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     

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.      

A simplified version of strong connectivity in geometries

If there are less than b distinct blocks in a BIB design with b blocks then we say the design has repeated blocks. The set of distinct blocks of a design is called the support of the design. BIB designs with repeated blocks, besides being optimal, have special applications in the design of experiments and controlled samplings. Construction of BIB(ν, b, r, k, λ) designs with repeated blocks becomes complicated whenever the three parameters b, r, and λ are relatively prime. BIB(8, 56, 21, 3, 6) designs are examples of such designs with the smallest number of varieties. BIB(10, 30, 9, 3, 2) designs are such designs with the smallest number of blocks. We make an interesting observation about BIB(8, 56, 21, 3, 6) designs and give a table of such designs with 30 different support sizes. We prove, by construction, that a BIB(10, 30, 9, 3, 2) design exists if and only if the support size belongs to {21, 23, 24, 25, 26, 27, 28, 29, 30}. Other results are also given.     

On the maximum cocliques of the rank 3 graph of 211 : M24

In this article, we consider the maximum cocliques of the 2¹¹: M₂₄ ‐graph Λ. We show that the maximum cocliques of size 24 of Λ can be obtained from two Hadamard matrices of size 24, and that there are exactly two maximum cocliques up to equivalence. We verify that the two nonisomorphic designs with parameters 5‐(24,9,6) can be constructed from the maximum cocliques of Λ, and that these designs are isomorphic to the support designs of minimum weights of the ternary extended quadratic residue and Pless symmetry [24,12,9] codes. Further, we give a new construction of Λ from these 5‐(24,9,6) designs. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 323–332, 2009     

Alias balanced and alias partially balanced fractional 2 m factorial designs of resolution 2l+1

As a generalization of alias balanced designs due to Hedayat, Raktoe and Federer [5], we introduce the concept of alias partially balanced designs for fractional 2 ^m factorial designs of resolution 2l+1. All orthogonal arrays of strength 2l yield alias balanced designs. Some balanced arrays of strength 2l yield alias balanced and alias partially balanced designs. In particular, simple arrays which are a special case of balanced arrays yield alias partially balanced designs. At most 2 ^m −1 alias balanced (or alias partially balanced) designs are generated from an alias balanced (or alias partially balanced) design by level permutations. This implies that alias balanced or alias partially balanced designs need not be orthogonal arrays or balanced arrays of strength 2l.     

Construction of Hamilton Path Tournament Designs

A Hamilton path tournament design involving n teams and n/2 stadiums, is a round robin schedule on n − 1 days in which each team plays in each stadium at most twice, and the set of games played in each stadium induce a Hamilton path on n teams. Previously, Hamilton path tournament designs were shown to exist for all even n not divisible by 4, 6, or 10. Here, we give an inductive procedure for the construction of Hamilton path tournament designs for n = 2 ^p ≥ 8 teams.     

2-Designs overGF(q)

We give a construction of a series of 2-(n, 3,q ²+q+1;q) designs of vector spaces over a finite fieldGF(q) of odd characteristic. These designs correspond to those constructed by Thomas and the author for even characteristic. As a natural generalization we give a collection ofm-dimensional subspaces which possibly become a 2-(n, m, λ; q) design.     

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.     

BOUND ON THE MAXIMUM NUMBER OF CLEAR TWO-FACTOR INTERACTIONS FOR 2^n-（n-k） DESIGNS

Clear effects criterion is an important criterion for selecting fractional factorial designs[1].Tang et al.[2]derived upper and lower bounds on the maximum number of clear two-factor interactions（2fi＇s）in 2^n-（n-k）designs of resolution Ⅲ and Ⅳ by constructing 2^n-（n-k）designs.But the method in[2]does not perform well sometimes when the resolution is Ⅲ.This article modifies the construction method for 2^n-（n-k） designs of resolution Ⅲ in[2].The modified method is a great improvement on that used in[2].     

What is an infinite design?

It is usually assumed that an infinite design is a design with infinitely many points. This encompasses a myriad of structures, some nice and others not. In this paper we consider examples of structures that we would not like to call designs, and investigate additional conditions that exclude such anomalous structures. In particular, we expect a design to be regular, the complement of a design to be a design, and a t‐design to be an s‐design, for all 0 < s ≤ t. These are all properties that can be taken for granted with finite designs, and for infinite Steiner systems. We present a new definition of an infinite t‐design, and give examples of structures that satisfy this definition. We note that infinite designs considered in the literature to date satisfy our definition. We show that infinite design theory does not always mirror finite design theory, for example there are examples of designs with υ > b. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 79–91, 2002; DOI 10.1002/jcd.10005     

A construction of balanced arrays of strengtht and some related incomplete block designs

Summary Saha [6] has shown the equivalence between a ‘tactical system’ (or at-design) and a 2-symbol balanced array (BA) of strengtht. The implicit method of construction of BA in that paper has been generalized herein to that of ans-symbol BA of strengtht. Some BIB and PBIB designs are also constructed from these arrays. Majindar [2], Vanstone [8] and Saha [6] have all shown that the existence of a symmetrical BIBD forv treatments implies the existence of six more BIBD's forv treatments in (v/2) blocks. An analogue of this result has been obtained for a large class of PBIB designs in this paper.     

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 construction of [q 4 + q 2 − q, 5,q 4 − q 3 + q 2 − 2q; q]-codes meeting the Griesmer bound

It is unknown whether or not there exists an [87, 5, 57; 3]-code. Such a code would meet the Griesmer bound. The purpose of this paper is to give a constructive proof of the existence of [q ⁴ + q ² – q, 5, q ⁴ – q ³ + q ² – 2q; q]-codes for any prime power q 3. As a special case, it is shown that there exists an [87, 5, 57; 3]-code with weight enumerator 1 + 156z ³⁷ + 82z ⁶⁰ + 2z ⁶³ + 2z ⁷⁸. The new construction settles an open problem due to Hill and Newton [10].     

On a series of Hadamard matrices of order 2 t and the maximal excess of Hadamard matrices of order 22t

In this paper we give a new series of Hadamard matrices of order 2 ^t . When the order is 16, Hadamard matrices obtained here belong to class II, class V or to class IV of Hall's classification [3]. By combining our matrices with the matrices belonging to class I, class II or class III obtained before, we can say that we have direct construction, namely without resorting to block designs, for all classes of Hadamard matrices of order 16.Furthermore we show that the maximal excess of Hadamard matrices of order 2^2t is 2^3t , which was proved by J. Hammer, R. Levingston and J. Seberry [4]. We believe that our matrices are inequivalent to the matrices used by the above authors. More generally, if there is an Hadamard matrix of order 4n ² with the maximal excess 8n ³, then there exist more than one inequivalent Hadamard matrices of order 2^2t n ² with the maximal excess 2^3t n ³ for anyt 2.     

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.     

Interval Routing onk-Trees

A graph has an optimall-interval routing scheme if it is possible to direct messages along shortest paths by labeling each edge with at mostlpairwise-disjoint subintervals of the cyclic interval [1…n] (where each node of the graph is labeled by an integer in the range). Although much progress has been made forl = 1, there is as yet no general tight characterization of the classes of graphs associated with largerl. Bodlaenderet al. have shown that under the assumption of dynamic cost links, each graph with an optimall-interval routing scheme has treewidth of at most 4l. For the setting without dynamic cost links, this paper addresses the complementary question of the number of intervals required to label classes of graphs of treewidthk. Although it has been shown that there exist graphs of treewidth 2 that require a nonconstant number of intervals, our work demonstrates a class of graphs of treewidth 2, namely 2-trees, that are guaranteed to allow 3-interval routing schemes. In contrast, this paper presents a 2-tree that cannot have a 2-interval routing scheme. For generalk, anyk-tree is shown to have an optimal interval routing scheme using 2^{k + 1}intervals per edge.     

There are exactly five biplanes with k = 11

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 < 11, at least five biplanes with k = 11, and at least two biplanes with k = 13. It is here shown by exhaustive computer search that the list of five known biplanes with k = 11 is complete. This result further implies that there exists no 3‐(57, 12, 2) design, no 112₁₁ symmetric configuration, and no (324, 57, 0, 12) strongly regular graph. The five biplanes have 16 residual designs, which by the Hall–Connor theorem constitute a complete classification of the 2‐(45, 9, 2) designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 117–127, 2008     

On the powers of 3/2 and other rational numbers

Letp > q > 1 be two coprime integers. In this paper, we prove several results about subsets of the interval [0, 1) which does or does not contain all the fractional parts {ξ (p /q)ⁿ }, n = 0, 1, 2, …, for certain non‐zero real number ξ. We show, for instance, that there are no real ξ for which the union of two intervals [8/39, 18/39] ∪ [21/39, 31/39] contains the set {ξ (3/2)ⁿ }, n ∈ N . The most important aspect of this result is that the total length of both intervals 20/39 is greater than 1/2: the same result as above for [0, 1/2) would imply that there are no Mahler's Z ‐numbers which the best known unsolved problem in this area. On the other hand, it is shown that there are infinitely many ξ for which {ξ (3/2)ⁿ } ∈ (5/48, 43/48) for each integer n ≥ 0. We also give simpler proofs of few recent results in this area. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal balanced 27 fractional factorial designs of resolutionV,withN≦42

Summary In this paper, we present a class of fractional factorial designs of the 2⁷ series, which are of resolutionV. Such designs allow the estimation of the general mean, the main effects and the two factors interactions (29 parameters in all for the 2⁷ factorial) assuming that the higher order effects are negligible. For every value ofN (the number of runs) such that 29≦N≦42, we give a resolutionV design that is optimal (with respect to the trace criterion) within the subclass of balanced designs. Also, for convenience of analysis, we present for each design, the covariance matrix of the estimates of the various parameters. As a by product, we establish many interesting combinatorial theorems concerning balanced arrays of strength four (which are generalizations of orthogonal arrays of strength four, and also of balanced incomplete block designs with block sizes not necessarily equal).     

A class of three-replicate three-associate p.b.i.b. designs

Summary Partially balanced incomplete block (p.b.i.b.) designs of two and more associate classes have been constructed by Saha and Das [9] through the use of confounded designs for 2 ⁿ factorals. By dualising one series of two-associated p.b.i.b. designs given by them, a class of three-replicate three-associate p.b.i.b. designs have been obtained. This can be taken to be a new class of designs in the sense that they are not included by Nair [7] in his table of three-replicate three-associate p.b.i.b. designs.