We have constructed three new symmetric (71,35,17) designs having $Frob_{17 cdot 8}$ as an automorphism group.
Also constructed are symmetric designs with parameters (43,21,10), (67,33,16) and (71,35,17) admitting an
automorphism group isomorphic to $Frob_{43 cdot 7}$, $Frob_{67 cdot 11}$ and $Frob_{71 cdot 7}$, 
respectively, which turn out to be isomorphic to previously known designs, constructed via cyclic 
difference sets. Each constructed Hadamard design leads to a series of Hadamard designs

S. Crnković

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Mathematical Communications 7(2002), 65-70 65Some Hadamard designs admitting a faithfulaction of Frobenius groupsDean Crnkovic´∗Abstract. We have constructed three new symmetric (71,35,17)designs having Frob17·8 as an automorphism group. Also constructed aresymmetric designs with parameters (43,21,10), (67,33,16) and (71,35,17)admitting an automorphism group isomorphic to Frob43·7, Frob67·11 andFrob71·7, respectively, which turn out to be isomorphic to previouslyknown designs, constructed via cyclic difference sets. Each constructedHadamard design leads to a series of Hadamard designs.Key words: symmetric design, Hadamard design, Frobenius group,automorphism groupAMS subject classifications: 05B05, 05E20Received November 16, 2001 Accepted June 27, 20021. Introduction and preliminariesFor the definition of a symmetric (v, k, λ) design (P ,B, I), its automorphism group,and isomorphisms between such structures, we refer the reader to [5, Chapter 1,sections 1.1 and 1.2]. As usual, P is the point set, B the block set, and I ⊆ P × Bthe incidence relation. The point and the block sets are mutually disjoint.Let D = (P ,B, I) be a symmetric (v, k, λ) design and G ≤ AutD. The groupaction of G produces the same number of point and block orbits. We denote thatnumber by t, the point orbits by P1, . . . ,Pt, the block orbits by B1, . . . ,Bt, and put|Pr| = ωr and |Bi| = Ωi. We shall denote the points of the orbit Pr by r0, . . . , rωr−1,(i.e. Pr = {r0, . . . , rωr−1}). Further, we denote by γir the number of points of Prwhich are incident with a representative of the block orbit Bi. For those numbersthe following equalities hold:t∑r=1γir = k , (1)t∑r=1Ωjωrγirγjr = λΩj + δij · (k − λ) . (2)∗Department of Mathematics, Faculty of Philosophy, Omladinska 14, HR-51 000 Rijeka, Croa-tia, e-mail: deanc@mapef.pefri.hr66 D.Crnkovic´Definition 1. A (t×t)-matrix (γir) with entries satisfying conditions (1) and (2)is called an orbit structure for the parameters (v, k, λ) and orbit lengths distributions(ω1, . . . , ωt), (Ω1, . . . ,Ωt).Definition 2. The set of those indices of points of the orbit Pr, which areincident with a representative of the block orbit Bi, is called the index set for theposition (i, r) of the orbit structure and the given representative.Determining the index sets when constructing symmetric designs is called in-dexing.A Hadamard matrix of order m is an (m×m)-matrix H = (hi,j), hi,j ∈ {−1, 1},satisfying HHT = HTH = mI, where I is the unit matrix. From each Hadamardmatrix of order m one can obtain a symmetric (m− 1, 12m− 1, 14m− 1) design (see[5]). Also, from any symmetric (m − 1, 12m − 1, 14m − 1) design we can recover aHadamard matrix. Symmetric designs with parameters (m−1, 12m−1, 14m−1) arecalled Hadamard designs.Definition 3. Let G be an additively written group of order v. A k−subset Dof G is a (v, k, λ;n)−difference set of order n = k − λ if every nonzero element ofG has exactly λ representations as a difference d − d′ with elements from D. Thedifference set is abelian, cyclic etc. if the group G has the respective property.A faithful action of a group G on a design D is a monomorphism from the groupG into the automorphism group AutD.For further basic definitions and construction procedures we refer the reader to[3], [4] and [9].2. Hadamard (71,35,17) designsThe first symmetric (71, 35, 17) design was constructed via a cyclic difference set(see [4], [6], [9]). Eight symmetric (71, 35, 17) designs admitting an automorphismgroup isomorphic to Frob21 were constructed recently (see [1]). None of those eightdesigns are isomorphic to the one constructed using a difference set.A Frobenius group G ∼= Frob17·8 is presented as follows:G = 〈ρ, σ | ρ17 = 1, σ8 = 1, ρσ = ρ2〉.Theorem 1. Up to isomorphism there exist three symmetric (71, 35, 17) designsadmitting an automorphism group isomorphic to Frob17·8 acting in such a way thatthe cyclic group of order 8 acts standardly. Full automorphism groups of thesedesigns are all isomorphic to the group Frob17·8 of order 136. One design is self-dual, and the other two are mutually dual. The 2-rank of these designs equals 36.Proof. Solving equations (1) and (2) it is easy to conclude that the permutationof order 17 must act on the symmetric (71, 35, 17) design fixing 3 points, with orbitstructures:Hadamard designs and Frobenius groups 67OS1 1 1 1 17 17 17 171 1 0 0 17 17 0 01 0 1 0 17 0 17 01 0 0 1 17 0 0 1717 1 1 1 8 8 8 817 1 0 0 8 8 9 917 0 1 0 8 9 8 917 0 0 1 8 9 9 8OS2 1 1 1 17 17 17 171 1 0 0 17 17 0 01 0 1 0 17 0 17 01 0 0 1 17 0 0 1717 1 1 0 8 8 8 917 1 0 1 8 8 9 817 0 1 1 8 9 8 817 0 0 0 8 9 9 9Orbit structure OS1 is self-dual, and OS2 is non-self-dual.We denote the points by 10, 20, 30, I0, . . . , I16, I = 4, 5, 6, 7. Generators of theFrobenius group G = 〈ρ, σ〉 are defined as follows:ρ = (10)(20)(30)(I0, . . . I16) I = 4, 5, 6, 7,σ = (10)(20)(30)(I0)(I1, I2, I4, I8, I16, I15, I13, I9)(I3, I6, I12, I7, I14, I11, I5, I10), I =4, 5, 6, 7.Index sets which could occur in designs constructed from OS1 and OS2 are:0 . . . 1, 2, 4, 8, 9, 13, 15, 16, 1 . . . 3, 5, 6, 7, 10, 11, 12, 14,2 . . . 0, 1, 2, 4, 8, 9, 13, 15, 16, 3 . . . 0, 3, 5, 6, 7, 10, 11, 12, 14.Indexing of the columns and rows corresponding to the fixed points and blocksis trivial. Therefore, we shall take into consideration only right-lower (4 × 4)-submatrices of the orbit structures. The orbit structure OS1 produces the self-dual design D1, and OS2 produces the non-self-dual design D2. Those designs arepresented by (4× 4)-matrices of index sets as follows:D1 D20 0 1 11 1 3 30 3 1 21 2 3 00 0 1 21 1 3 00 3 1 11 2 3 3The orbit structure dual to OS2 produces a design dual to D2.A computer program by Vladimir D. Tonchev [8] computes the orders as wellas the generators of the full automorphism groups of these designs. Application ofanother computer program by Tonchev [8] yields the assertion about the 2-ranks.✷Theorem 1 implies that there are at least 12 pairwise nonisomorphic symmetric(71,35,17) designs.68 D.Crnkovic´3. Series of Hadamard designsFrom each Hadamard design, using the following well known theorem, one canconstruct a series of Hadamard designs.Theorem 2. Let M be an incidence matrix of a Hadamard design D with para-meters (m−1, 12m−1, 14m−1), andM be the incidence matrix of the complementarydesign Dc. Then0 1 · · · 1 0 · · · 01... M M10... M M0is an incidence matrix of a Hadamard design D1 with parameters (2m − 1,m −1, 12m− 1). Each automorphism of the design D is an automorphism of the designD1. If the design D is self-dual, so is the design D1.Proof. It is easy to verify properties 1, 2, and 3 from the definition of sym-metric designs. Statements about automorphism and self-duality are clear from theconstruction of the incidence matrix of the design D1. ✷Applying Theorem 2 on the result from Theorem 1 we get the following corollary:Corollary 1. For each non-negative integer k there exist one dual and oneself-dual Hadamard design with parameters (72 ·2k−1, 36 ·2k−1, 18 ·2k−1) havingFrob17·8 as an automorphism group.Using the described method one can construct other Hadamard designs and thecorresponding series of Hadamard designs.Example 1. A Frobenius group G1 of order 497 is presented as follows:G1 = 〈ρ, σ | ρ71 = 1, σ7 = 1, ρσ = ρ20〉.The group G1 must act on a symmetric (71,35,17) design D standardly withorbit lengths distribution (71), (71). The only possible orbit structure for parameters(71,35,17) and the group G1 is OS3:OS3 7171 35Up to isomorphism there is only one symmetric (71, 35, 17) design admitting a faith-ful action of a Frobenius group of order 497. The full automorphism group of thisdesign is isomorphic to the group Frob71·35 of order 2485. This design is self-dual.The 2-rank of the design equals 36. This design is isomorphic to the one describedin [4].Applying Theorem2, we get that for each non-negative integer k there exists aself-dual Hadamard design with parameters (72 ·2k−1, 36 ·2k−1, 18 ·2k−1) havingFrob71·35 as an automorphism group.Hadamard designs and Frobenius groups 69Example 2. According to [6], until 1996 there were two known symmetric(43, 21, 10) designs. These designs were constructed via difference sets. Later on,S. Topalova constructed further 80 symmetric (43, 21, 10) designs admitting an au-tomorphism group isomorphic to Z21 (see [2] and [7]).Indexing of the orbit structureOS4 4343 21with an assumption that a Frobenius group of order 301 acts as an automorphismgroup, produces exactly two mutually nonisomorphic designs, isomorphic to the onesconstructed via cyclic difference sets. So, we have proved that up to isomorphismthere exist precisely two symmetric designs with parameters (43, 21, 10) admitting afaithful action of a Frobenius group of order 301. The full automorphism groups ofthese designs are isomorphic to Frob43·7, and Frob43·21. Both designs are self-dual.The 2-rank of these two designs equals 43.Theorem2 yields that for each non-negative integer k there exists a self-dualHadamard design with parameters (44 ·2k−1, 22 ·2k−1, 11 ·2k−1) having Frob43·21as an automorphism group.Example 3. There is only one known Hadamard (67, 33, 16) design (see [6]).This design is constructed using a cyclic difference set (see [4]). In the same wayas in the case of symmetric (71, 35, 17) design with the group Frob71·7, one canprove that up to isomorphism there exists only one symmetric design with parame-ters (67, 33, 16) admitting a faithful action of a Frobenius group of order 737. Fullautomorphism group of this design is isomorphic to Frob67·33 of order 2211. Thisdesign is self-dual. The 2-rank of the design equals 67. This design is isomorphicto the one described in [4].For each non-negative integer k there exists a self-dual Hadamard design withparameters (68 · 2k− 1, 34 · 2k− 1, 17 · 2k− 1) having Frob67·33 as an automorphismgroup.References[1] D.Crnkovic´, D.Held, Some Hadamard designs with parameters (71,35,17),J. Combin Designs 9(2001), 1-6.[2] J.Dinitz, New Results in Combinatorial Designs, Part 1, el. document, July2001, http://www.emba.uvm.edu/~dinitz/newresults.part1.html[3] Z. Janko, Coset enumeration in groups and constructions of symmetric de-signs, Combinatorics 90(1992), 275–277.[4] D. Jungnickel, A.Pott, Difference sets: Abelian, in: The CRC Handbookof Combinatorial Designs, (C. J.Colbourn and J.H. Dinitz, Eds.), CRC press,1996, 297-307.[5] E. Lander, Symmetric Designs: An Algebraic Approach, Cambridge Univer-sity Press, 11983.70 D.Crnkovic´[6] R.Mathon, A.Rosa, 2−(v, k, λ) designs of small order, in: The CRC Hand-book of Combinatorial Designs, (C. J.Colbourn and J.H.Dinitz, Eds.), CRCpress, 1996, 3-41.[7] S.Topalova, Hadamard 2-(43,21,10) designs with automorphisms of order21 and binary self-dual codes of lengths 86 and 88, Proceedings of the II Int.Workshop on Optimal Codes and Related Topics, Sozopol, Bulgaria, June,1998.[8] V.Tonchev, Michigan Technological University, private communication viaZ. Janko of Universita¨t Heidelberg.[9] W.D.Wallis, A.P. Street, J. S.Wallis, Combinatorics: Room Squares,Sum-Free Sets, Hadamard matrices, Springer Verlag, Berlin-Heidelberg-NewYork, 1972.

Some Hadamard designs admitting a faithful action of Frobenius groups

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