Super-simple balanced incomplete block designs with block size 4 and index 5

In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple designs are also useful in other constructions, such as superimposed codes and perfect hash families etc. The existence of super-simple (v,4,λ)-BIBDs have been determined for λ=2,3,4 and 6. When λ=5, the necessary conditions of such a design are that and v≥13. In this paper, we show that there exists a super-simple (v,4,5)-BIBD for each and v≥13.     

Existence of (2, 8) GWhD( v) and (4, 8) GWhD( v) with $${ v \equi v 0,1 (mod 8)}$$

(2, 8) Generalized Whist tournament Designs (GWhD) on v players exist only if . We establish that these necessary conditions are sufficient for all but a relatively small number of (possibly) exceptional cases. For there are at most 12 possible exceptions: {177, 249, 305, 377, 385, 465, 473, 489, 497, 537, 553, 897}. For there are at most 98 possible exceptions the largest of which is v = 3696. The materials in this paper also enable us to obtain four previously unknown (4, 8)GWhD(8n+1), namely for n = 16,60,191,192 and to reduce the list of unknown (4, 8) GWhD(8n) to 124 values of v the largest of which is v = 3696.      

The existence of doubly near resolvable (v,3,2)-BIBDs

The existence of doubly near resolvable (v,2,1)-BIBDs was established by Mullin and Wallis in 1975. In this article, we determine the spectrum of a second class of doubly near resolvable balanced incomplete block designs. We prove the existence of DNR(v,3,2)-BIBDs for v ≡ 1 (mod 3), v ≥ 10 and v ? {34,70,85,88,115,124,133,142}. The main construction is a frame construction, and similar constructions can be used to prove the existence of doubly resolvable (v,3,2)-BIBDs and a class of Kirkman squares with block size 3, KS₃(v,2,4). © 1994 John Wiley & Sons, Inc.     

Balanced incomplete block designs with block size 8

The necessary conditions for the existence of a balanced incomplete block design on υ ≥ k points, with index λ and block size k, are that: For k = 8, these conditions are known to be sufficient when λ = 1, with 38 possible exceptions, the largest of which is υ = 3,753. For these 38 values of υ, we show (υ, 8, λ ) BIBDs exist whenever λ > 1 for all but five possible values of υ, the largest of which is υ = 1,177, and these five υ's are the only values for which more than one value of λ is open. For λ>1, we show the necessary conditions are sufficient with the definite exception of two further values of υ, and the possible exception of 7 further values of υ, the largest of which is υ=589. In particular, we show the necessary conditions are sufficient for all λ> 5 and for λ = 4 when υ ≠ 22. We also look at (8, λ) GDDs of type 7^m. Our grouplet divisible design construction is also refined, and we construct and exploit α ‐ frames in constructing several other BIBDs. In addition, we give a PBD basis result for {n: n ≡ 0, 1; mod 8, n ≥ 8}, and construct a few new TDs with index > 1. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 233–268, 2001     

Super-simple （5, 4）-GDDs of group type g^u

In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super- simple group divisible designs are useful in constructing other types of super- simple designs which can be applied to codes and designs. In this article, the existence of a super-simple （5, 4）-GDD of group type gU is investigated and it is shown that such a design exists if and only if u ≥ 5, g（u - 2） ≥ 12, and u（u - 1）g^2≡ 0 （mod 5） with some possible exceptions.     

Existence of super-simple OA_{\lambda }(3, 5, v)^{\prime }s

It was proved recently that a super-simple orthogonal array (SSOA) of strength \(t\) and index \(\lambda \ge 2\) is equivalent to a minimum detecting array (DTA). In computer software tests in component-based systems, such a DTA can be used to generate test suites that are capable of locating \(d=\lambda -1\) \(t\) -way interaction faults and detect whether there are more than \(d\) interaction faults. It is proved in this paper that an SSOA of strength \(t=3\) , index \(\lambda \ge 2\) and degree \(k=5\) , or an SSOA \(_{\lambda }(3,5,v)\) , exists if and only if \(\lambda \le v\) excepting possibly a handful of cases.     

4‐*GDDs(3n) and generalized Steiner systems GS(2, 4, v, 3)

Generalized Steiner systems GS(2, k, v, g) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size g + 1 with minimum Hamming distance 2k ? 3, in which each codeword has length v and weight k. As to the existence of a GS(2, k, v, g), a lot of work has been done for k = 3, while not so much is known for k = 4. The notion k‐*GDD was first introduced and used to construct GS(2, 3, v, 6). In this paper, singular indirect product (SIP) construction for GDDs is modified to construct GS(2, 4, v, g) via 4‐*GDDs. Furthermore, it is proved that the necessary conditions for the existence of a 4‐*GDD(3ⁿ), namely, n ≡ 0, 1 (mod 4) and n ≥ 8 are also sufficient. The known results on the existence of a GS(2, 4, v, 3) are then extended. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 381–393, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10047     

A 2-(22, 8, 4) Design Cannot Have a 2-(10, 4, 4) Subdesign

The smallest BIBD, as for the number of points and blocks, whose existence is still undecided is 2-(22, 8, 4). Possible subconfigurations of such a design, namely 2-(10, 4, 4) designs, are here ruled out. The result is obtained by classifying all 2-(10, 4, 4) designs and trying to find 2-(22, 8, 4) designs by solving instances of the maximum clique problem.     

Existence results for doubly near resolvable (υ, 3, 2)-BIBDs

Let V be a set of υ elements. A (1, 2; 3, υ, 1)-frame F is a square array of side v which satisfies the following properties. We index the rows and columns of F with the elements of V, V={x₁,x₂,…,x_υ}. (1) Each cell is either empty or contains a 3-subset of V. (2) Cell (x_i, x_i) is empty for i=1, 2,…, υ. (3) Row x_i of F contains each element of V−{x_i} once and column x_i of F contains each element of V−{x_i} once. (4) The collection of blocks obtained from the nonempty cells of F is a (υ, 3, 2)-BIBD. A (1, 2; 3, υ, 1)-frame is a doubly near resolvable (υ, 3, 2)-BIBD. In this paper, we first present a survey of existence results on doubly near resolvable (υ, 3, 2)-BIBDs and (1, 2; 3, υ, 1)-frames. We then use frame constructions to provide a new infinite class of doubly near resolvable (υ, 3, 2)-BIBDs by constructing (1, 2; 3, υ, 1)-frames.     

(v,k, λ)-Graphs and polarities of (v,k, λ)-designs

In 2.1 it is established that there is a one-to-one correspondence between (v, k, )-graphs and polarities, with no absolute points, of (v, k, )-designs. This is used to show that the parameters of a (v, k, )-graph are of the form ((s/a)((s + a)²–1), s(s+a), sa) where s and a are positive integers with a dividing s(s²–1) (Theorem 3.4) but strictly less than s(s²–1) (Proposition 4.3). Some consequences of this parametrization are discussed and in particular, it is shown that for fixed 2 there are only finitely many non-isomorphic (v, k, )-graphs. In 4. it is shown that (v, k, )-graphs can also be constructed using polarities, with all points absolute, of certain designs. In 5. isomorphisms and automorphisms of graphs and designs are discussed. Many examples of (v, k, )-graphs, including some apparently new ones, are given.Dedicated to Peter Dembowski, 28 January 1971     

Super‐simple holey Steiner pentagon systems and related designs

A Steiner pentagon system of order v (SPS(v)) is said to be super‐simple if its underlying (v, 5, 2)‐BIBD is super‐simple; that is, any two blocks of the BIBD intersect in at most two points. It is well known that the existence of a holey Steiner pentagon system (HSPS) of type T implies the existence of a (5, 2)‐GDD of type T. We shall call an HSPS of type T super‐simple if its underlying (5, 2)‐GDD of type T is super‐simple; that is, any two blocks of the GDD intersect in at most two points. The existence of HSPSs of uniform type hⁿ has previously been investigated by the authors and others. In this article, we focus our attention on the existence of super‐simple HSPSs of uniform type hⁿ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 301–328, 2008     

COMPUTATION OF THE E(s2) VALUES OF SOME E(s2) OPTIMAL SUPERSATURATED DESIGNS

1 IntroductionAn rerun design for m two-level faCtors is saturated if n = m 1. Such designs haveminimum number of runs for estimating all the main effects when the interactions are negligible,and are useful for screening experiments in the initial stage of an investigation where the primarygoal is to identify the few active faCtors from a large number of potential faCtors. And whelln < in 1, such designs are called supersaturated designs, which provide more flexibility andcost saving. No…     

Classification of (v, 3)-configurations

A (v, 3)-configuration is a nondegenerate matrix of dimension v over the field GF(2) considered up to permutation of rows and columns and containing exactly three 1’s in the rows and columns, while the inverse matrix has also exactly three 1’s in the rows and columns. It is proved that, for each even v ≥ 4, there is only one indecomposable (v, 3)-configuration, while, for odd v, there are no such configurations, the only exception being the unique (5, 3)-configuration.     

广义(u,v)幂等矩阵

首先定义了一类新的矩阵一广义(u,v)幂等矩阵,然后研究了它的等价刻画,从而推广了(u,v)幂等矩阵、m幂幺矩阵、m幂等矩阵的一些相应结果.此外,也探讨了广义(u,v)幂等矩阵的性质,以及广义(u,v)幂等矩阵与广义m幂矩阵的关系.     

The Existence of (ν,6, λ)-Perfect Mendelsohn Designs with λ > 1

The basic necessary conditions for the existence of a (v, k, λ)-perfect Mendelsohn design (briefly (v, k, λ)-PMD) are v ≥ k and λ v(v − 1) ≡ 0 (mod k). These conditions are known to be sufficient in most cases, but certainly not in all. For k = 3, 4, 5, 7, very extensive investigations of (v, k, λ)-PMDs have resulted in some fairly conclusive results. However, for k = 6 the results have been far from conclusive, especially for the case of λ = 1, which was given some attention in papers by Miao and Zhu [34], and subsequently by Abel et al. [1]. Here we investigate the situation for k = 6 and λ > 1. We find that the necessary conditions, namely v ≥ 6 and λ v(v − 1)≡0 (mod 6) are sufficient except for the known impossible cases v = 6 and either λ = 2 or λ odd. Researcher F.E. Bennett supported by NSERC Grant OGP 0005320.     

On 2-chromatic (v, 5, 1)-designs

In this note, we completely settle the existence of 2-chromatic (v, 5, 1)-designs. This settles a problem posed by Rosa and Colbourn.     

A property of (v,k, λ)-designs

It is shown that if the subsetsX ₁,...,X _v of a setX form a (v, k, λ)-design, then there does not exist another subsetX _v+1 ofX havingany cardinalityk ₁ and intersecting each of theX _j, 1≦j≦v, inany number λ₁ of elements, where 0<k ₁<v and 0<λ₁<k (in order to avoid uninteresting cases).