# On Symmetric Transversal Designs STD8[24; 3]'s

Authors: Teppei Hatono; Chihiro Suetake
Journal: International Journal of Combinatorics
 (2010)

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## Abstract

The existence of a class regular symmetric transversal designSTDλ[3λ;3] is equivalent to a generalized Hadamard matrix of order 3u over GF(3). Let nλ be the number of nonisomorphic STDλ[3λ;3]'s. It is known that n1=1, n2=1, n3=4, n4=1, n5=0, n6≥20, and n7≥5. In this paper, it is shown that n8≥24.


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## Body

## 1. Introduction

Asymmetric transversal designSTDλ[k;u] (STD) is an incidence structure 𝒟=(𝒫,ℬ,I) satisfying the following three conditions, where k≥2,u≥2, and λ≥1. (i)
Each block contains exactlyk points.(ii)
The point set𝒫 is partitioned into k point sets 𝒫0,𝒫1,…,𝒫k-1 of equal size u such that any two distinct points are incident with exactly λ blocks or no block according as they are contained in different 𝒫i’s or not. 𝒫0,𝒫1,…,𝒫k-1 are said to be the point classes of 𝒟.(iii)
The dual structure of𝒟 also satisfies the above conditions (i) and (ii). The point classes of the dual structure of 𝒟 are said to be the block classes of 𝒟.Let𝒟=(𝒫,ℬ,I) be an STD with the set of point classes Ω and the set of block classes Δ. Let G be an automorphism group. Then, by definition of STD, G induces a permutation group on Ω∪Δ. If G fixes any element of Ω∪Δ, then G is said to be anelation group and any element of G is said to be an elation. In this case, it is known that G acts semiregularly on each point class and on each block class. Especially, if G is an elation group of order u, then 𝒟 is said to be class regular with respect toG.If𝒟 is a class regular STDλ[k;u] with respect to a group U of order u, we can construct a generalized Hadamard matrix H of order k over U (GH(k,U)) from the incidence matrix of 𝒟, and vice versa.The existence of anSTDλ[k;2](k=2λ) is equivalent to the existence of a Hadamard matrix of order k. Therefore, in this case k≡0(mod4). It is easily showed that the number of nonisomorphic STDλ[k;2]’s is the same as the number of inequivalent Hadamard matrices of order k. Thus, the numbers of nonisomorphic STDλ[k;2]’s are now known for k’s up through 28 [1]. We are interested in the numbers of nonisomorphic STDλ[k;3]’s (k=3λ) with the next class size. Let nλ be the number of nonisomorphic STDλ[k;3]’s. Then, n1=1,n2=1,n3=4,n4=1, and n5=0 [2–4]. We remark that n5=0, that is, the nonexistence of STD5[15;3] is the special case of a theorem proved by Haemers [4]. But, for λ≥6 the value nλ is not determined except λ with known nλ=0. Recently de Launey informed us that one of his colleagues showed that n6≥40 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, STD8[24;3]'s exist. It is difficult to know how many nonisomorphic STD8[24;3]'s were constructed in [7], because their construction is very indirect. We calculated the full automorphism group of the STD8[24;3] 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 STD8[24;3]'s corresponding to GH(24,GF(3))’s constructed by de Launey.In this paper we construct 22 class regularSTD8[24;3]'s which have a noncyclic automorphism group of order 9 containing an elation of order 3. Here, G 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 G on points and blocks. Also any STD8[24;3] which we constructed is not isomorphic to anyone of two STD's constructed by Zhang et al. stated above. Thus, we have n8≥24.It is an interesting problem to construct a class regularSTD8[8u;u] or GH(8u,GF(u)) for a prime power u with u≤19, because the existence of GH(8u,GF(u))'s is known for a prime power u with 19<u<200 or for a prime u with u>19. For u∈{5,7,11,13,17,19}, 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 ofSTDλ[3λ; 3]’s

Let𝒟=(𝒫,ℬ,I) be an STDλ[k;3], where k=3λ. Let Ω={𝒫0,𝒫1,…,𝒫k-1} be the set of point classes of 𝒟 and Δ={ℬ0,ℬ1,…,ℬk-1} the set of block classes of 𝒟. Let 𝒫i={p3i,p3i+1,…,p3(i+1)-1} and ℬj={B3j,B3j+1,…,B3(j+1)-1}(0≤i,j≤k-1).On the other hand let𝒟′=(𝒫′,ℬ′,I′) be an STDλ[k;3]. Let Ω′={𝒫0′,𝒫1′,…,𝒫k-1′} be the set of point classes of 𝒟′ and Δ′={ℬ0′,ℬ1′,…,ℬk-1′} the set of block classes of 𝒟′. Let 𝒫i′={p3i′,p3i+1′,…,p3(i+1)-1′} and ℬj′={B3j′,B3j+1′,…,B3(j+1)-1′}(0≤i,j≤k-1).LetΛ be the set of permutation matrices of degree 3. Let (2.1)L=(L00⋯L0k-1⋮⋮Lk-10⋯Lk-1k-1),L′=(L00′⋯L0k-1′⋮⋮Lk-10′⋯Lk-1k-1′)be the incidence matrices of𝒟 and 𝒟′ corresponding to these numberings of the point sets and the block sets, where Lij,Lij′∈Λ(0≤i,j≤k-1), respectively. Let E be the identity matrix of degree 3. Then, we may assume that Li0=Li0′=E(0≤i≤k-1) and L0j=L0j′=E(0≤j≤k-1) after interchanging some rows of (3r)th row, (3r+1)th row, …,(3(r+1)-1)th row and interchanging some columns of (3s)th column, (3s+1)th column, …,(3(s+1)-1)th column of L and L′ for 0≤r,s≤k-1.Definition 2.1.
LetS={0,1,…,k-1}. We denote the symmetric group on S by Sym S. Let f=(01⋯k-1f(0)f(1)⋯f(k-1))∈SymS and X0,X1,…,Xk-1∈Λ.
(i) We define(2.2)(f,(X0,X1,…,Xk-1))=(X00⋯X0k-1⋮⋯⋮Xk-10⋯Xk-1k-1)
by
(2.3)Xij={Xiifj=f(i),Ootherwise,
where O is the 3×3 zero matrix.
(ii) We define(2.4)(f,(X0X1⋮Xk-1))=(X00⋯X0k-1⋮⋯⋮Xk-10⋯Xk-1k-1)
by
(2.5)Xij={Xjifi=f(j),Ootherwise,
where O is the 3×3 zero matrix.
From [14, Lemma 3.2], it follows that an isomorphism from 𝒟 to 𝒟′ is given by f,g∈SymS and X0,X1,…,Xk-1,Y0,Y1,…,Yk-1∈Λ satisfying(2.6)(f,(X0,X1,…,Xk-1))L(g,(Y0Y1⋮Yk-1))=L′.
Set(2.7)Γ={(100010001),(010001100),(001100010)}.Lemma 2.2 (see [15, Corollary 3.4]).
LetLij,Lij′∈Γ for 0≤i,j≤k-1. Then, two STDλ[3λ,3]’s 𝒟 and 𝒟′ are isomorphic if and only if there exists (f,g)∈SymS×SymS such that
(2.8)Lf(i)g(0)-1Lf(i)g(j)Lf(0)g(j)-1Lf(0)g(0)=Lij′
for 0≤i≤k-1 and 1≤j≤k-1 or there exists (f,g)∈SymS×SymS such that
(2.9)Lf(i)g(0)-1Lf(i)g(j)Lf(0)g(j)-1Lf(0)g(0)=Lij′-1
for 0≤i≤k-1 and 1≤j≤k-1.Lemma 2.3 (see [15, Corollary 3.5]).
LetLij∈Γ for 0≤i,j≤k-1. Then, any automorphism of 𝒟 is given (f,g,Y)∈SymS×SymS×Γ such that
(2.10)Lf(i)g(0)-1Lf(i)g(j)Lf(0)g(j)-1Lf(0)g(0)=Lij
for 0≤i≤k-1 and 1≤j≤k-1 or (f,g,Y)∈SymS×SymS×(Λ-Γ) such that
(2.11)Lf(i)g(0)-1Lf(i)g(j)Lf(0)g(j)-1Lf(0)g(0)=Lij-1
for 0≤i≤k-1 and 1≤j≤k-1. Here Xi=Y0-1Lf(i)g(0)-1(0≤i≤k-1) and Yj=Lf(0)g(j)-1Lf(0)g(0)Y0(1≤j≤k-1).Remark 2.4.
LetH=(hij)0≤i,j≤k be a generalized Hadamard matrix over GF(3) of order k=3λ. Set
(2.12)d(0)=(100010001),d(1)=(010001100),d(2)=(001100010).
Then, the incidence matrix of STDλ[3λ;3]𝒟(H) corresponding to H is given by d(H)=(d(hij))0≤i,j≤k-1.


## 3. CLass RegularSTD6[24; 3]’s and GH(24,GF(3))’s

The constructions by de Launey [6] and Zhang et al. [7] are known about GH(24,GF(3))'s. We give one of GH(24,GF(3))'s constructed by Zhang et al.Lemma 3.1 (see [7]).
Consider the following:(3.1)H1=(112211222200220000110011112211220022002211001100221122112200220011001100221122110022002200110011112222110011110022000022112222111100001100222200221111220011110000222200221111221100001122000022202001012020010120200101202001010202101002021010020210102020010102021010020210100202101020200101202010100110200201102002202010101001022010010220020201010110200210010220020201011001022001102002011020022020101001100220011020020202010110012002100102202020101010012002100102200202010101100220011002200110022020201010011002201001200202020101100120020110022002020101100120021001200220201010)
is a GH(24,GF(3)) and |Aut𝒟(H1)|=32×3.Proof.
LetH1=(hij)0≤i,j≤23 and we normalize H1. That is, set H1′=(hij-h0j)0≤i,j≤23=(hij′)0≤i,j≤23 and then set H1′′=(hij′-hi0′)0≤i,j≤23=(hij′′)0≤i,j≤23. Then, 𝒟(H1)≅𝒟(H1′′). We calculate the full automorphism group of 𝒟(H1′′) by Lemma 2.2 using a computer. Then, we have |𝒟(H1)|=|𝒟(H1′′)|=32×3.Example 3.2.
LetH1′′ be the GH(24,GF(3)) stated in the proof of Lemma 3.1. Then,  (3.2)H1′′=(000000000000000000000000000000001122112211221122001100112222222200110011001100110011001122222222000011221111220022220011000011222200111100112222001122220000112222001111001122221122000011112200012012012012012012012012012012010221100221100221022110021201201210022110022110022110022101201201012021100102202120211210012021101020021202122101022101202021121021011020022101200212210112100102010220211201022112101020010220212110201221010102021221010120211010202021021221011002120101020212010202122021210101202110010202120212121010021201021212101210102002210120021212102101010220121002).
SetH1′′=(wij)0≤i,j≤23. H1′′ is a normalized GH(24,GF(3)). We use the notations used in Lemma 2.2. Set S={0,1,2,…,23}. Let(3.3)f=(01234567891032105476191817|11121314151617181920212223162021222398111014151213)=(0,3)(1,2)(4,5)(6,7)(8,19,10,17)(9,18,11,16)(12,20,14,22)(13,21,15,23),g=(0123456789106754231011109|111213141516171819202122238141512132120222316171918)=(8,11)(9,10)(12,14)(13,15)(0,6,1,7)(2,5,3,4)(16,21,17,20)(18,22,19,23)∈SymS.
Then, it follows that -wf(i)g(0)+wf(i)g(j)-wf(0)g(j)+wf(0)g(0)=wij for 0≤i≤23,1≤j≤23 on GF(3). Therefore, by Lemma 2.2, for any Y0∈Γ(f,g,Y0) one yields an automorphism of 𝒟(H1′′). Here Xi=Y0-1d(wf(i)g(0))-1(0≤i≤23),  Yi=d(wf(0)g(j))-1d(wf(0)g(0))Y0(1≤j≤23).In the rest of this section we deal withGH(24,GF(3))’s with the following form, where for GF(3)={0,1,2} set 0′=1,1′=2, and 2′=0:(3.4)H=(a0a1a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a23a2a0a1a5a3a4a8a6a7a11a9a10a14a12a13a17a15a16a20a18a19a23a21a22a1a2a0a4a5a3a7a8a6a10a11a9a13a14a12a16a17a15a19a20a18a22a23a21Ψ2b0b1b2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19b20b21b22b23b2b0b1b5b3b4b8b6b7b11b9b10b14b12b13b17b15b16b20b18b19b23b21b22b1b2b0b4b5b3b7b8b6b10b11b9b13b14b12b16b17b15b19b20b18b22b23b21Ψ2c0c1c2c3c4c5c6c7c8c9c10c11c12c13c14c15c16c17c18c19c20c21c22c23c2c0c1c5c3c4c8c6c7c11c9c10c14c12c13c17c15c16c20c18c19c23c21c22c1c2c0c4c5c3c7c8c6c10c11c9c13c14c12c16c17c15c19c20c18c22c23c21Ψ2d0d1d2d3d4d5d6d7d8d9d10d11d12d13d14d15d16d17d18d19d20d21d22d23d2d0d1d5d3d4d8d6d7d11d9d10d14d12d13d17d15d16d20d18d19d23d21d22d1d2d0d4d5d3d7d8d6d10d11d9d13d14d12d16d17d15d19d20d18d22d23d21Ψ2e0e1e2e3e4e5e6e7e8e9e10e11e12e13e14e15e16e17e18e19e20e21e22e23e2e0e1e5e3e4e8e6e7e11e9e10e14e12e13e17e15e16e20e18e19e23e21e22e1e2e0e4e5e3e7e8e6e10e11e9e13e14e12e16e17e15e19e20e18e22e23e21Ψ2f0f1f2f3f4f5f6f7f8f9f10f11f12f13f14f15f16f17f18f19f20f21f22f23f2f0f1f5f3f4f8f6f7f11f9f10f14f12f13f17f15f16f20f18f19f23f21f22f1f2f0f4f5f3f7f8f6f10f11f9f13f14f12f16f17f15f19f20f18f22f23f21Ψ2g0g1g2g3g4g5g6g7g8g9g10g11g12g13g14g15g16g17g18g19g20g21g22g23g2g0g1g5g3g4g8g6g7g11g9g10g14g12g13g17g15g16g20g18g19g23g21g22g1g2g0g4g5g3g7g8g6g10g11g9g13g14g12g16g17g15g19g20g18g22g23g21Ψ2h0h0′h0′′h1h1′h1′′h2h2′h2′′h3h3′h3′′h4h4′h4′′h5h5′h5′′h6h6′h6′′h7h7′h7′′k0k0′k0′′k1k1′k1′′k2h2′k2′′k3k3′k3′′k4k4′k4′′k5k5′k5′′k6h6′k6′′k7k7′k7′′l0l0′l0′′l1l1′l1′′l2l2′l2′′l3l3′l3′′l4l4′l4′′l5l5′l5′′l6l6′l6′′l7l7′l7′′)The existence of such generalized Hadamard matrixH is equivalent to the existence of STD8[24;3]𝒟(H) with the following automorphism group G.(i)
G is an elementary abelian group of order 9.(ii)
G contains an elation of order 3.(iii)
G acts semiregularly on blocks.(iv)
Ginvariesonly one point class.Theorem 3.3.
All matricesHi(i=2,3,…,12) in the appendix are GH(24,GF(3))'s. Then, GH(24,GF(3))'s 𝒟(Hi)'s (i=1,2,…,12) are not isomorphic to each other and are not self dual. Moreover one also has the following table, where Ωi and Δi are a set of the point classes and a set of the block class of 𝒟(Hi), respectively.Corollary 3.4.
Letnλ be the number of nonisomorphic STDλ[3λ;3]'s. Then, n8≥24.



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