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.     

Unifying some known infinite families of combinatorial 3-designs

In this paper we present a construction of 3-designs by using a 3-design with resolvability. The basic construction generalizes a well-known construction of simple 3-(v,4,3) designs by Jungnickel and Vanstone (1986). We investigate the conditions under which the designs obtained by the basic construction are simple. Many infinite families of simple 3-designs are presented, which are closely related to some known families by Iwasaki and Meixner (1995), Laue (2004) and van Tran (2000, 2001). On the other hand, the designs obtained by the basic construction possess various properties: A theory of constructing simple cyclic 3-(v,4,3) designs by Köhler (1981) can be readily rebuilt from the context of this paper. Moreover many infinite families of simple resolvable 3-designs are presented in comparison with some known families. We also show that for any prime power q and any odd integer n there exists a resolvable 3-(qn+1,q+1,1) design. As far as the authors know, this is the first and the only known infinite family of resolvable t-(v,k,1) designs with t?3 and k?5. Those resolvable designs can again be used to obtain more infinite families of simple 3-designs through the basic construction.     

Permutation designs

An (n, k)-permutation design is a collection of k! permutations on n elements such that for any choice of k elements all k! permutations of these elements appear as subpermutations in the collection. For every n we construct an (n, n ? 1)-permutation design. There is a rather unique (6, 4)-design but no (5, 3)- and (7, 4)-design. Permutation designs and ordered triple systems are special cases of ordered designs. For all n ≠ 2 (mod 3) we have an ordered triple system containing no triples twice.     

Infinite families of 3-designs from a type of five-weight code

It has been known for a long time that t-designs can be employed to construct both linear and nonlinear codes and that the codewords of a fixed weight in a code may hold a t-design. While a lot of progress in the direction of constructing codes from t-designs has been made, only a small amount of work on the construction of t-designs from codes has been done. The objective of this paper is to construct infinite families of 2-designs and 3-designs from a type of binary linear codes with five weights. The total number of 2-designs and 3-designs obtained in this paper are exponential in any odd m and the block size of the designs varies in a huge range.     

Strongly uniform extensions of dual 2-designs

A strongly α-uniform partial line space of order (s, t) is called an α-partial geometry. If α = t+1, then the geometry is a dual 2-design. Locally triangular and locally Grassman graphs correspond to triangular extensions of certain dual 2-designs, and the class of strongly uniform quasi-biplanes coincides with the class of strongly uniform extensions of dual 2-designs. We study strongly uniform extensions of dual 2-designs.     

On Minimal Defining Sets of Full Designs and Self-Complementary Designs, and a New Algorithm for Finding Defining Sets of t-Designs

A defining set of a t-(v, k, λ) design is a partial design which is contained in a unique t-design with the given parameters. A minimal defining set is a defining set, none of whose proper partial designs is a defining set. This paper proposes a new and more efficient algorithm that finds all non-isomorphic minimal defining sets of a given t-design. The complete list of minimal defining sets of 2-(6, 3, 6) designs, 2-(7, 3, 4) designs, the full 2-(7, 3, 5) design, a 2-(10, 4, 4) design, 2-(10, 5, 4) designs, 2-(13, 3, 1) designs, 2-(15, 3, 1) designs, the 2-(25, 5, 1) design, 3-(8, 4, 2) designs, the 3-(12, 6, 2) design, and 3-(16, 8, 3) designs are given to illustrate the efficiency of the algorithm. Also, corrections to the literature are made for the minimal defining sets of four 2-(7, 3, 3) designs, two 2-(6, 3, 4) designs and the 2-(21, 5, 1) design. Moreover, an infinite class of minimal defining sets for 2-((v) || 3){v\choose3} designs, where v ≥ 5, has been constructed which helped to show that the difference between the sizes of the largest and the smallest minimal defining sets of 2-((v) || 3){v\choose3} designs gets arbitrarily large as v → ∞. Some results in the literature for the smallest defining sets of t-designs have been generalized to all minimal defining sets of these designs. We have also shown that all minimal defining sets of t-(2n, n, λ) designs can be constructed from the minimal defining sets of their restrictions when t is odd and all t-(2n, n, λ) designs are self-complementary. This theorem can be applied to 3-(8, 4, 3) designs, 3-(8, 4, 4) designs and the full 3-(8 || 4)3-{8 \choose 4} design using the previous results on minimal defining sets of their restrictions. Furthermore we proved that when n is even all (n − 1)-(2n, n, λ) designs are self-complementary.     

Extended (2, 4)-designs

In this paper, the concept of an extended (2, 4)-design is introduced. An extended (2, 4)-design is a pair (X, $\text{B}$) where X is a finite set and $\text{B}$ is a collection of 4-tuples of not necessarily distinct elements of X, such that every pair of not necessarily distinct elements of X is contained in exactly one member of $\text{B}$. It is shown that an extended (2,4)-design of order n exists for every positive integer n except n = 6, 8 and 9. Several inequivalent designs of order n are obtained.     

Flag-transitive 2- $$(v,k,\lambda )$$ ( v,k , λ ) designs with $$r>\lambda (k-3)$$ r > λ ( k - 3 )

Designs, Codes and Cryptography - In this paper, we study the flag-transitive automorphism groups of 2-designs and prove that if G is a flag-transitive automorphism group of a 2-design $$\mathcal...     

A symmetric design with parameters 2-(49, 16, 5)

A symmetric 2-design with parameters (v, k, λ) = (49, 16, 5) is constructed. Both this design and its residual, a design with parameters (v, b, r, k, λ) = (33, 48, 16, 11, 5), seem to be new. The derived designs do not have repeated blocks. The group of the design is cyclic of order 15. There is no polarity.     

On the structure of 1-designs with at most two block intersection numbers

We introduce the notion of an unrefinable decomposition of a 1-design with at most two block intersection numbers, which is a certain decomposition of the 1-designs collection of blocks into other 1-designs. We discover an infinite family of 1-designs with at most two block intersection numbers that each have a unique unrefinable decomposition, and we give a polynomial-time algorithm to compute an unrefinable decomposition for each such design from the family. Combinatorial designs from this family include: finite projective planes of order n; SOMAs, and more generally, partial linear spaces of order (s, t) on (s + 1)² points; as well as affine designs, and more generally, strongly resolvable designs with no repeated blocks.      

Relative <Emphasis Type="Italic">t</Emphasis>-designs in binary Hamming association scheme <Emphasis Type="Italic">H</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, 2)

A relative t-design in the binary Hamming association schemes H(n, 2) is equivalent to a weighted regular t-wise balanced design, i.e., certain combinatorial t-design which allows different sizes of blocks and a weight function on blocks. In this paper, we study relative t-designs in H(n, 2), putting emphasis on Fisher type inequalities and the existence of tight relative t-designs. We mostly consider relative t-designs on two shells. We prove that if the weight function is constant on each shell of a relative t-design on two shells then the subset in each shell must be a combinatorial \((t-1)\)-design. This is a generalization of the result of Kageyama who proved this under the stronger assumption that the weight function is constant on the whole block set. Using this, we define tight relative t-designs for odd t, and a strong restriction on the possible parameters of tight relative t-designs in H(n, 2). We obtain a new family of such tight relative t-designs, which were unnoticed before. We will give a list of feasible parameters of such relative 3-designs with \(n \le 100\), and then we discuss the existence and/or the non-existence of such tight relative 3-designs. We also discuss feasible parameters of tight relative 4-designs on two shells in H(n, 2) with \(n \le 50\). In this study we come up with the connection on the topics of classical design theory, such as symmetric 2-designs (in particular 2-\((4u-1,2u-1,u-1)\) Hadamard designs) and Driessen’s result on the non-existence of certain 3-designs. We believe Problems 1 and 2 presented in Sect. 5.2 open a new way to study relative t-designs in H(n, 2). We conclude our paper listing several open problems.     

Inequalities and bounds for quasi-symmetric 3-designs

Quasi-symmetric 3-designs with block intersection numbers x and y(0x<y<k) are studied, several inequalities satisfied by the parameters of a quasi-symmetric 3-designs are obtained. Let D be a quasi-symmetric 3-design with the block size k and intersection numbers x, y; y>x1 and suppose D′ denote the complement of D with the block size k′ and intersection numbers x′ and y′. If k −1 x + y then it is proved that x′ + y′ k′. Using this it is shown that the quasi-symmetric 3-designs corresponding to y = x + 1, x + 2 are either extensions of symmetric designs or designs corresponding to the Witt-design (or trivial design, i.e., v = k + 2) or the complement of above designs.     

Resolvable t-Designs

A general group theoretic approach is used to find resolvable designs. Infinitely many resolvable 3-designs are obtained where each is block transitive under some PSL(2, p ^f ) or PGL(2, p ^f ). Some known Steiner 5-designs are assembled from such resolvable 3-designs such that they are also resolvable. We give some visualizations of Steiner systems which make resolvability obvious.     

3-齐次群与旗传递5-（v，k，3）设计

旗传递t-设计的分类是代数组合学的一个重要课题．本文主要讨论了旗传递5-（v,k,3）设计．由P．J．Cameron和C．E．Praeger的结论可知,此时设计的自同构群是3-齐次群．本文利用3-齐次群的分类,证明了设计的自同构群不能是仿射型群．     

Classifications of certain t-designs

The nonextendability and extendability of both a t-(v, k, λ_t) design (with $\text{k ?}\text{v}\text{2}$ and its complementary t-design are discussed in four cases: (A) a t-design is extendable and its complement is not extendable; (B) both a t-design and its complement are extendable; (C) a t-design is not extendable and its complement is extendable; (D) both a t-design and its complement are not extendable. Nontrivial examples for each case are presented. Furthermore, some series of t-designs belonging to cases (A), (B), and (D) are also given.     

搜索区传递2-（q，4，1）设计

对于区传递但非旗传递的可解2-（q，4，1）设计，Camina指出，当q=13，37，61，109，157，181时有具体的例子，但是否有更多的q产生具体例子有待研究。主要结果：设q是素数幂且q=13（mod24），则对于每个q〈2000，总存在区传递但非旗传递的2-（q，4，1）设计。     

Quasi-symmetric 2-(31, 7, 7) designs and a revision of Hamada's conjecture

It is proved by use of the classification of the doubly even (32, 16) codes, that in addition to the design formed by the planes in PG(4, 2), there are exactly four other nonisomorphic quasi-symmetric 2-(31, 7, 7) designs, and they all have 2-rank 16. This shows that the “only if” part of the following conjecture due to Hamada, is not true in general: “If N(D) is an incidence matrix of a design D with the parameters of a design G defined by the flats of a given dimension in PG(t, q) or AG(t, q), then rank_q N(D) ⩾ rank_q N(G), with equality if and only if D is isomorphic with G.” The five quasi-symmetric 2-(31, 7, 7) designs are extendable to nonisomorphic 3-(32, 8, 7) designs having 2-rank 16, one of which is formed by the 3-flats in AG(5, 2), thus the designs arising from a finite affine geometry also are not characterized by their ranks in general. A quasi-symmetric 2-(45, 9, 8) design yielding a pseudo-geometric strongly regular graph with parameters (r, k, t) = (15, 10, 6) is also constructed on the base of the known extremal doubly even (48, 24) code.     

McLaren’s improved snub cube and other new spherical designs in three dimensions

Evidence is presented to suggest that, in three dimensions, spherical 6-designs withN points exist forN=24, 26,≥28; 7-designs forN=24, 30, 32, 34,≥36; 8-designs forN=36, 40, 42,≥44; 9-designs forN=48, 50, 52,≥54; 10-designs forN=60, 62, ≥64; 11-designs forN=70, 72,≥74; and 12-designs forN=84,≥86. The existence of some of these designs is established analytically, while others are given by very accurate numerical coordinates. The 24-point 7-design was first found by McLaren in 1963, and—although not identified as such by McLaren—consists of the vertices of an “improved” snub cube, obtained from Archimedes' regular snub cube (which is only a 3-design) by slightly shrinking each square face and expanding each triangular face. 5-designs with 23 and 25 points are presented which, taken together with earlier work of Reznick, show that 5 designs exist forN=12, 16, 18, 20,≥22. It is conjectured, albeit with decreasing confidence fort≥9, that these lists oft-designs are complete and that no other exist. One of the constructions gives a sequence of putative sphericalt-designs withN=12m points (m≥2) whereN=1/2t ²(1+o(1)) ast→∞.     

Designs with block-size 6 in projective planes of characteristic 2

We construct simple 3-designs and 4-designs of block-size 6 in the classical projective planesPG(2,q),q a power of 2. All of our designs are invariant under the projective groupPGL(3,q). Aside from several infinite series of 3-designs we get some relatively small designs of independent interest, e.g. designs with parameters 4-(21, 6, 16) and 4-(73, 6, 330) defined in the planes of orders 4 and 8, respectively.     

USING FINITE GEOMETRIES TO CONSTRUCT 3-PBIB(2) DESIGNS AND 3-DESIGNS

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