are contained in different ’s or not. are said to be the point classes of .(iii)The dual structure of also satisfies the above conditions (i) and (ii). The point classes of the dual structure of are said to be the block classes of .

Let be an STD with the set of point classes and the set of block classes . Let be an automorphism group. Then, by definition of STD, induces a permutation group on . If fixes any element of , then is said to be an elation group and any element of is said to be an elation. In this case, it is known that acts semiregularly on each point class and on each block class. Especially, if is an elation group of order , then is said to be class regular with respect to .

If is a class regular with respect to a group of order , we can construct a generalized Hadamard matrix of order over () from the incidence matrix of , and vice versa.

The existence of an is equivalent to the existence of a Hadamard matrix of order . Therefore, in this case . It is easily showed that the number of nonisomorphic ’s is the same as the number of inequivalent Hadamard matrices of order . Thus, the numbers of nonisomorphic ’s are now known for ’s up through 28 [1]. We are interested in the numbers of nonisomorphic ’s with the next class size. Let be the number of nonisomorphic ’s. Then, , and [2–4]. We remark that , that is, the nonexistence of is the special case of a theorem proved by Haemers [4]. But, for the value is not determined except with known . Recently de Launey informed us that one of his colleagues showed that using a computer [5]. The combinatorial constructions of GH(24,GF(3))’s by de Launey [6] and Zhang et al. [7] are known. Therefore, 's exist. It is difficult to know how many nonisomorphic 's were constructed in [7], because their construction is very indirect. We calculated the full automorphism group of the corresponding to only one GH(24,GF(3)) constructed in [7]. This STD is not self-dual. On the other hand, since we were not able to secure de Launey's Ph.D. thesis [6], we did not calculate the full automorphism groups of 's corresponding to GH(24,GF(3))’s constructed by de Launey.

In this paper we construct 22 class regular 's which have a noncyclic automorphism group of order 9 containing an elation of order 3. Here, acts semiregularly on blocks (points), but does not act semiregularly on points (blocks) for any STD of those. We used orbit theorems [8, 9] of STD's to determine such action of on points and blocks. Also any which we constructed is not isomorphic to anyone of two STD's constructed by Zhang et al. stated above. Thus, we have .

It is an interesting problem to construct a class regular or for a prime power with , because the existence of 's is known for a prime power with or for a prime with . For , this problem is open. We expect that our construction is useful to solve this problem.

For general notation and concepts in design theory, we refer the reader to basic textbooks in the subject such as [10–12] or [13].

2. Isomorphisms and Automorphisms of ’s

Let be an , where . Let be the set of point classes of and the set of block classes of . Let and .

On the other hand let be an . Let be the set of point classes of and the set of block classes of . Let , and ,.

Let be the set of permutation matrices of degree 3. Let

be the incidence matrices of and corresponding to these numberings of the point sets and the block sets, where , respectively. Let be the identity matrix of degree 3. Then, we may assume that and after interchanging some rows of th row, th row, th row and interchanging some columns of th column, th column, th column of and for .

Definition 2.1. Let . We denote the symmetric group on by Sym . Let and .
(i) We define by where is the zero matrix.
(ii) We define by where is the zero matrix.
From [14, Lemma ], it follows that an isomorphism from to is given by and satisfying
Set

Lemma 2.2 (see [15, Corollary ]). Let for . Then, two ’s and are isomorphic if and only if there exists such that for and or there exists such that for and .

Lemma 2.3 (see [15, Corollary ]). Let for . Then, any automorphism of is given such that for and or such that for and . Here and .

Remark 2.4. Let be a generalized Hadamard matrix over of order . Set Then, the incidence matrix of corresponding to is given by .

3. CLass Regular ’s and GH(24,GF(3))’s

The constructions by de Launey [6] and Zhang et al. [7] are known about 's. We give one of 's constructed by Zhang et al.

Lemma 3.1 (see [7]). Consider the following: is a and .

Proof. Let and we normalize . That is, set and then set . Then, . We calculate the full automorphism group of by Lemma 2.2 using a computer. Then, we have .

Example 3.2. Let be the stated in the proof of Lemma 3.1. Then,
Set . is a normalized . We use the notations used in Lemma 2.2. Set . Let Then, it follows that for on . Therefore, by Lemma 2.2, for any one yields an automorphism of . Here , .