New difference sets in nonabelian groups of order 100

In two groups of order 100 new difference sets are constructed. The existence of a difference set in one of them has not been known. The correspondence between a (100, 45, 20) symmetric design having regular automorphism group and a difference set with the same parameters has been used for the construction. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 424–434, 2001     

Ruling out (160, 54, 18) difference sets in some nonabelian groups

We prove the following theorems. Theorem A. Let G be a group of order 160 satisfying one of the following conditions. (1) G has an image isomorphic to D₂₀ × Z₂ (for example, if G ≃ D₂₀ × K). (2) G has a normal 5‐Sylow subgroup and an elementary abelian 2‐Sylow subgroup. (3) G has an abelian image of exponent 2, 4, 5, or 10 and order greater than 20. Then G cannot contain a (160, 54, 18) difference set. Theorem B. Suppose G is a nonabelian group with 2‐Sylow subgroup S and 5‐Sylow subgroup T and contains a (160, 54, 18) difference set. Then we have one of three possibilities. (1) T is normal, |ϕ(S)| = 8, and one of the following is true: (a) G = S × T and S is nonabelian; (b) G has a D₁₀ image; or (c) G has a Frobenius image of order 20. (2) G has a Frobenius image of order 80. (3) G is of index 6 in A Γ L(1, 16). To prove the first case of Theorem A, we find the possible distribution of a putative difference set with the stipulated parameters among the cosets of a normal subgroup using irreducible representations of the quotient; we show that no such distribution is possible. The other two cases are due to others. In the second (due to Pott) irreducible representations of the elementary abelian quotient of order 32 give a contradiction. In the third (due to an anonymous referee), the contradiction derives from a theorem of Lander together with Dillon's “dihedral trick.” Theorem B summarizes the open nonabelian cases based on this work. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 221–231, 2000     

A new symmetric design with parameters (176, 50, 14)

A new symmetric non‐self‐dual design with parameters (176, 50, 14) is constructed, where its full automorphism group of order 1000 is a faithful extension of the non‐abelian group of order 125 and exponent 5 by the cyclic group of order 8. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 387–390, 2000     

Primitive symmetric designs with prime power number of points

In this paper we either prove the non‐existence or give explicit construction of primitive symmetric (v, k, λ) designs with v=p^m<2500, p prime and m>1. The method of design construction is based on an automorphism group action; non‐existence results additionally include the theory of difference sets, multiplier theorems in particular. The research involves programming and wide‐range computations. We make use of software package GAP and the library of primitive groups which it contains. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 141–154, 2010     

Some Hadamard designs with parameters (71,35,17)

Up to isomorphisms there are precisely eight symmetric designs with parameters (71, 35, 17) admitting a faithful action of a Frobenius group of order 21 in such a way that an element of order 3 fixes precisely 11 points. Five of these designs have 84 and three have 420 as the order of the full automorphism group G. If |G| = 420, then the structure of G is unique and we have G = (Frob₂₁ × Z5):Z4. In this case Z(G) = 〈1〉, G′ has order 35, and G induces an automorphism group of order 6 of Z7. If |G| = 84, then Z(G) is of order 2, and in precisely one case a Sylow 2‐subgroup is elementary abelian. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 144–149, 2002; DOI 10.1002/jcd.996     

Symmetric designs with parameters (69, 17, 4) and F39 as a group of automorphisms

According to Mathon and Rosa [The CRC handbook of combinatorial designs, CRC Press, 1996] there is only one known symmetric design with parameters (69, 17, 4). This known design is given in Beth, Jungnickel, and Lenz [Design theory, B. I. Mannheim, 1985]; the Frobenius group F₃₉ of order 39 acts on this design, where Z₁₃ has exactly 4 fixed points and Z₃ has exactly 9 fixed points. The purpose of this article is to investigate the converse of this fact with the hope of obtaining in this way at least one more design with these parameters. In fact we obtain exactly one new such design. In this article we have classified all such designs invariant under F₃₉. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 231–233, 1998     

Primitive symmetric designs with up to 2500 points

By this article we conclude the construction of all primitive ( v, k,λ ) symmetric designs with v < 2500 , up to a few unsolved cases. Complementary to the designs with prime power number of points published previously, here we give 55 primitive symmetric designs with v ≠ p ^m , p prime and m positive integer, together with the analysis of their full automorphism groups. The research involves programming and wide‐range computations. We make use of the software package GAP and the library of primitive groups which it contains. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:463‐474, 2011     

A classification of all symmetric block designs of order nine with an automorphism of order six

We complete the classification of all symmetric designs of order nine admitting an automorphism of order six. As a matter of fact, the classification for the parameters (35,17,8), (56,11,2), and (91,10,1) had already been done, and in this paper we present the results for the parameters (36,15,6), (40,13,4), and (45,12,3). We also provide information about the order and the structure of the full automorphism groups of the constructed designs. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 301–312, 2006     

New symmetric 2-(176,50,14) designs

In this paper, we construct two new symmetric designs with parameters 2-$(176,50,14)$ as designs invariant under certain subgroups of the full automorphism group of the Higman design. One is self-dual and has the full automorphism group of order 11520 and the other one is not self-dual and has the full automorphism group of order 2520.     

Investigation of solvable (120, 35, 10) difference sets

Abelian difference sets with parameters (120, 35, 10) were ruled out by Turyn in 1965. Turyn's techniques do not apply to nonabelian groups. We attempt to determine the existence of (120, 35, 10) difference sets in the 44 nonabelian groups of order 120. We prove that if a solvable group admits a (120, 35, 10) difference set, then it admits a quotient group isomorphic to the cyclic group of order 24 or to U₂₄ ? 〈x,y : x⁸ = y³ = 1, xyx^?1 = y^?1〉. We describe a computer search, which rules out solutions with a ?₂₄ quotient. The existence question remains undecided in the three solvable groups admitting a U₂₄ quotient. The question also remains undecided for the three nonsolvable groups of order 120. © 2004 Wiley Periodicals, Inc.     

Linking systems of difference sets

A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2‐groups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a linking system of difference sets in non‐2‐groups. We make a connection to Kerdock and bent sets, which provides large linking systems of difference sets in elementary abelian 2‐groups. We give a new construction for linking systems of difference sets in 2‐groups, taking advantage of a previously unrecognized connection with group difference matrices. This construction simplifies and extends prior results, producing larger linking systems than before in certain 2‐groups, new linking systems in other 2‐groups for which no system was previously known, and the first known examples in nonabelian groups.     

Restrictions on parameters of partial difference sets in nonabelian groups

A partial difference set $S$ in a finite group $G$ satisfying $1?S$ and $S=S^{?1}$ corresponds to an undirected strongly regular Cayley graph $\text{Cay}\left(G,S\right)$. While the case when $G$ is abelian has been thoroughly studied, there are comparatively few results when $G$ is nonabelian. In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian groups and are especially effective in groups with a nontrivial center. In particular, these results apply to $p$‐groups, and we are able to rule out the existence of partial difference sets in many instances.     

Flag-transitive 2-(v, k, λ) symmetric designs with (k, λ) = 1 and alternating socle

Consider the flag-transitive 2-(v, k, λ) symmetric designs with (k, λ) = 1. We prove that if {D} is a nontrivial 2-(v, k, λ) symmetric design with (k, λ) = 1 and G≤Aut({D}) is flag-transitive with Soc(G) = A_n for n≥5, then {D} is the projective space PG₂(3,2) and G = A₇.     

The existence of symmetric designs with parameters (105, 40, 15)

The existence of symmetric designs with parameters (105, 40, 15) was shown. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 17–19, 1999     

Alternating groups and flag‐transitive 2‐(v,k, 4) symmetric designs

In this article, we study the classification of flag‐transitive, point‐primitive 2‐ (v, k, 4) symmetric designs. We prove that if the socle of the automorphism group G of a flag‐transitive, point‐primitive nontrivial 2‐ (v, k, 4) symmetric design ?? is an alternating group A_n for n≥5, then (v, k) = (15, 8) and ?? is one of the following: (i) The points of ?? are those of the projective space PG(3, 2) and the blocks are the complements of the planes of PG(3, 2), G = A₇ or A₈, and the stabilizer G_x of a point x of ?? is L₃(2) or AGL₃(2), respectively. (ii) The points of ?? are the edges of the complete graph K₆ and the blocks are the complete bipartite subgraphs K_{2, 4} of K₆, G = A₆ or S₆, and G_x = S₄ or S₄ × Z₂, respectively. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:475‐483, 2011     

One (96,20,4)-symmetric Design and related Nonabelian Difference Sets

New (96,20,4)-symmetric design has been constructed, unique under the assumption of an automorphism group of order 576 action. The correspondence between a (96,20,4)-symmetric design having regular automorphism group and a difference set with the same parameters has been used to obtain difference sets in five nonabelian groups of order 96. None of them belongs to the class of groups that allow the application of so far known construction (McFarland, Dillon) for McFarland difference sets.AMS lassification: 05B05     

Quasi‐symmetric designs with fixed difference of block intersection numbers

The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.

• (1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z³ then λ < x + 1 + z + z³.
• (2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
• (3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z².
• (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
• (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.

© 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007     

Classification of 6‐(14,7,4) designs with nontrivial automorphism groups

In this paper, we introduce some intersection matrices for t‐designs. Utilizing these matrices together with a modified version of a backtracking algorithm, we classify all 6‐(14,7,4) and 5‐(13,6,4) designs with nontrivial automorphism groups and obtain 13 and 21 such designs, respectively. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 180–194, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ) DOI 10.1002/jcd.10004     

Some infinite families of 2‐designs from PG (n,q)

In this paper, we establish the existence of some infinite families of 2‐designs from ‐dimensional projective geometry , which admit ‐dimensional projective special linear group as their flag‐transitive automorphism group.