Quasi-symmetric 2-(28, 12, 11) designs with an automorphism of order 7

All quasi-symmetric 2-(28, 12, 11) designs with an automorphism of order 7 without fixed points or blocks are enumerated. Up to isomorphism, there are exactly 246 such designs. All but four of these designs are embeddable as derived designs in symmetric 2-(64, 28, 12) designs, producing in this way at least 8784 nonisomorphic symmetric 2-(64, 28, 12) designs. The remaining four 2-(28, 12, 11) designs are the first known examples of nonembeddable quasi-symmetric quasi-derived designs. These symmetric 2-(64, 28, 12) designs also produce at least 8784 nonisomorphic quasi-symmetric 2-(36, 16, 12) designs with intersection numbers 6 and 8, including the first known examples of quasi-symmetric 2-(36, 16, 12) designs with a trivial automorphism group. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 213–223, 1998     

Block designs with v = 10, k = 5, λ = 4

It was shown by Singhi that there are 21 nonisomorphic block designs BD (10, 5; 18, 9, 4) which are residual designs of (19, 9, 4) Hadamard designs. In this paper we show that there are no other block designs with these parameters, i.e., each such design is embeddable in a symmetric design. We give a complete list of these designs and their automorphism groups.     

Quasi-residual quasi-symmetric designs

It is shown that for each λ ? 3, there are only finitely many quasi-residual quasi-symmetric (QRQS) designs and that for each pair of intersection numbers (x, y) not equal to (0, 1) or (1, 2), there are only finitely many QRQS designs.A design is shown to be affine if and only if it is QRQS with x = 0. A projective design is defined as a symmetric design which has an affine residual. For a projective design, the block-derived design and the dual of the point-derivate of the residual are multiples of symmetric designs.     

Nonisomorphic Hadamard designs

A block b of a Hadamard design is called a good block if the symmetric difference b + b₁ is also a block for all nonparallel blocks b₁. The isomorphism classes of such designs having a good block are shown to be related to a double coset decomposition of a symmetric group. As an example, over one million mutually nonisomorphic 3-(32, 16, 7) designs of a certain type are constructed.Equivalence of Hadamard matrices is described in terms of designs and it is shown that nonisomorphic designs may arise from the same matrix.     

A Class of 2‐(3n7, 3n−17, (3n−17−1)/2) Designs

Generalized Hadamard matrices are used for the construction of a class of quasi‐residual nonresolvable BIBD's with parameters . The designs are not embeddable as residual designs into symmetric designs if n is even. The construction yields many nonisomorphic designs for every given n ≥ 2, including more than 10¹⁷ nonisomorphic 2‐(63,21,10) designs. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 460–464, 2007     

Pseudo quasi‐3 designs and their applications to coding theory

We define a pseudo quasi‐3 design as a symmetric design with the property that the derived and residual designs with respect to at least one block are quasi‐symmetric. Quasi‐symmetric designs can be used to construct optimal self complementary codes. In this article we give a construction of an infinite family of pseudo quasi‐3 designs whose residual designs allow us to construct a family of codes with a new parameter set that meet the Grey Rankin bound. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 411–418, 2009     

Symmetric (41,16,6)-designs with a nontrivial automorphism of odd order

If a symmetric (41,16,6)-design has an automorphism σ of odd prime order q then q = 3 or 5. In the case q = 5 we determine all such designs and find a total of 419 nonisomorphic ones, of which 15 are self-dual. When q = 3 a combinatorial explosion occurs and the complete classification becomes impracticable. However, we give a characterization in the particular case when σ has order 3 and fixes 11 points, and find that there are 3,076 nonisomorphic designs with this property, all of them being non self-dual. The other remaining possibility is that σ, of order 3, fixes 5 points. In this case there are 960 orbit matrices (up to isomorphism and duality) and all but one of them yield designs. Here an incomplete investigation shows that in total there are at least 112,000 nonisomorphic designs. © 1993 John Wiley & Sons, Inc.     

A complete classification of symmetric (31, 10, 3) designs

In recent years several authors have determined all symmetric (31, 10, 3) designs with a nontrivial automorphism. Here we describe an algorithm for the generation by computer of all symmetric (31, 10, 3) designs and find that there are precisely 151 such nonisomorphic designs. Of these, 107 have a trivial automorphism group.     

New symmetric designs and nonabelian difference sets with parameters(100, 45, 20)

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     

Quasi-symmetric designs,codes, quadrics,and hyperplane sections

It is proved that a quasi-symmetric design with theSymmetric Difference Property (SDP) is uniquely embeddable as a derived or a residual design into a symmetric SDP design. Alternatively, any quasi-symmetric SDP design is characterized as the design formed by the minimum weight vectors in a binary code spanned by the simplex code and the incidence vector of a point set in PG(2m-1, 2) that intersects every hyperplane in one of two prescribed numbers of points. Applications of these results for the classification of point sets in PG(2m-1, 2) with the same intersection properties as an elliptic or a hyperbolic quadric, as well as the classification of codes achieving the Grey-Rankin bound are discussed.     

Decomposable symmetric designs

The first infinite families of symmetric designs were obtained from finite projective geometries, Hadamard matrices, and difference sets. In this paper we describe two general methods of constructing symmetric designs that give rise to the parameters of all other known infinite families of symmetric designs. The method of global decomposition produces an incidence matrix of a symmetric design as a block matrix with each block being a zero matrix or an incidence matrix of a smaller symmetric design. The method of local decomposition represents incidence matrices of a residual and a derived design of a symmetric design as block matrices with each block being a zero matrix or an incidence matrix of a smaller residual or derived design, respectively.     

On quasi-symmetric 2-(28,12,11) and 2-(36,16,12) designs

New quasi-symmetric 2-(28,12,11) and 2-(36,16,12) designs are constructed by embedding known designs into symmetric designs.Partially supported by an NSA Research Grant.     

Admissible parameters of symmetric designs satisfying v=4(k-\lambda )+2 and symmetric designs with inner balance

The admissible parameters of symmetric \((v,k,\lambda )\) designs satisfying \(v=4(k-\lambda )+2\) are shown to correspond with the solutions of a certain Pell equation. We then determine the feasible parameters of such designs that could have a quasi-symmetric residual design with respect to a block, and classify them into two possible families. Finally, we consider the feasible parameters of symmetric designs with inner balance as defined by Nilson and Heidtmann (Des. Codes Cryptogr. doi:10.1007/s10623-012-9730-2, (2012)), and show that (with one exception) they must all belong to one of these families.     

On 2-(45, 12, 3) designs

Under the assumption that the incidence matrix of a 2-(45, 12, 3) design has a certain block structure, we determine completely the number of nonisomorphic designs involved. We discover 1136 such designs with trivial automorphism group. In addition we analyze all 2-(45, 12, 3) designs having an automorphism of order 5 or 11. Altogether, the total number of nonisomorphic 2-(45, 12, 3) designs found in 3752. Many of these designs are self-dual and each of these self-dual designs possess a polarity. Some have polarities with no absolute points, giving rise to strongly regular (45, 12, 3, 3) graphs. In total we discovered 58 pairwise nonisomorphic strongly regular graphs, one of which has a trivial automorphism group. Further, we analyzed completely all the designs for subdesigns with parameters 2-(12, 4, 3), 2-(9, 3, 3), and 2-(5, 4, 3). In the first case, the number of 2-(12, 4, 3) subdesigns that a design possessed, if non-zero, turned out to be a multiple of 3, whereas 2-(9, 3, 3) subdesigns were so abundant it was more unusual to find a design without them. Finally, in the case of 2-(5, 4, 3) subdesigns there is a design, unique amongst the ones discovered, that has precisely 9 such subdesigns and these form a partition of the point set of the design. This design has a transitive group of automorphisms of order 360. © 1996 John Wiley & Sons, Inc.     

A Characterization of Designs Related to an Extremal Doubly-Even Self-Dual Code of Length 48

The uniqueness of a binary doubly-even self-dual [48, 24, 12] code is used to prove that a self-orthogonal 5-(48, 12, 8) design, as well as some of its derived and residual designs, including a quasi-symmetric 2-(45, 9, 8) design, are all unique up to isomorphism.Received November 5, 2003     

Some Observations on Quasi-3 Designs and Hadamard Matrices

We consider Hadamard matrices with the quasi-3 property, and their implications for the existence of certain quasi-symmetric designs. We find necessary conditions on the parameters, and show that the quasi-3 property is preserved by the Kronecker product.     

Weak minimum aberration and maximum number of clear two-factor interactions in 2m-p Ⅳ designs

Both the clear effects and minimum aberration criteria are the important rules for the design selection. In this paper, it is proved that some 2m-p Ⅳ designs have weak minimum aberration, by considering the number of clear two-factor interactions in the designs.And some conditions are provided, under which a 2m-p Ⅳ design can have the maximum number of clear two-factor interactions and weak minimum aberration at the same time.Some weak minimum aberration 2m-p Ⅳ designs are provided for illustrations and two nonisomorphic weak minimum aberration 213-6 Ⅳ designs are constructed at the end of this paper.     

Inner balance of symmetric designs

A triple array is a row-column design which carries two balanced incomplete block designs (BIBDs) as substructures. McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005), Section 8, gave one example of a triple array that also carries a third BIBD, formed by its row-column intersections. This triple array was said to be balanced for intersection, and they made a search for more such triple arrays among all potential parameter sets up to some limit. No more examples were found, but some candidates with suitable parameters were suggested. We define the notion of an inner design with respect to a block for a symmetric BIBD and present criteria for when this inner design can be balanced. As triple arrays in the canonical case correspond to SBIBDs, this in turn yields new existence criteria for triple arrays balanced for intersection. In particular, we prove that the residual design of the related SBIBD with respect to the defining block must be quasi-symmetric, and give necessary and sufficient conditions on the intersection numbers. This, together with our parameter bounds enable us to exclude the suggested triple array candidates in McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005) and many others in a wide search. Further we investigate the existence of SBIBDs whose inner designs are balanced with respect to every block. We show as a key result that such SBIBDs must possess the quasi-3 property, and we answer the existence question for all known classes of these designs.     

A characterization of projective subspaces of codimension two as quasi-symmetric designs with good blocks

Consider an incidence structure whose points are the points of a PG_n(n+2,q) and whose block are the subspaces of codimension two, where n?2. Since every two subspaces of codimension two intersect in a subspace of codimension three or codimension four, it is easily seen that this incidence structure is a quasi-symmetric design. The aim of this paper is to prove a characterization of such designs (that are constructed using projective geometries) among the class of all the quasi-symmetric designs with correct parameters and with every block a good block. The paper also improves an earlier result for the special case of n=2 and obtains a Dembowski-Wagner-type result for the class of all such quasi-symmetric designs.     

Some remarks on permutation designs

A definition of isomorphism of two permutation designs is proposed, which differs from the definition in Bandt [J. Combinatorial Theory Ser. A21 (1976), 384–392]. The proposed definition has the (generally required) property that the allowed permutations always transform a permutation design into a permutation design. It is shown that the n permutation designs coming from the partitioning of S_n into permutation designs, as constructed in Bandt [J. Combinatorial Theory Ser. A21 (1976), 384–392] are all isomorphic. Further we find that this modified definition does not increase the number of nonisomorphic (6, 4) permutation designs. The same investigation showed that one of the designs, claimed to be a (6, 4) permutation design in [J. Combinatorial Theory Ser. A21 (1976), 384–392], is actually not a (6, 4) permutation design.