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     

Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10

All Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10 are constructed and classified up to isomorphism together with related Hadamard matrices of order 64. Affine 2-(64,16,5) designs can be obtained from Hadamard 2-(63,31,15) designs having line spreads by Rahilly’s construction [A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991) 291-303]. The parameter set 2-(64,16,5) is one of two known sets when there exists several nonisomorphic designs with the same parameters and p-rank as the design obtained from the points and subspaces of a given dimension in affine geometry AG(n,p^m) (p a prime). It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2-(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3,4), or is one of the two exceptional designs found by Harada, Lam and Tonchev [M. Harada, C. Lam, V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs Codes Cryptogr. 34 (2005) 71-87].     

2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order,and their related Hadamard matrices and codes

We present the full classification of Hadamard 2-(31,15,7), Hadamard 2-(35, 17,8) and Menon 2-(36,15,6) designs with automorphisms of odd prime order. We also give partial classifications of such designs with automorphisms of order 2. These classifications lead to related Hadamard matrices and self-dual codes. We found 76166 Hadamard matrices of order 32 and 38332 Hadamard matrices of order 36, arising from the classified designs. Remarkably, all constructed Hadamard matrices of order 36 are Hadamard equivalent to a regular Hadamard matrix. From our constructed designs, we obtained 37352 doubly-even [72,36,12] codes, which are the best known self-dual codes of this length until now.      

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     

M_(12)作用在396个点上的SPBIB设计的分类

研究Mathieu群M_(12)作用在396个点上所构成的对称的部分平衡不完全区组设计(即SPBIB设计)的分类情况.首先,证明了以M_(12)作为自同构群的非平凡的2-(396,k,λ)对称设计是不存在的.然后,得到了同构意义下的3个点数为396且区组长度为80的SPBIB设计.最后,给出了396个点上以M_(12)作为自同构群的SPBIB设计的完全分类.     

Enumeration of the doubles of the projective plane of order 4

A classification of the doubles of the projective plane of order 4 with respect to the order of the automorphism group is presented and it is established that, up to isomorphism, there are 1 746 461 307 doubles. We start with the designs possessing non-trivial automorphisms. Since the designs with automorphisms of odd prime orders have been constructed previously, we are left with the construction of the designs with automorphisms of order 2. Moreover, we establish that a 2-(21,5,2) design cannot be reducible in two inequivalent ways. This makes it possible to calculate the number of designs with only the trivial automorphism, and consequently the number of all double designs. Most of the computer results are obtained by two different approaches and implementations.     

Centered -discrepancy of random sampling and Latin hypercube design, and construction of uniform designs

In this paper properties and construction of designs under a centered version of the -discrepancy are analyzed. The theoretic expectation and variance of this discrepancy are derived for random designs and Latin hypercube designs. The expectation and variance of Latin hypercube designs are significantly lower than that of random designs. While in dimension one the unique uniform design is also a set of equidistant points, low-discrepancy designs in higher dimension have to be generated by explicit optimization. Optimization is performed using the threshold accepting heuristic which produces low discrepancy designs compared to theoretic expectation and variance.

     

Small Box-Behnken design

Box-Behnken design has been popularly used for the second-order response surface model. It is formed by combining two-level factorial designs with incomplete block designs in a special manner—the treatments in each block are replaced by an identical design. In this paper, we construct small Box-Behnken design. These designs can fit the second-order response surface model with reasonably high efficiencies but with only a much smaller run size. The newly constructed designs make use of balanced incomplete block design (BIBD) or partial BIBD, and replace treatments partly by 2_III³⁻¹ designs and partly by full factorial designs. It is shown that the orthogonality properties in the original Box and Behnken designs will be kept in the new designs. Furthermore, we classify the parameters into groups and introduce Group Moment Matrix (GMM) to estimate all the parameters in each group. This allows us to significantly reduce the amount of computational costs in the construction of the designs.     

Unitals and Unitary Polarities in Symmetric Designs

We extend the notion of unital as well as unitary polarity from finite projective planes to arbitrary symmetric designs. The existence of unitals in several families of symmetric designs has been proved. It is shown that if a unital in a point-hyperplane design PG _d-1(d,q) exists, then d = 2 or 3; in particular, unitals and ovoids are equivalent in case d = 3. Moreover, unitals have been found in two designs having the same parameters as the PG ₄(5,2), although the latter does not have a unital. It had been not known whether or not a nonclassical design exists, which has a unitary polarity. Fortunately, we have discovered a unitary polarity in a symmetric 2-(45,12,3) design. To a certain extent this example seems to be exceptional for designs with these parameters.     

On the connectedness of partially balanced block designs

The necessary and sufficient conditions for m-associate partially balanced block (PBB) designs to be connected are given. This generalizes the criterion for m-associate partially balanced incomplete block (PBIB) designs, which has originally been established by Ogawa, Ikeda and Kageyama (1984, Proceedings of the Seminar on Combinatorics and Applications, 248–255, Statistical Publishing Society, Calcutta).This work was partially supported by the Polish Academy of Sciences Grant No. MR I.1-2/2.     

Linear Perfect Codes and a Characterization of the Classical Designs

A new definition for the dimension of a combinatorial t-(v,k,) design over a finite field is proposed. The complementary designs of the hyperplanes in a finite projective or affine geometry, and the finite Desarguesian planes in particular, are characterized as the unique (up to isomorphism) designs with the given parameters and minimum dimension. This generalizes a well-known characterization of the binary hyperplane designs in terms of their minimum 2-rank. The proof utilizes the q-ary analogue of the Hamming code, and a group-theoretic characterization of the classical designs.     

Characterizing geometric designs, II

We provide a characterization of the classical point-line designs PG₁(n,q), where n?3, among all non-symmetric 2-(v,k,1)-designs as those with the maximal number of hyperplanes. As an application of this result, we characterize the classical quasi-symmetric designs PGn−2(n,q), where n?4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q+1 and all intersection numbers at least qn−4+?+q+1. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as PG₁(n,q); in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.     

Partially balanced incomplete block designs from weakly divisible nearrings

In [[6] Riv. Mat. Univ. Parma 11 (2) (1970) 79-96] Ferrero demonstrates a connection between a restricted class of planar nearrings and balanced incomplete block designs. In this paper, bearing in mind the links between planar nearrings and weakly divisible nearrings (wd-nearrings), first we show the construction of a family of partially balanced incomplete block designs from a special class of wd-nearrings; consequently, we are able to give some formulas for calculating the design parameters.     

Extremal Self-Dual [50, 25, 10] Codes with Automorphisms of Order 3 and Quasi-Symmetric 2-(49, 9,6) Designs

Five non-isomorphic quasi-symmetric 2-(49, 9, 6) designs are known. They arise from extremal self-dual [50, 25, 10] codes with a certain weight enumerator. Four of them have an automorphism of order 3 fixing two points. In this paper, it is shown that there are exactly 48 inequivalent extremal self-dual [50, 25, 10] code with this weight enumerator and an automorphism of order 3 fixing two points. 44 new quasi-symmetric 2-(49, 9, 6) designs with an automorphism of order 3 are constructed from these codes.     

D‐optimal designs and group divisible designs

We obtain new conditions on the existence of a square matrix whose Gram matrix has a block structure with certain properties, including D‐optimal designs of order , and investigate relations to group divisible designs. We also find a matrix with large determinant for n = 39. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 451–462, 2006     

Point-weight designs with design conditions on t points

This paper examines some of the properties of point-weight incidence structures, i.e. incidence structures for which every point is assigned a positive integer weight. In particular it examines point-weight designs with a design condition that stipulates that any two “identical” sets of t points must lie on the same number of blocks. We introduce a new class of designs with this property: row-sum designs, and examine the basic properties of row-sum point-weight designs and their similarities to classical (non-point-weight) designs and the point-weight designs of Horne [On point-weighted designs, Ph.D. Thesis, Royal Holloway, University of London, 1996].     

Comparing different fractions of a factorial design: a metal cutting case study

Full factorial designs of a significant size are very rarely performed in industry due to the number of trials involved and unavailable time and resources. The data in this paper were obtained from a six‐factor full factorial (2⁶) designed experiment that was conducted to determine the optimum operating conditions for a steel milling operation. Fractional‐factorial designs 2 (one‐eighth) and 2 (one‐fourth, using a fold‐over from the one‐eighth) are compared with the full 2⁶ design. Four of the 2 are de‐aliased by adding four more runs. In addition, two 12‐run Plackett–Burman experiments and their combination into a fold‐over 24‐run experiment are considered. Many of the one‐eighth fractional‐factorial designs reveal some significant effects, but the size of the estimates varies much due to aliasing. Adding four more runs improves the estimation considerably. The one‐quarter fraction designs yield satisfactory results, compared to the full factorial, if the ‘correct’ parameterization is assumed. The Plackett–Burman experiments, estimating all main effects, always perform worse than the equivalent regular designs (which have fewer runs). When considering a reduced model many of the different designs are more or less identical. The paper provides empirical evidence for managers and engineers that the choice of an experimental design is very important and highlights how designs of a minimal size may not always result in productive findings. Copyright © 2006 John Wiley & Sons, Ltd.     

Adding more observations to saturated D‐optimal resolution III two‐level factorial designs

We consider the class of saturated main effect plans for the 2^k factorial. With these saturated designs, the overall mean and all main effects can be unbiasedly estimated provided that there are no interactions. However, there is no way to estimate the error variance with such designs. Because of this and other reasons, we like to add some additional runs to the set of (k+1) runs in the D‐optimal design in this class. Our goals here are: (1) to search for s additional runs so that the resulting design based on (k+s+1) runs yields a D‐optimal design in the class of augmented designs; (2) to classify all the runs into equivalent classes so that the runs in the same equivalent class give us the same value of the determinant of the information matrix. This allows us to trade runs for runs if this becomes necessary; (3) to obtain upper bounds for determinant of the information matrices of augmented designs. In this article we shall address these approaches and present some new results. © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 51–77, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10026     

Connectedness of PBIB designs having asymmetrical association schemes

Summary A necessary and sufficient condition for the connectedness ofm-associate partially balanced incomplete block (PBIB) designs having an asymmetrical association scheme is given, only in terms of design parameters, without inner structure parameters of designs. Supported in part by Grant 321-6066-58530013 (Japan).     

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.