Search

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 size 11520 and other is not self-dual and has the full automorphism group of size 2520., Comment: 5pages

The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a of possible biplane ${\mathcal D}$ of order $14$ divides $2^7\cdot3^2\cdot5\cdot7\cdot11\cdot13$. In this paper we show that such a biplane do not have an automorphism of order $11$ or $13$, and thereby establish that $|Aut({\mathcal D})|$ divides $2^7\cdot3^2\cdot5\cdot7.$ Further, we study a possible action of an automorphism of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters $(121,16,2)$., Comment: 11 pages

Available online 8 November 2016, Our aim is to study the modular representation theory of coherent configurations. Let p be a prime. We consider structures of modular adjacency algebras of coherent configurations obtained from combinatorial designs. The structures of standard modules of modular adjacency algebras provide more information than p-ranks of incidence matrices of combinatorial designs. (C) 2016 Elsevier Inc. All rights reserved., Article, LINEAR ALGEBRA AND ITS APPLICATIONS. 514:174-197 (2017)

We show that symmetric block designs $${\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})$$D=(P,B) can be embedded in a suitable commutative group $${\mathfrak {G}}_{\mathcal {D}}$$GD in such a way that the sum of the elements in each block is zero, whereas the only Steiner triple systems with this property are the point-line designs of $${\mathrm {PG}}(d,2)$$PG(d,2) and $${\mathrm {AG}}(d,3)$$AG(d,3). In both cases, the blocks can be characterized as the only k-subsets of $$\mathcal {P}$$P whose elements sum to zero. It follows that the group of automorphisms of any such design $$\mathcal {D}$$D is the group of automorphisms of $${\mathfrak {G}}_\mathcal {D}$$GD that leave $$\mathcal {P}$$P invariant. In some special cases, the group $${\mathfrak {G}}_\mathcal {D}$$GD can be determined uniquely by the parameters of $$\mathcal {D}$$D. For instance, if $$\mathcal {D}$$D is a 2-$$(v,k,\lambda )$$(v,k,ź) symmetric design of prime order p not dividing k, then $${\mathfrak {G}}_\mathcal {D}$$GD is (essentially) isomorphic to $$({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}$$(Z/pZ)v-12, and the embedding of the design in the group can be described explicitly. Moreover, in this case, the blocks of $$\mathcal {B}$$B can be characterized also as the v intersections of $$\mathcal {P}$$P with v suitable hyperplanes of $$({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}$$(Z/pZ)v-12.

Let be a nontrivial 2- symmetric design admitting a flag-transitive, point-primitive automorphism group G of almost simple type with sporadic socle. We prove that there are up to isomorphism six designs, and must be one of the following: a 2-(144, 66, 30) design with or , a 2-(176, 50, 14) design with , a 2-(176, 126, 90) design with or , or a 2-(14,080, 12,636, 11,340) design with .

Let G be a finite graph with μ as an eigenvalue of multiplicity k . A star set for μ is a set X of k vertices in G such that μ is not an eigenvalue of G − X . We investigate independent star sets of largest possible size in a variety of situations. We note connections with symmetric designs, codes, strongly regular graphs, and graphs with least eigenvalue −2.

Let be a symmetric (ν,κ,λ) design with λ ≤ 100. If G is a flag-transitive and point-primitive automorphism group of , then G must be an affine or almost simple group.

We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means that there are at least 5421 symmetric (45,12,3) designs. Further, we discuss trigeodetic graphs obtained from the symmetric $(45,12,3)$ designs. We prove that $k$-geodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs obtained from symmetric $(45,12,3)$ designs.

Recall that combinatorial 2s-designs admit a classical lower bound $b \ge\binom{v}{s}$ on their number of blocks, and that a design meeting this bound is called tight. A long-standing result of Bannai is that there exist only finitely many nontrivial tight 2s-designs for each fixed s?5, although no concrete understanding of `finitely many' is given. Here, we use the Smith Bound on approximate polynomial zeros to quantify this asymptotic nonexistence. Then, we outline and employ a computer search over the remaining parameter sets to establish (as expected) that there are in fact no such designs for 5?s?9, although the same analysis could in principle be extended to larger s. Additionally, we obtain strong necessary conditions for existence in the difficult case s=4.

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 An 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 Gx 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 K2, 4 of K6, G = A6 or S6, and Gx = S4 or S4 × Z2, respectively. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:475-483, 2011

Using the list of 2607 so far constructed (96, 20, 4) difference sets as a source, we checked the related symmetric designs upon isomorphism and analised their full automorphism groups. New (96, 20, 4, 4) and (96, 19, 2, 4) regular partial difference sets are constructed, together with the corresponding strongly regular graphs.

Let D be a finite nontrivial triplane, i.e. a 2-(v,k,3) symmetric design, with a flag-transitive, point-primitive automorphism group G. If G is almost simple, with the simple socle X of G being a classical group, then D is either the unique (11, 6, 3)-triplane, with G=PSL"2(11) and G"@a=A"5, or the unique (45, 12, 3)-triplane, with G=X:2=PSp"4(3):[email protected]?PSU"4(2):2 and G"@a=X"@a:2=(2^@?(A"4xA"4).2):2, where @a is a point of D.

We construct four flag-transitive symmetric designs having 96 points, blocks of size 20, and 4 blocks on each point-pair. Moreover we prove that these are the only such designs. Our classification completes the classification of flag-transitive, point-imprimitive, symmetric designs with the (constant) number of blocks on a point-pair at most four.

A $\{; ; 0, \pm 1\}; ; $-matrix $S$ is called a Siamese twin design sharing the entries of $I$, if $S=I+K-L$, where $I, K, L$ are non-zero $\{; ; 0, 1\}; ; $-matrices and both $I+K$ and $I+L$ are incidence matrices of symmetric designs with the same parameters. Let $p$ and $2p+3$ be prime powers and $p \equiv 3\ (mod\ 4)$. We construct a Siamese twin design with parameters $(4(p+1)^2, 2p^2 +3p+1, p^2 + p)$.

This article describes a general method of construction of supersaturated designs for asymmetric factorials obtained by exploiting the concept of resolvable orthogonal arrays and Hadamard matrices. The supersaturated design constructed here has a restricted form of t:q z n, where z factors are at q-levels each and one factor is at t-levels and the number of runs is n. The designs obtained have the factor at t-levels always orthogonal to the z factors with q levels each in the symmetric design q z n. The method of construction is illustrated with the help of examples. A catalogue of designs obtained is prepared and fNOD-efficiency and c 2 -efficiency of the designs are given. Many designs are optimal while other designs have high efficiencies. The efficiency of the resulting design is better than that of the symmetric design q z n. AMS Subject Classification: 62K15, 62K10

A biplane is a 2-(k(k − 1)/2 + 1,k,2) symmetric design. Only sixteen nontrivial biplanes are known: there are exactly nine biplanes with k

Let $p$ and $2p+3$ be prime powers and $p \equiv 3\ (mod\ 4)$. We describe a construction of a symmetric design D with parameters $(4(p+1)^2, 2p^2 +3p+1, p^2 + p)$. If $p$ and $2p+3$ are primes, then a derived design of D is 1-rotational.

A set of type-(m,n)S is a set of points of a design with the property that each block of the design meets either m points or n points of S. The notions of type and of parameters of a k-set (there called characters) were introduced for the first time by Tallini Scafati in [M. Tallini Scafati, {k,n}-archi di un piano grafico finito, con particolare riguardo a quelli con due caratteri. Note I and Note II, Rend. Accad. Naz. Lincei 40 (8) (1996) 812-818 (1020-1025)]. If m=1, S gives rise to a subdesign of the design. Under weaker conditions for the order of each symmetric design, the parameters of sets of type-(1,n) in projective planes were characterised by G. Tallini and the biplane case was dealt with by S. Kim, by solving the corresponding Diophantine equation for each case, separately. In this paper, we first characterise the parameters of sets of type-(1,n) in the triplane with more generalised order conditions than prime power order. Next, we generalise the result on triplanes to arbitrary symmetric designs for @l>=3. As results, a non-existence condition for special parameter sets and a characterisation of parameters for the existence, restricted by some derived bounds, are given.

We describe a construction of symmetric designs with parameters (324, 153, 72), (576, 276, 132), and (900, 435, 210), admitting an automorphism group isomorphic to $Frob_{; ; 17 \cdot 8}; ; \times Z_9$, $Frob_{; ; 23 \cdot 11}; ; \times Z_{; ; 12}; ; $, and $Frob_{; ; 29 \cdot 14}; ; \times Z_{; ; 15}; ; $ respectively. The derived designs of the constructed designs, with respect to the fixed block, are cyclic.

A block design we call primitive if it has an automorphism group acting primitively on both point and block set. Taking the projective line X={;∞};∪GF(q) as the set of points, our research aims to determine, up to isomorphism and complementation, all primitive block designs with PSL(2, q) as an automorphism group. The number of such designs we denote by npd(q). In dealing with primitive permutation representations of almost simple groups with socle PSL(2, q) we make use of the study [1] of their maximal subgroups. The obtained designs we describe by their base block (a union of orbits of a block stabilizer) and the full automorphism group. Our results so far include completely solving the problem in case when a block stabilizer is not in the fifth Aschbacher's class (in particular, for q a prime), and assertions such as the following. Lemma 1. Let q>=4. Then npd(q)=0 if and only if q=7, 11, 23 or q=2^r, r a prime. Lemma 2. Let q>=13 and let there exist a block design D, the socle of AutD being PSL(2, q). If the base block stabilizer is in the second Aschbacher's class, then q is congruent to 1(mod4), D is 2-(q+1, (q-1)/2, (q-1)(q-3)/8) design up to complementation, and AutD=PΣL(2, q). [1] M. Giudici, Maximal subgroups of almost simple groups with socle PSL(2, q), arXiv:math/0703685v1 [math.GR], 2007.

In this paper, we study when the incidence matrix of a symmetric (v, k, λ)-BIBD is a 3-frameproof code. We show the existence of infinite families of symmetric BIBDs that are 3-frameproof codes, as well as infinite families of symmetric BIBDs that are not 3-frameproof codes.

A symmetric design with parameters v = q 2(q + 2), k = q(q + 1), ? = q, q ? 2, is called a quasi-affine design if its point set can be partitioned into q + 2 subsets P 0, P 1,..., P q , P q+1 such that the induced structure in every point neighborhood is an affine plane of order q (repeated q times). A quasi-affine design with q ? 3 determines its point neighborhoods uniquely and dual of such a design is also a quasi-affine design. These structural properties pave way for definition of a strongly quasi-affine design and it is also shown that associated with every quasi-affine design is a unique strongly quasi-affine design from which the given quasi-affine design is obtained by certain unique cutting and pasting operation. This investigation also enables us to associate a unique 2-regular graph with q + 2 vertices and in turn, a unique colored partition of the integer q + 2. These combinatorial consequences are finally used to obtain an exponential lower bound on the number of non-isomorphic solutions of such symmetric designs improving the earlier lower bound of 2.

Constructions are given for the production of symmetric block designs whose point set is divisible (partitionable) in several ways. Some properties of the dual design are also considered. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 415–422, 2006

We introduce a recursive construction of regular Handamard matrices with row sum 2h for h = ±3n Whenever q = (2h - 1)2 is a prime power, we construct, for every positive integer m, a symmetric designs with parameters (4h2(qm+1 - 1)/(q - 1), (2h2 - h)qm, (h2 - h)qm).

A theorem of Bannai and Sawano shows that certain four-weight spin models exist if and only if certain quasi-3 designs exist. We verify that the known quasi-3 designs, other than SDP designs, do not give rise to four-weight spin models.

It is known that the designs PGn-1(n,q) in some cases have spreads of maximal α-arcs. Here a α-arc is a non-empty subset of points that meets every hyperplane in 0 or α points. The situation for designs in general is not so well known. This paper establishes an equivalence between the existence of a spread of α-arcs in the complement of a Hadamard design and the existence of an affine design and a symmetric design which is also the complement of a Hadamard design.

We investigate several bounds for both K 2, m − K 1, n Ramsey numbers and K 2, m − K 1, n bipartite Ramsey numbers, extending some previous results. Constructions based on certain geometric structures (designs, projective planes, unitals) yield classes of near-optimal bounds or even exact values. Moreover, relationships between these numbers are also discussed.

A failed symmetric design is a symmetric incidence structure in which any two points are joined by exactly one or λ blocks, and dually. We construct a failed biplane with block size k whenever k - 1 is a prime power and a failed triplane with block size k whenever k + 1 is a prime power congruent to 1 modulo 3. In fact, our examples admit cyclic Singer groups and thus belong to failed difference sets. Finally, we also exhibit a related infinite series of partially symmetric designs with indices λ1 = 2 and λ2 = 3.

All groups of type E25·E4 are considered as possible automorphism groups of a (100,45,20) symmetric design. New designs are constructed, and those among them leading to Bush-type Hadamard matrices of order 100 are singled out. Further, the full automorphism groups of the constructed designs are given.

Let M be a set of incidence matrices of symmetric (v,k,λ)-designs and G a group of mappings M→ M. We give a sufficient condition for the matrix W⊗ M, where M∈ M and W is a balanced generalized weighing matrix over G, to be the incidence matrix of a larger symmetric design. This condition is then applied to the designs corresponding to McFarland and Spence difference sets, and it results in four families of symmetric (v,k,λ )-designs with the following parameters k and λ (m and d are positive integers, p and q are prime powers): (i) k = q^2m-1p^d, λ = (q-1)q^2m-2p^d-1, q = p^d+1-1/p-1; (ii) k = (q ^2m-1p^d-1)p^d/(p-1)(p ^d+1), λ = (q^2m-2p^2d-1)p^d/(p-1)(p ^d+1), q = p^d+1+p-1; (iii) k = 3^d q ^2m-1, λ = 3^d(3^d+1)q^2m-2/2, q = 3^d+1+1/2; (iv) k = 3^d(3^dq^2m-1-1/2(3^d-1), λ = 3^d(3^2dq^2m-2-1)/2(3^d-1) , q = 3^d+1-2.

In this paper, we will give a construction of a family of(4q^{2n+2}\frac{q^{2n+2}-1}{q^2-1} ,q^{2n+1}[\frac{2(q^{2n+2}-1)}{q+1}+1] , ( q^{2n+2}-q^{2n+1})\frac{q^{2n+1}+1}{q+1})-difference sets in the group K\times G, where q is any power of2 , K is any group with|K|=\frac{q^{2n+2}-1}{q^2-1} and G is an abelian2 -group of order 4q^{2n+2} which contains an elementary abelian subgroup of index 2.

In this paper we present the classification of symmetric designs with parameters (79, 27, 9) on which a non-abelian group of order 39 acts faithfully. In particular, we show that such a group acts semi-standardly with 7 orbits. Using the method of tactical decompositions, we are able to construct exactly 1320 non-isomorphic designs. The orders of the full automorphism groups of these designs all divide 8 · 3 · 13.

We study difference sets with parameters(v, k, λ) = (p^s(r^2m - 1)/(r - 1), p^{s-1}r^{2m-2}(r - 1)r^{2m -2}, where r = (p^s - 1)/(p - 1) and p is a prime. Examples for such difference sets are known from a construction of McFarland which works for m = 1 and all p, s. We will prove a structural theorem on difference sets with the above parameters; it will include the result, that under the self-conjugacy assumption McFarland‘s construction yields all difference sets in the underlying groups. We also show that no abelian .160; 54; 18/-difference set exists. Finally, we give a new nonexistence prove of (189, 48, 12)-difference sets in Z_3 × Z_9 × Z_7.

It is shown that a quasi-residual 2-(v, k, λ) design is the residuum of a symmetric design provided that k > cλ4 for a constant number c. This result improves earlier results of Bose et al. (1976) and Neumaier (1982), who proved the result for k >12λ5 + 0(λ4). This embedding theorem will be a consequence of more general characterization theorems for certain strongly regular multigraphs (see Theorem 2 and its corollary in the introduction).

It is shown for a non-identity automorphism α of a symmetric 2-( v , k , λ ) design fixing f points that f⩽ 1 4 (ν+3κ−6) if |α|⩾3 , f⩽ 1 3 (ν+2κ−4) if |α|⩾2 where |α| denotes the order of α.

The main aim of this paper is to construct symmetric designs with trivial automorphism groups. Being aware of the fact that an exhaustive search for parameters $(36,15,6)$ and $(41,16,6)$ is still impossible, we assume that these designs admit a tactical decomposition which would correspond to an orbit structure achieved under an action of an automorphism of order $3$. This constraint proves to be fruitful and allows us to classify simultaneously those symmetric designs with mentioned parameters which admit an automorphism of order $3$ as well as to construct new designs with a trivial automorphism group.

We construct self-orthogonal codes from the row span over GF(2) or GF(3) of the adjacency matrices of some strongly regular graphs defined by the rank-3 action of the simple unitary group U(4, 3) on the conjugacy classes of some of its maximal subgroups. We establish some properties of these codes and the nature of some classes of codewords.

In this paper we either prove the non-existence or give explicit construction of primitive symmetric (v, k, λ) designs with v=pm 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

Let $q$ be a prime power, $q \equiv 1\ (mod\ 4)$, and let $p=\frac{; ; q+1}; ; {; ; 2}; ; $. We construct an orbit structure for the parameters $(4p^2, 2p^2 - p, p^2 - p)$ and the orbit size distribution with $q+1$ fixed points and $2q$ orbits of size $p$. Further, we describe a construction of eight symmetric designs with parameters (196, 91, 42) admitting an automorphism group isomorphic to $Frob_{; ; 7 \cdot 3}; ; \times Z_6$.

New (96,20,4,4) and (96,19,2,4) regular partial difference sets are constructed, together with the corresponding strongly regular graphs. Our source are (96,20,4) regular symmetric designs.

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

Let $p$ and $2p-1$ be prime powers and $p \equiv 3\ (mod\ 4)$. Then there exists a symmetric design with parameters $(4p^2, 2p^2 - p, p^2 - p)$. Thus there exists a regular Hadamard matrix of order $4p^2$.

Interest in special sets of a symmetric design has been growing for a number of years. The purpose of this paper is to deal with sets of points which determine sets of blocks with the same incidence properties in the dual design.

Since the advent of design theory that began with constructive results of R.C. Bose and M. Hall, symmetric designs have occupied a very special status. This is due to several reasons, two most important of which are the structural symmetry and the difficulty in constructions of these designs (compared to other classes of designs). The initial part of this exposition will concentrate on new results on symmetric designs. Quasi-symmetric designs are closely related to symmetric designs. These are designs that have (at the most) two block intersection numbers. One can associate a block graph with a quasi-symmetric design and in many cases of interest, this also turns out to be a graph with special properties, and is called a strongly regular graph. One trivial way of constructing quasi-symmetric designs is to take multiple copies of a symmetric design. Since a symmetric design has exactly one block intersection number, the resulting design will have two block intersection numbers. Such quasi-symmetric designs are called improper. Only one example of a proper quasi-multiple quasi-symmetric design seems to be known so far. The question of the existence of a quasi-multiple of a symmetric design is of some importance particularly when the corresponding symmetric design is known not to exist. In that case, obtaining a quasi-multiple with least multiplicity has attracted attention of combinatorialists in the last thirty years.

A block negacyclic Bush-type Hadamard matrix of order 36 is used in a symmetric BGW(26, 25, 24) with zero diagonal over a cyclic group of order 12 to construct a twin strongly regular graph with parameters v=936, k=375, lambda=mi=150 whose points can be partitioned in 26 cocliques of size 36. The same Hadamard matrix then is used in symmetric BGW(50, 49, 48) with zero diagonal over a cyclic group of order 12 to construct a Siamese twin strongly regular graph with parameters v=1800, k=1029, lambda=mi=588.

Fortysix mutually nonisomorphic symmetric (64,28,12)-designs have been constructed by means of tactical decompositions. They all admit an action of the nonabelian group of order 21. The computation of their full automorphism groups as well as their derived (28,12,11)-designs proves that none of them can be isomorphic to any of the known (64,28,12)-designs.

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

Six nonisomorphic new symmetric designs with parameters (100, 45, 20) are constructed by action of the Frobenius group E25 · Z12. This group proves to be their full automorphism group. Its Frobenius subgroup of order 100 acts on the designs as their nonabelian Singer group. The result is presented through six nonisomorphic new nonabelian (100, 45, 20) difference sets as well. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 291–299, 2000