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.     

PBIBDs from weakly divisible nearrings and related codes

In [5] the authors are able to give a method for the construction of a family of partially balanced incomplete block designs from a special class of wd-nearrings (wd-designs). In this paper the wd-design incidence matrix and the connected row and column codes are studied. The parameters of two special classes of wd-designs and those of the related row and column codes are calculated.     

Resolvable balanced incomplete block designs with subdesigns of block size 4

In this paper, we look at resolvable balanced incomplete block designs on v points having blocks of size 4, briefly (v,4,1) RBIBDs. The problem we investigate is the existence of (v,4,1) RBIBDs containing a (w,4,1) RBIBD as a subdesign. We also require that each parallel class of the subdesign should be in a single parallel class of the containing design. Removing the subdesign gives an incomplete RBIBD, i.e., an IRB(v,w). The necessary conditions for the existence of an IRB(v,w) are that v?4w and . We show these conditions are sufficient with a finite number (179) of exceptions, and in particular whenever and whenever w?1852.We also give some results on pairwise balanced designs on v points containing (at least one) block of size w, i.e., a (v,{K,w^*},1)-PBD.If the list of permitted block sizes, K₅, contains all integers of size 5 or more, and v,w∈K₅, then a necessary condition on this PBD is v?4w+1. We show this condition is not sufficient for any w?5 and give the complete spectrum (in v) for 5?w?8, as well as showing the condition v?5w is sufficient with some definite exceptions for w=5 and 6, and some possible exceptions when w=15, namely 77?v?79. The existence of this PBD implies the existence of an IRB(12v+4,12w+4).If the list of permitted block sizes, K₁₍₄₎, contains all integers , and v,w∈K₁₍₄₎, then a necessary condition on this PBD is v?4w+1. We show this condition is sufficient with a finite number of possible exceptions, and in particular is sufficient when w?1037. The existence of this PBD implies the existence of an IRB(3v+1,3w+1).     

More balanced incomplete block designs from frobenius groups

From a Frobenius group G = N ×_σ Φ with kernel N and complement Φ, one can sometimes construct new balanced incomplete block designs by taking as blocks the translations in N of blocks made from the union of the nontrivial orbits of Φ of a and −a in N together with the trivial orbit {0}.     

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.     

Pairwise-balanced,variance-balanced and resistant incomplete block designs revisited

A general solution is derived to the problem of characterizing block designs that are simultaneously pairwise-balanced and variance-balanced. Applications of the characterizations obtained to some problems concerned with the local resistance of BIB designs are presented.This research was begun while the second author was visiting the Department of Mathematical and Statistical Methods, Academy of Agriculture in Pozna, Poland.The work of this author was partially supported by the Polish Academy of Sciences Grant No. CPBP 01.01.2/2.     

High-rate LDPC codes from partially balanced incomplete block designs

Journal of Algebraic Combinatorics - This paper presents a combinatorial construction of low-density parity-check (LDPC) codes from partially balanced incomplete block designs. Since...     

Pairwise and variance balanced incomplete block designs

Summary The purpose of this paper is three-fold. The first purpose is to compile and to systematize published and dispersed results on two aspects of balancing in incomplete block designs, i.e., pairwise balance and variance balance. This was done in order to establish the status of these two concepts of balance in published literature and to put them in a form which is useful for further work in this area. Also, the results in this form are necessary for the development of the remainder of the paper. The second purpose of this paper is to present a method of constructing unequal replicate and/or unequal block size experiment designs for which the variance balance property is achieved. The method of construction involves the union of blocks from two or more block designs and the augmentation of some of the blocks with additional treatments; the method is denoted asunionizing block designs. A straight-forward extension of the method would produce a partially balanced block design with unequal replicate and/or unequal block designs. The enlargement of the concept and availability of variance blanced block designs to accommodate unequal replication and/or unequal block sizes is important to the researcher, the teacher, and the experimenter needing such designs. For example, an animal nutritionist or a psychologist is no longer required to have constant litter or family sizes for the blocks and may have unequal replication on the treatments for those treatments with insufficient material and still attain the goal of equal variances on all normalized treatment contracts. The third purpose of the paper is to apply the unionizing block designs method to construct a family of unequal replicate and unequal block size variance balanced designs. Some comments are given on the extension of the unionizing block designs method to construct other families of variance balanced or partially balanced block designs. This investigation was supported in part by PHS Research Grant No. R01-GM-05900 from General Medical Services.     

Improved inequalities for balanced incomplete block designs

It is known that there are some lower bounds for the number of blocks in a balanced incomplete block design (BIBD). Especially, Fisher's inequality b?v is well-known for a BIBD with parameters v, b, r, k and λ. Fisher's inequality can be improved upon if one puts additional restrictions on a BIBD. Artificial restrictions are infinite in number so is the number of new bounds. The condition of non-symmetry on the design discussed here is a very simple restriction. The main purpose of this paper is to give improvements of inequalities for BIBDs with the only condition of non-symmetry. Improved inequalities appear to be the best for any non-symmetrical BIBD.     

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.     

New designs from circular nearrings

Nearrings are generalized rings in which addition is not in general abelian and only one distributive law holds. Some interesting combinatorial structures, as tactical configurations and balanced incomplete block designs (BIBDs) naturally arise when considering the class of planar and circular nearrings. We define the concept of disk and prove that in the case of field-generated planar circular nearrings it yields a BIBD. Such designs can be used in the construction of some classes of codes for which we are able to calculate the parameters.     

Frame difference families and resolvable balanced incomplete block designs

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational \((pq+1,p+1,1)\)-RBIBDs over \({\mathbb {F}}_{p}^+ \times {\mathbb {F}}_{q}^+\) are derived, and the existence of \((125q+1,6,1)\)-RBIBDs is discussed. We construct (v, 8, 1)-RBIBDs for \(v\in \{624,\) \(1576,2976,5720,5776,10200,14176,24480\}\), whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences.     

Resolvable modified group divisible designs with block size three

A resolvable modified group divisible design (RMGDD) is an MGDD whose blocks can be partitioned into parallel classes. In this article, we investigate the existence of RMGDDs with block size three and show that the necessary conditions are also sufficient with two exceptions. © 2005 Wiley Periodicals, Inc. J Combin Designs 15: 2–14, 2007     

Some results on resolvable incomplete block designs

This paper is concerned with the uniformity of a certain kind of resolvable incomplete block (RIB for simplicity) design which is called the PRIB design here. A sufficient and necessary condition is obtained, under which a PRIB design is the most uniform in the sense of a discrete discrepancy measure, and the uniform PRIB design is shown to be connected. A construction method for such designs via a kind of U-type designs is proposed, and an existence result of these designs is given. This method sets up an important bridge between PRIB designs and U-type designs.     

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