We have constructed 377 non-isomorphic 2-(17,4,3) designs as derived and 649 non-isomorphic 2-(52,13,4) designs as residual designs of symmetric (69,17,4) designs admitting an action of Z6

Dean Crnković

Sanja Rukavina

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GLASNIK MATEMATICKIVol. 36(56)(2001), 169 { 175SOME NEW 2-(17,4,3) AND 2-(52,13,4) DESIGNSDean Crnkovic and Sanja RukavinaUniversity of Rijeka, CroatiaAbstract. We have constructed three hundred and seventy-sevennon-isomorphic 2-(17,4,3) designs as derived and six hundred and forty-nine non-isomorphic 2-(52,13,4) designs as residual designs of symmetric(69,17,4) designs admitting an action of Z6.1. Introduction and preliminariesA 2  (v; k; ) design is a nite incidence structure (P ;B; I), where P andB are disjoint sets and I  P  B, with the following properties:1. jPj = v,2. every element of B is incident with exactly k elements of P ,3. every pair of elements of P is incident with exactly  elements of B.Elements of B are called blocks, and elements of P are called points. For x 2 Band P 2 P , denote hxi = fQ 2 Pj(Q; x) 2 Ig, hP i = fy 2 Bj(P; y) 2 Ig. Fortwo designs D1 = (P1;B1; I1) and D2 = (P2;B2; I2) an isomorphism from D1onto D2 is a bijection which maps points onto points and blocks onto blockspreserving the incidence. If there exists an isomorphism from D1 onto D2, weshall say that D1 and D2 are isomorphic and write D1 = D2. An isomorphismfrom D onto D is an automorphism of D. The set of all automorphism ofthe design D is a group called the full automorphism group of D, and it isdenoted by AutD. Each subgroup of the AutD is an automorphism group ofD. A symmetric (v; k; ) design is a 2  (v; k; ) design with jPj = jBj = v.First known symmetric (69,17,4) design was constructed by Shrikhandeand Singhi (see [7]). Designs with these parameters belong to the familyof symmetric designs satisfying conditions v = m3 + m + 1, k = m2 + 12000 Mathematics Subject Classication. 05B05.Key words and phrases. symmetric design, residual design, derived design, automor-phism, orbit structure.169170 DEAN CRNKOVIC AND SANJA RUKAVINAand  = m, where both m   1 and m2  m + 1 are prime powers (see [8]).Recently Z. Bozikov (see [1]) constructed two symmetric designs with param-eters (69,17,4) having Frobenius group F39 as an automorphism group, out ofwhich one was new and the other was previously constructed by Shrikhandeand Singhi. In august 1998., S. Topalova reported that there are four sym-metric (69,17,4) designs with automorphisms of order 13 (see [2]). Finally, S.Rukavina constructed all symmetric (69,17,4) designs admitting an action ofcyclic group of order six (see [6]), out of which fty-seven designs were newand two designs admitting an action of Frobenius group F39 were previouslyknown.2. Derived and residual designsLet D be a symmetric (v; k; ) design. The derived design Dx with respectto a block x is the incidence structure whose points are the points of x andwhose blocks are the sets hyi \ hxi, where y is any block other than x. Dx isa 2   (k; ;    1) design. The residual design Dx with respect to a block xis the incidence structure whose points are those points of D which are notincident with x and whose blocks are the sets hyi   hxi, where y ranges overall blocks other than x. It is a 2  (v   k; k   ; ) design.Theorem 1. Let  be an isomorphism of symmetric designs D = (P ;B; I)and D0 = (P 0;B0; I 0). Further, let x 2 B and x = x0 2 B0. Then 1 = jDxis an isomorphism from Dx onto D0x0 and 2 = jDx is an isomorphism fromDx onto D0x0 .Proof. Let x1 2 B. (hx1i \ hxi)1 = hx1i1 \ hxi1 = hx11i \ hx0iand (hx1i   hxi)2 = hx1i2   hxi2 = hx12i   hx0i; which means that 1maps blocks from Dx onto blocks from D0x0 and 2 maps blocks from Dx ontoblocks from D0x0 .Corollary 1. Let D = (P ;B; I) be a symmetric design, x; x0 2 B andG  AutD. If x0 2 xG, then Dx = Dx0 and Dx = Dx0 .Proof. There exists  2 G such that x0 = x. jDx is an isomorphismfrom Dx onto Dx0 and jDx is an isomorphism from Dx onto Dx0 .Corollary 2. Let D = (P ;B; I) be a symmetric design, x 2 B and 2 AutD. If x = x then jDx 2 AutDx and jDx 2 AutDx.3. Some new 2-(17,4,3) designsExcluding one block and all points that do not belong to that block fromsymmetric (69,17,4) design we obtain 2-(17,4,3) design.In [6] fty-nine mutually non-isomorphic symmetric (69,17,4) designs arepresented. In that article, representatives of block orbits for constructedSOME NEW 2-(17,4,3) AND 2-(52,13,4) DESIGNS 171designs and corresponding automorphisms are given. From corollary 1 fol-lows that construction of derived designs corresponding to block orbit repre-sentatives led us, after eliminating isomorphic copies, to all mutually non-isomorphic 2-(17,4,3) designs that can be obtained from given symmetric(69,17,4) design. For all designs in [6], except designs 54 and 58, repre-sentatives of block orbits of full automorphism groups of those designs aregiven. Designs 54 and 58 are presented in [6] by representatives of subgroupsof full automorphism groups. Therefore, except in those two cases, from each(69,17,4) design we shall construct derived designs with respect to the repre-sentatives of block orbits given in [6]. On designs 54 and 58 full automorphismgroups of those designs act in three block orbits. For construction of deriveddesigns we shall consider one block from each of those three orbits. Precisely,from design 54 we shall exclude the following blocks:0 1 2 3 4 21 22 23 24 25 26 27 28 29 30 31 321 5 6 7 8 21 22 23 36 37 38 51 52 53 60 61 620 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47From design 58 we shall exclude the following blocks:1 5 6 7 8 21 22 23 36 37 38 51 52 53 57 58 590 1 2 3 4 21 22 23 24 25 26 27 28 29 30 31 320 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47Described process gave us six hundred and forty-nine 2-(17,4,3) designs.Using computer program by V. Krcadinac (see [3], [5]) we obtain that betweenthem there are three hundred and seventy-seven mutually non-isomorphicdesigns.For each of three hundred and seventy-seven mutually non-isomorphic 2-(17,4,3) design we present (or describe) block that should be excluded fromsymmetric (69,17,4) design. Ordinal numbers of symmetric (69,17,4) designscorrespond to those in [6].design 1:excluding xed block from (69,17,4) design 1designs 2 - 54:excluding block 0 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47from (69,17,4) designs 1 -53designs 55 - 107:excluding block 1 5 6 7 8 21 22 23 36 37 38 51 52 53 60 61 62from (69,17,4) designs 1 - 53designs 108:excluding block 0 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47from (69,17,4) design 1designs 109:172 DEAN CRNKOVIC AND SANJA RUKAVINAexcluding block 1 5 6 7 8 21 22 23 36 37 38 51 52 53 60 61 62from (69,17,4) design 1designs 110 - 321:excluding representatives of block orbits of length 12 from (69,17,4) de-signs 1-53design 322:excluding block 1 5 6 7 8 21 22 23 36 37 38 51 52 53 60 61 62from (69,17,4) design 54design 323:excluding block 0 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47from (69,17,4) design 54design 324:excluding xed block from (69,17,4) design 55designs 325 - 336:excluding representatives of block orbits of length 2 from (69,17,4) designs55 - 57design 337:excluding block 1 2 9 27 30 33 36 39 42 45 48 51 52 54 55 57 60from (69,17,4) design 55design 338:excluding block 3 4 12 21 24 39 42 46 47 49 50 53 56 57 60 63 66from (69,17,4) design 55designs 339 - 341:excluding block 18 21 24 27 30 33 36 40 41 43 44 58 59 61 62 63 66from (69,17,4) designs 55 - 57designs 342 - 365:excluding representatives of block orbits of length 6 from (69,17,4) designs55 - 57designs 366 - 368:excluding previously presented blocks from (69,17,4) design 58designs 369 - 377:excluding all nine block representatives from (69,17,4) design 59Distribution of orders of full automorphism groups of obtained 2-(17,4,3)designs is: (240,1), (4,2), (122,3), (3,4), (1,6), (2,12), (1,16), (1,32), (1,39),(1,48) and (1,78), where the rst number is the number of designs and thesecond number is the order of an automorphism group.Full automorphism groups of designs 110 - 321, 342 - 365 and 374 -377are trivial and full automorphism groups of designs 337, 338, 372 and 373 areisomorphic the group Z2. Furthermore, full automorphism groups of designs2 - 107, 323, 325 - 336, 367, 370 and 371 are isomorphic to the group Z3 andfull automorphism groups of designs 339 - 341 are isomorphic to the groupZ4. Full automorphism group of design 324 is isomorphic to the group Z6.SOME NEW 2-(17,4,3) AND 2-(52,13,4) DESIGNS 173Full automorphism groups of designs 366 and 369 are isomorphic to the groupZ12. Full automorphism group of design 108 is non-abelian group of order 16.Generators a, b, c and d of that group satisfy conditions: a2 = b2 = c2 = d2 =1, ab = ac = ad = a, bc = ab, bd = c and cd = b. Full automorphism group ofdesign 109 is non-abelian group of order 32. Generators a, b, c, d and e of thatgroup satisfy conditions: a2 = b2 = c2 = d2 = e2 = 1, ab = ac = ad = ae = a,bc = bd = be = b, cd = ce = c and de = bd. Full automorphism group of design322 is isomorphic to the group F39. Full automorphism group of design 1 isnon-abelian group of order 48. Generators a, b, c, d and e of that group satisfyconditions: a2 = b2 = c2 = d3 = e2 = 1, ab = ac = ad = ae = a, bc = bd = b,be = ab, cd = ce = c and de = d2. Full automorphism group of design 368 isisomorphic to the group Z2  F39.4. Some new 2-(52,13,4) designsExcluding one block and all points that belong to that block from asymmetric (69,17,4) design we obtain a 2-(52,13,4) design.From fty-nine symmetric (69,17,4) designs presented in [6] we have con-structed six hundred forty-nine residual 2-(52,13,4) designs excluding blockorbit representatives from all designs as we did in previous section for de-rived designs. Corollary 1 imply that such a construction of residual designsled us, after eliminating isomorphic copies, to all mutually non-isomorphic2-(52,13,4) designs that can be obtained from given symmetric (69,17,4) de-signs. Using computer program by V. Krcadinac (see [3], [5]) we obtain thatall of six hundred and forty-nine constructed 2-(52,13,4) designs are mutuallynon-isomorphic.Distribution of orders of full automorphism groups of obtained 2-(52,13,4)designs is: (240,1), (14,2), (122,3), (212,4), (3,6), (56,12), and (2,39), wherethe rst number is the number of designs and the second number is the orderof an automorphism group. Full automorphism groups are presented below.Trivial full automorphism groups: designs obtained from (69,17,4) designs 1 - 53 excluding block orbitrepresentatives of block orbits of length twelfth, designs obtained from (69,17,4) designs 55 - 57 excluding block orbitrepresentatives of block orbits of length six, designs obtained from (69,17,4) design 59 excluding block orbit repre-sentatives of block orbits of length twelfth.Full automorphism groups isomorphic to the group Z2: designs obtained from (69,17,4) designs 55 - 57 excluding block orbitrepresentatives of block orbits of length three, designs obtained from (69,17,4) design 59 excluding block orbit repre-sentatives of block orbits of length six.Full automorphism groups isomorphic to the group Z3:174 DEAN CRNKOVIC AND SANJA RUKAVINA designs obtained from (69,17,4) designs 1 - 53 excluding block orbitrepresentatives of block orbits of length four, designs obtained from (69,17,4) designs 55 - 57 excluding block orbitrepresentatives of block orbits of length two, designs obtained from (69,17,4) design 59 excluding block orbit repre-sentatives of block orbits of length four, design obtained from (69,17,4) design 54 excluding block1 5 6 7 8 21 22 23 36 37 38 51 52 53 60 61 62, design obtained from (69,17,4) design 58 excluding block1 5 6 7 8 21 22 23 36 37 38 51 52 53 57 58 59.Full automorphism groups isomorphic to the group Z4: designs obtained from (69,17,4) designs 1 - 53 excluding block orbitrepresentatives of block orbits of length four.Full automorphism groups isomorphic to the group Z6: designs obtained from (69,17,4) designs 55 - 57 excluding xed blocks.Full automorphism groups isomorphic to the group Z12: designs obtained from (69,17,4) designs 1 - 53 and 55 - 57 excludingxed blocks, designs obtained from (69,17,4) designs 54 and 58 excluding block0 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47.Full automorphism groups isomorphic to the group F39: designs obtained from (69,17,4) designs 54 and 58 excluding block0 1 2 3 4 21 22 23 24 25 26 27 28 29 30 31 32 .Remark 4.1. We constructed up to isomorphism all 2-(52,13,4) designsthat can be obtained as residual designs of symmetric (69,17,4) designs withan automorphism of order six. However, an explicit construction of non-embeddable 2-(52,13,4) design, given by Tran van Trung (see [9]) proves theexistence of 2-(52,13,4) design that cannot be obtained as residual design ofany (69,17,4) design..References[1] Z. Bozikov, Symmetric designs with parameters (69,17,4) and F39 as a group of auto-morphisms, J. Comb. Des. 6, No. 4, (1998), 231-233.[2] J. Dinitz, New Results in Combinatorial Designs, Part 1, el. document, August 2000,http://www.emba.uvm.edu/~dinitz/newresults.part1.html[3] V. Krcadinac, Steiner 2-designs S k; 2k2   2k + 1, MSc thesis, University of Zagreb(1999).[4] E. Lander, Symmetric Designs: An Algebraic Approach, Cambridge University Press(1983).[5] B.D. McKay, Nauty Users Guide (version 1.5) Technical Report TR-CS-90-02, Depart-ment of Computer Science, Australian National University, 1990.[6] S. Rukavina, Enumeration of symmetric (69,17,4) designs admitting Z6 as an automor-phism group, Glasnik matematicki, 34 (54) (1999), 129-145.SOME NEW 2-(17,4,3) AND 2-(52,13,4) DESIGNS 175[7] S. S. Shrikhande and N. M. Singhi, Construction of geometroids, Utilitas Math. 8,(1975), 187-192.[8] Tran van Trung, Symmetric Designs, in The CRC Handbook of Combinatorial Designs,(ed. C. J. Colbourn and J. H. Dinitz), CRC press (1996), 75-87.[9] Tran van Trung, Nonembeddable quasi-residual designs, Contemp. Math. 111, (1990),237-278.Department of mathematicsFaculty of philosophy in RijekaOmladinska 14, 51000 Rijeka, CroatiaE-mail : deanc@mapef.pefri.hrDepartment of mathematicsFaculty of philosophy in RijekaOmladinska 14, 51000 Rijeka, CroatiaE-mail : sanjar@mapef.pefri.hrReceived : 10.05.2000.Revised : 18.09.2000.

Some new 2-(17,4,3) and 2-(52,13,4) design

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GLASNIK MATEMATIČKIVol. 36(56)(2001), 169 – 175SOME NEW 2-(17,4,3) AND 2-(52,13,4) DESIGNSDean Crnković and Sanja RukavinaUniversity of Rijeka, CroatiaAbstract. We have constructed three hundred and seventy-sevennon-isomorphic 2-(17,4,3) designs as derived and six hundred and forty-nine non-isomorphic 2-(52,13,4) designs as residual designs of symmetric(69,17,4) designs admitting an action of Z6.1. Introduction and preliminariesA 2− (v, k, λ) design is a finite incidence structure (P ,B, I), where P andB are disjoint sets and I ⊆ P × B, with the following properties:1. |P| = v,2. every element of B is incident with exactly k elements of P ,3. every pair of elements of P is incident with exactly λ elements of B.Elements of B are called blocks, and elements of P are called points. For x ∈ Band P ∈ P , denote 〈x〉 = {Q ∈ P|(Q, x) ∈ I}, 〈P 〉 = {y ∈ B|(P, y) ∈ I}. Fortwo designs D1 = (P1,B1, I1) and D2 = (P2,B2, I2) an isomorphism from D1onto D2 is a bijection which maps points onto points and blocks onto blockspreserving the incidence. If there exists an isomorphism from D1 onto D2, weshall say that D1 and D2 are isomorphic and write D1 ∼= D2. An isomorphismfrom D onto D is an automorphism of D. The set of all automorphism ofthe design D is a group called the full automorphism group of D, and it isdenoted by AutD. Each subgroup of the AutD is an automorphism group ofD. A symmetric (v, k, λ) design is a 2− (v, k, λ) design with |P| = |B| = v.First known symmetric (69,17,4) design was constructed by Shrikhandeand Singhi (see [7]). Designs with these parameters belong to the familyof symmetric designs satisfying conditions v = m3 + m + 1, k = m2 + 12000 Mathematics Subject Classification. 05B05.Key words and phrases. symmetric design, residual design, derived design, automor-phism, orbit structure.169170 DEAN CRNKOVIĆ AND SANJA RUKAVINAand λ = m, where both m − 1 and m2 −m + 1 are prime powers (see [8]).Recently Z. Božikov (see [1]) constructed two symmetric designs with param-eters (69,17,4) having Frobenius group F39 as an automorphism group, out ofwhich one was new and the other was previously constructed by Shrikhandeand Singhi. In august 1998., S. Topalova reported that there are four sym-metric (69,17,4) designs with automorphisms of order 13 (see [2]). Finally, S.Rukavina constructed all symmetric (69,17,4) designs admitting an action ofcyclic group of order six (see [6]), out of which fifty-seven designs were newand two designs admitting an action of Frobenius group F39 were previouslyknown.2. Derived and residual designsLet D be a symmetric (v, k, λ) design. The derived design Dx with respectto a block x is the incidence structure whose points are the points of x andwhose blocks are the sets 〈y〉 ∩ 〈x〉, where y is any block other than x. Dx isa 2 − (k, λ, λ − 1) design. The residual design Dx with respect to a block xis the incidence structure whose points are those points of D which are notincident with x and whose blocks are the sets 〈y〉 − 〈x〉, where y ranges overall blocks other than x. It is a 2− (v − k, k − λ, λ) design.Theorem 1. Let α be an isomorphism of symmetric designs D = (P ,B, I)and D′ = (P ′,B′, I ′). Further, let x ∈ B and xα = x′ ∈ B′. Then α1 = α|Dxis an isomorphism from Dx onto D′x′ and α2 = α|Dx is an isomorphism fromDx onto D′x′.Proof. Let x1 ∈ B. (〈x1〉 ∩ 〈x〉)α1 = 〈x1〉α1 ∩ 〈x〉α1 = 〈x1α1〉 ∩ 〈x′〉and (〈x1〉 − 〈x〉)α2 = 〈x1〉α2 − 〈x〉α2 = 〈x1α2〉 − 〈x′〉, which means that α1maps blocks from Dx onto blocks from D′x′ and α2 maps blocks from Dx ontoblocks from D′x′.Corollary 1. Let D = (P ,B, I) be a symmetric design, x, x′ ∈ B andG ≤ AutD. If x′ ∈ xG, then Dx ∼= Dx′ and Dx ∼= Dx′.Proof. There exists α ∈ G such that x′ = xα. α|Dx is an isomorphismfrom Dx onto Dx′ and α|Dx is an isomorphism from Dx onto Dx′.Corollary 2. Let D = (P ,B, I) be a symmetric design, x ∈ B andα ∈ AutD. If xα = x then α|Dx ∈ AutDx and α|Dx ∈ AutDx.3. Some new 2-(17,4,3) designsExcluding one block and all points that do not belong to that block fromsymmetric (69,17,4) design we obtain 2-(17,4,3) design.In [6] fifty-nine mutually non-isomorphic symmetric (69,17,4) designs arepresented. In that article, representatives of block orbits for constructedSOME NEW 2-(17,4,3) AND 2-(52,13,4) DESIGNS 171designs and corresponding automorphisms are given. From corollary 1 fol-lows that construction of derived designs corresponding to block orbit repre-sentatives led us, after eliminating isomorphic copies, to all mutually non-isomorphic 2-(17,4,3) designs that can be obtained from given symmetric(69,17,4) design. For all designs in [6], except designs 54 and 58, repre-sentatives of block orbits of full automorphism groups of those designs aregiven. Designs 54 and 58 are presented in [6] by representatives of subgroupsof full automorphism groups. Therefore, except in those two cases, from each(69,17,4) design we shall construct derived designs with respect to the repre-sentatives of block orbits given in [6]. On designs 54 and 58 full automorphismgroups of those designs act in three block orbits. For construction of deriveddesigns we shall consider one block from each of those three orbits. Precisely,from design 54 we shall exclude the following blocks:0 1 2 3 4 21 22 23 24 25 26 27 28 29 30 31 321 5 6 7 8 21 22 23 36 37 38 51 52 53 60 61 620 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47From design 58 we shall exclude the following blocks:1 5 6 7 8 21 22 23 36 37 38 51 52 53 57 58 590 1 2 3 4 21 22 23 24 25 26 27 28 29 30 31 320 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47Described process gave us six hundred and forty-nine 2-(17,4,3) designs.Using computer program by V. Krčadinac (see [3], [5]) we obtain that betweenthem there are three hundred and seventy-seven mutually non-isomorphicdesigns.For each of three hundred and seventy-seven mutually non-isomorphic 2-(17,4,3) design we present (or describe) block that should be excluded fromsymmetric (69,17,4) design. Ordinal numbers of symmetric (69,17,4) designscorrespond to those in [6].design 1:excluding fixed block from (69,17,4) design 1designs 2 - 54:excluding block 0 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47from (69,17,4) designs 1 -53designs 55 - 107:excluding block 1 5 6 7 8 21 22 23 36 37 38 51 52 53 60 61 62from (69,17,4) designs 1 - 53designs 108:excluding block 0 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47from (69,17,4) design 1designs 109:172 DEAN CRNKOVIĆ AND SANJA RUKAVINAexcluding block 1 5 6 7 8 21 22 23 36 37 38 51 52 53 60 61 62from (69,17,4) design 1designs 110 - 321:excluding representatives of block orbits of length 12 from (69,17,4) de-signs 1-53design 322:excluding block 1 5 6 7 8 21 22 23 36 37 38 51 52 53 60 61 62from (69,17,4) design 54design 323:excluding block 0 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47from (69,17,4) design 54design 324:excluding fixed block from (69,17,4) design 55designs 325 - 336:excluding representatives of block orbits of length 2 from (69,17,4) designs55 - 57design 337:excluding block 1 2 9 27 30 33 36 39 42 45 48 51 52 54 55 57 60from (69,17,4) design 55design 338:excluding block 3 4 12 21 24 39 42 46 47 49 50 53 56 57 60 63 66from (69,17,4) design 55designs 339 - 341:excluding block 18 21 24 27 30 33 36 40 41 43 44 58 59 61 62 63 66from (69,17,4) designs 55 - 57designs 342 - 365:excluding representatives of block orbits of length 6 from (69,17,4) designs55 - 57designs 366 - 368:excluding previously presented blocks from (69,17,4) design 58designs 369 - 377:excluding all nine block representatives from (69,17,4) design 59Distribution of orders of full automorphism groups of obtained 2-(17,4,3)designs is: (240,1), (4,2), (122,3), (3,4), (1,6), (2,12), (1,16), (1,32), (1,39),(1,48) and (1,78), where the first number is the number of designs and thesecond number is the order of an automorphism group.Full automorphism groups of designs 110 - 321, 342 - 365 and 374 -377are trivial and full automorphism groups of designs 337, 338, 372 and 373 areisomorphic the group Z2. Furthermore, full automorphism groups of designs2 - 107, 323, 325 - 336, 367, 370 and 371 are isomorphic to the group Z3 andfull automorphism groups of designs 339 - 341 are isomorphic to the groupZ4. Full automorphism group of design 324 is isomorphic to the group Z6.SOME NEW 2-(17,4,3) AND 2-(52,13,4) DESIGNS 173Full automorphism groups of designs 366 and 369 are isomorphic to the groupZ12. Full automorphism group of design 108 is non-abelian group of order 16.Generators a, b, c and d of that group satisfy conditions: a2 = b2 = c2 = d2 =1, ab = ac = ad = a, bc = ab, bd = c and cd = b. Full automorphism group ofdesign 109 is non-abelian group of order 32. Generators a, b, c, d and e of thatgroup satisfy conditions: a2 = b2 = c2 = d2 = e2 = 1, ab = ac = ad = ae = a,bc = bd = be = b, cd = ce = c and de = bd. Full automorphism group of design322 is isomorphic to the group F39. Full automorphism group of design 1 isnon-abelian group of order 48. Generators a, b, c, d and e of that group satisfyconditions: a2 = b2 = c2 = d3 = e2 = 1, ab = ac = ad = ae = a, bc = bd = b,be = ab, cd = ce = c and de = d2. Full automorphism group of design 368 isisomorphic to the group Z2 × F39.4. Some new 2-(52,13,4) designsExcluding one block and all points that belong to that block from asymmetric (69,17,4) design we obtain a 2-(52,13,4) design.From fifty-nine symmetric (69,17,4) designs presented in [6] we have con-structed six hundred forty-nine residual 2-(52,13,4) designs excluding blockorbit representatives from all designs as we did in previous section for de-rived designs. Corollary 1 imply that such a construction of residual designsled us, after eliminating isomorphic copies, to all mutually non-isomorphic2-(52,13,4) designs that can be obtained from given symmetric (69,17,4) de-signs. Using computer program by V. Krčadinac (see [3], [5]) we obtain thatall of six hundred and forty-nine constructed 2-(52,13,4) designs are mutuallynon-isomorphic.Distribution of orders of full automorphism groups of obtained 2-(52,13,4)designs is: (240,1), (14,2), (122,3), (212,4), (3,6), (56,12), and (2,39), wherethe first number is the number of designs and the second number is the orderof an automorphism group. Full automorphism groups are presented below.Trivial full automorphism groups:• designs obtained from (69,17,4) designs 1 - 53 excluding block orbitrepresentatives of block orbits of length twelfth,• designs obtained from (69,17,4) designs 55 - 57 excluding block orbitrepresentatives of block orbits of length six,• designs obtained from (69,17,4) design 59 excluding block orbit repre-sentatives of block orbits of length twelfth.Full automorphism groups isomorphic to the group Z2:• designs obtained from (69,17,4) designs 55 - 57 excluding block orbitrepresentatives of block orbits of length three,• designs obtained from (69,17,4) design 59 excluding block orbit repre-sentatives of block orbits of length six.Full automorphism groups isomorphic to the group Z3:174 DEAN CRNKOVIĆ AND SANJA RUKAVINA• designs obtained from (69,17,4) designs 1 - 53 excluding block orbitrepresentatives of block orbits of length four,• designs obtained from (69,17,4) designs 55 - 57 excluding block orbitrepresentatives of block orbits of length two,• designs obtained from (69,17,4) design 59 excluding block orbit repre-sentatives of block orbits of length four,• design obtained from (69,17,4) design 54 excluding block1 5 6 7 8 21 22 23 36 37 38 51 52 53 60 61 62,• design obtained from (69,17,4) design 58 excluding block1 5 6 7 8 21 22 23 36 37 38 51 52 53 57 58 59.Full automorphism groups isomorphic to the group Z4:• designs obtained from (69,17,4) designs 1 - 53 excluding block orbitrepresentatives of block orbits of length four.Full automorphism groups isomorphic to the group Z6:• designs obtained from (69,17,4) designs 55 - 57 excluding fixed blocks.Full automorphism groups isomorphic to the group Z12:• designs obtained from (69,17,4) designs 1 - 53 and 55 - 57 excludingfixed blocks,• designs obtained from (69,17,4) designs 54 and 58 excluding block0 5 9 10 11 21 22 23 33 34 35 39 40 41 45 46 47.Full automorphism groups isomorphic to the group F39:• designs obtained from (69,17,4) designs 54 and 58 excluding block0 1 2 3 4 21 22 23 24 25 26 27 28 29 30 31 32 .Remark 4.1. We constructed up to isomorphism all 2-(52,13,4) designsthat can be obtained as residual designs of symmetric (69,17,4) designs withan automorphism of order six. However, an explicit construction of non-embeddable 2-(52,13,4) design, given by Tran van Trung (see [9]) proves theexistence of 2-(52,13,4) design that cannot be obtained as residual design ofany (69,17,4) design..References[1] Z. Božikov, Symmetric designs with parameters (69,17,4) and F39 as a group of auto-morphisms, J. Comb. Des. 6, No. 4, (1998), 231-233.[2] J. Dinitz, New Results in Combinatorial Designs, Part 1, el. document, August 2000,http://www.emba.uvm.edu/˜dinitz/newresults.part1.html[3] V. Krčadinac, Steiner 2-designs S(k, 2k2 − 2k + 1), MSc thesis, University of Zagreb(1999).[4] E. Lander, Symmetric Designs: An Algebraic Approach, Cambridge University Press(1983).[5] B.D. McKay, Nauty Users Guide (version 1.5) Technical Report TR-CS-90-02, Depart-ment of Computer Science, Australian National University, 1990.[6] S. Rukavina, Enumeration of symmetric (69,17,4) designs admitting Z6 as an automor-phism group, Glasnik matematički, 34 (54) (1999), 129-145.SOME NEW 2-(17,4,3) AND 2-(52,13,4) DESIGNS 175[7] S. S. Shrikhande and N. M. Singhi, Construction of geometroids, Utilitas Math. 8,(1975), 187-192.[8] Tran van Trung, Symmetric Designs, in The CRC Handbook of Combinatorial Designs,(ed. C. J. Colbourn and J. H. Dinitz), CRC press (1996), 75-87.[9] Tran van Trung, Nonembeddable quasi-residual designs, Contemp. Math. 111, (1990),237-278.Department of mathematicsFaculty of philosophy in RijekaOmladinska 14, 51000 Rijeka, CroatiaE-mail : deanc@mapef.pefri.hrDepartment of mathematicsFaculty of philosophy in RijekaOmladinska 14, 51000 Rijeka, CroatiaE-mail : sanjar@mapef.pefri.hrReceived : 10.05.2000.Revised : 18.09.2000.

Some new 2-(17,4,3) and 2-(52,13,4) design

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