Construction of nested balanced block designs, rectangular designs andq-ary codes

Construction of nested balanced incomplete block (BIB) designs, nested balanced ternary designs and rectangular designs from given nested BIB designs and resolvable BIB designs are described. New constructions ofq-ary codes from nested BIB designs and balanced bipartite weighing designs are also given.     

On construction of partially balanced n-ary designs

A new method of constructing a series of partially balanced ternary (PBT) designs is presented. In the method, we have added the corresponding rows of incidence matrices of a BIB design and a PBIB design, both obtained from single initial block with at least one element in common between them. The BIB and PBIB designs above were obtained by method of differences. We have also constructed PBT designs and PB n-ary designs from a PBIB design alone based on NC_m-scheme as well as from a group divisible PBIB design with smaller number of blocks and moderate block sizes.     

A robust and powerful rank test of treatment effects in balanced incomplete block designs

This paper is concerned primarily with a Monte Carlo investigation into the small sample robustness and power of an aligned rank transformation statistic. The aligned rank transformation statistic (ART) is compared to the classical F-test and to Durbin's (1951) rank test for its ability to detect treatment effects in balanced incomplete block (BIB) designs.     

Some imbedding theorems for block designs balanced with respect to pairs

We prove imbedding theorems for block designs balanced with respect to pairs, and with the aid of these theorems we establish the existence of (v, k, )-resolvable BIB block designs with parameters v, k, such that =k–1 [and also such that =(k–l)/2 if k is odd], k ¦(p–1) for each prime divisor p of the number v/k; we also establish an imbedding theorem for Kirkman triple systems.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 173–184, July, 1974.     

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.     

A class of pbib-designs obtained from projective geometries

In this paper, a new class of partially balanced incomplete block designs is constructed over an association scheme obtained from a finite projective geometry. Further, a general method for deriving a balanced incomplete block design from another one is given.     

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.     

On the block structure of BIB designs with parameters v = 22, b = 33, r = 12, k = 8, and λ = 4

It is unknown whether a BIB design with parameters (22, 33, 12, 8, 4) exists or not. In this paper, a necessary condition for the block structure of BIB designs is given. Using this necessary condition, it is shown that the intersection numbers {n_i} for any block in any BIB design with parameters (22, 33, 12, 8, 4) must be one of the four types shown in Table I if the BIB design exists. It is also shown that if there exists a BIB design with parameters (22, 33, 12, 8, 4), the BIB design must contain at least one block of Type 1 or 3 in Table I.     

Efficiency of connected binary block designs when a single observation is unavailable

In this paper the problem of finding the design efficiency is considered when a single observation is unavailable in a connected binary block design. The explicit expression of efficiency is found for the resulting design when the original design is a balanced incomplete block design or a group divisible, singular or semiregular or regular with ₁>0, design. The efficiency does not depend on the position of the unavailable observation. For a regular group divisible design with ₁>0, the efficiency depends on the position of the unavailable observation. The bounds, both lower and upper, on the efficiency are given in this situation. The efficiencies of designs resulting from a balanced incomplete block design and a group divisible design are in fact high when a single observation is unavailable.The work of the first author is sponsored by the Air Force Office of Scientific Research under Grant AFOSR-90-0092.On leave from Indian Statistical Institute, Calcutta, India. The work of the third author was supported by a grant from the CMDS, Indian Institute of Management, Calcutta.     

Balanced Incomplete Block Designs with Block Size 9 and λ=2,4,8

The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: $$\begin{gathered} {\text{ }}\lambda (v - 1) \equiv 0{\text{ mod (}}k - 1{\text{)}} \hfill \\ \lambda v(v - 1) \equiv 0{\text{ mod (}}k - 1{\text{)}} \hfill \\ \end{gathered} $$ In this paper we study k=9 with λ=2,4 or 8. For λ=8, we show these conditions on v are sufficient, and for λ=2, 4 respectively there are 8 and 3 possible exceptions the largest of which are v=1845 and 783. We also give some examples of group divisible designs derived from balanced ternary designs.     

On construction ofT m -type PBIB designs

Summary Recently Saha and Das [10] constructed partially balanced incomplete block (PBIB) designs of two and more associate classes by using confounded designs for 2 ⁿ factorials. Several new designs of two associate classes were obtained through those methods. This paper generalizes one of the earlier methods of construction to obtain several series ofT _m -type (m≧2) PBIB designs, i.e., the designs havingm-dimensional triangular association schemes. Some more new designs of two associate classes (i.e.,T ₂-type) are obtained through the generalized methods of construction.     

Balanced Incomplete Block Designs with Block Size 7

The object of this paper is the construction of balanced incomplete block designs with k=7. This paper continues the work begun by Hanani, who solved the construction problem for designs with a block size of 7, and with =6, 7, 21 and 42. The construction problem is solved here for designs with > 2 except for v=253, = 4,5 ; also for = 2, the number of unconstructed designs is reduced to 9 (1 nonexistent, 8 unknown).     

Partial geometric designs and two-class partially balanced designs

It is shown that a partial geometric design with parameters (r, k, t, c) satisfying certain conditions is equivalent to a two-class partially balanced incomplete block design. This generalizes a result concerning partial geometric designs and balanced incomplete block designs.     

Balanced nested designs and balanced arrays

Balanced nested designs are closely related to other combinatorial structures such as balanced arrays and balanced n-ary designs. In particular, the existence of symmetric balanced nested designs is equivalent to the existence of some balanced arrays. In this paper, various constructions for symmetric balanced nested designs are provided. They are used to determine the spectrum of symmetric balanced nested balanced incomplete block designs with block size 3 and 4.     

An improved two‐stage variance balance approach for constructing partial profile designs for discrete choice experiments

In many discrete choice experiments set up for product innovation, the number of attributes is large, which results in a substantial cognitive burden for the respondents. To reduce the cognitive burden in such cases, Green suggested in the early '70s the use of partial profiles that vary only the levels of a subset of the attributes. In this paper, we present two new methods for constructing Bayesian ‐optimal partial profile designs for estimating main‐effects models. They involve alternative generalizations of Green's approach that makes use of balanced incomplete block designs and take into account the fact that attributes may have differing numbers of levels. We refer to our methods as variance balance I and II because they vary an attribute with a larger number of levels more often than an attribute with fewer levels to stabilize the variances of the individual part‐worth estimates. The two variance balance methods differ in the way attributes with differing numbers of levels are weighted. Both methods provide statistically more efficient partial profile designs for differing numbers of attribute levels than another generalization of Green's approach that does not weight the attributes. This method is called attribute balance. We show results from an actual experiment in software development demonstrating the usefulness of our methods. Copyright © 2014 John Wiley & Sons, Ltd.     

A generalization of combinatorial designs and related codes

In this paper we study a certain generalization of combinatorial designs related to almost difference sets, namely the t-adesign, which was coined by Ding (Codes from difference sets, 2015). It is clear that 2-adesigns are partially balanced incomplete block designs which naturally arise in many combinatorial and statistical problems. We discuss some of their basic properties and give several constructions of 2-adesigns (some of which correspond to new almost difference sets and some to new almost difference families), as well as two constructions of 3-adesigns. We discuss basic properties of the incidence matrices and make an initial investigation into the codes which they generate. We find that many of the codes have good parameters in the sense they are optimal or have relatively high minimum distance.     

某些可分解平衡不完全区组设计

本文证明存在某些可分解平衡不完全区组设计RB(k,λ;v)。例如,存在RB(6,5;42)及RB(5,2;55)。     

Inner balance of symmetric designs

A triple array is a row-column design which carries two balanced incomplete block designs (BIBDs) as substructures. McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005), Section 8, gave one example of a triple array that also carries a third BIBD, formed by its row-column intersections. This triple array was said to be balanced for intersection, and they made a search for more such triple arrays among all potential parameter sets up to some limit. No more examples were found, but some candidates with suitable parameters were suggested. We define the notion of an inner design with respect to a block for a symmetric BIBD and present criteria for when this inner design can be balanced. As triple arrays in the canonical case correspond to SBIBDs, this in turn yields new existence criteria for triple arrays balanced for intersection. In particular, we prove that the residual design of the related SBIBD with respect to the defining block must be quasi-symmetric, and give necessary and sufficient conditions on the intersection numbers. This, together with our parameter bounds enable us to exclude the suggested triple array candidates in McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005) and many others in a wide search. Further we investigate the existence of SBIBDs whose inner designs are balanced with respect to every block. We show as a key result that such SBIBDs must possess the quasi-3 property, and we answer the existence question for all known classes of these designs.     

Combinatorial characterizations of authentication codes

In this paper, we prove two new combinatorial characterizations of authentication codes. Authentication codes without secrecy are characterized in terms of orthogonal arrays; and general authentication codes are characterized in terms of balanced incomplete block designs. In both of these characterizations, it turns out that encoding rules must be equiprobable; in the second characteriztion, the source states must also be equiprobable.A preliminary version of this paper appeared in Advances in Cryptology—CRYPTO '91 Proceedings Lecture Notes in Computer Science 576 (1992) 62–73, Springer-Verlag.Research supported by NSERC grant A9287.     

Some characterization of locally resistant BIB designs of degree one

Summary This paper investigates locally resistant balanced incomplete block (LRBIB) designs of degree one. A new necessary condition for the existence of such an LRBIB design is presented. This condition yields a complete characterization of affine α-resolvable LRBIB designs of degree one. Furthermore, regarding construction methods of LRBIB designs of degree one, it is shown that Shah and Gujarathi's method (1977,Sankhy?, B39, 406–408) yields the same parameters as Hedayat and John's method (1974,Ann. Statist.,2, 148–158), but their block structures are different and interesting. Partially supported by Grants 59540043 (C) and 60530014 (C).