Combinatorial Problems

1950 ◽  
Vol 2 ◽  
pp. 93-99 ◽  
Author(s):  
S. Chowla ◽  
H. J. Ryser
Keyword(s):  
Combinatorial Problem ◽  
Difference Sets ◽  
Projective Planes ◽  
Combinatorial Problems ◽  
Block Designs ◽  
Incomplete Block ◽  
Incidence Matrices ◽  
Finite Projective Planes ◽  
Balanced Incomplete Block Designs ◽  
Balanced Incomplete Block

Let it be required to arrange v elements into v sets such that every set contains exactly k distinct elements and such that every pair of sets has exactly elements in common . This combinatorial problem is studied in conjunction with several similar problems, and these problems are proved impossible for an infinitude of v and k. An incidence matrix is associated with each of the combinatorial problems, and the problems are then studied almost entirely in terms of their incidence matrices. The techniques used are similar to those developed by Bruck and Ryser for finite projective planes [3]. The results obtained are of significance in the study of Hadamard matrices [6;8], finite projective planes [9], symmetrical balanced incomplete block designs [2; 5], and difference sets [7].

On Integer Matrices and Incidence Matrices of Certain Combinatorial Configurations, III: Rectangular Matrices

1966 ◽  
Vol 18 ◽  
pp. 9-17
Author(s):  
Kulendra N. Majindar
Keyword(s):  
Necessary Conditions ◽  
Block Designs ◽  
Incomplete Block ◽  
Incidence Matrices ◽  
Balanced Incomplete Block Designs ◽  
Balanced Incomplete Block ◽  
Definition Of ◽  
Combinatorial Configurations

In this paper, we give a connection between incidence matrices of affine resolvable balanced incomplete block designs and rectangular integer matrices subject to certain arithmetical conditions. The definition of these terms can be found in paper II of this series or in (2). For some necessary conditions on the parameters of affine resolvable balanced incomplete block designs and their properties see (2).

scholarly journals Some new construction methods of variance balanced block designs with repeated blocks

2008 ◽  
Vol 5 (1) ◽  
Author(s):  
Bronisław Ceranka ◽  
Małgorzata Graczyk
Keyword(s):  
Block Designs ◽  
Incomplete Block ◽  
Incidence Matrices ◽  
Balanced Incomplete Block Designs ◽  
Construction Methods ◽  
New Construction ◽  
Balanced Incomplete Block

Some new construction methods of the variance balanced block designs with repeated blocks are given. They are based on the specialized product of incidence matrices of the balanced incomplete block designs.

scholarly journals ON D-OPTIMAL CHEMICAL BALANCE WEIGHING DESIGNS

2015 ◽  
Vol 1 (311) ◽  
Author(s):  
Bronisław Ceranka ◽  
Małgorzata Graczyk
Keyword(s):  
Optimal Design ◽  
Measurement Errors ◽  
Block Designs ◽  
Incomplete Block ◽  
Chemical Balance ◽  
Incidence Matrices ◽  
Balanced Incomplete Block Designs ◽  
Construction Methods ◽  
Existence Conditions ◽  
Balanced Incomplete Block

The paper deals with the problem of determining the chemical balance weighing designs satisfying the criterion of D-optimality under assumption that the measurement errors are equal correlated and they have the same variances. The existence conditions and the form of the optimal design are given. Moreover, some construction methods of the design matrices based on the incidence matrices of the balanced incomplete block designs and ternary balanced block designs are presented. Any example of construction is given.

On Integer Matrices and Incidence Matrices of Certain Combinatorial Configurations, II: Rectangular Matrices

1966 ◽  
Vol 18 ◽  
pp. 6-8
Author(s):  
Kulendra N. Majindar
Keyword(s):  
Block Designs ◽  
Incomplete Block ◽  
Incidence Matrices ◽  
Balanced Incomplete Block Designs ◽  
Balanced Incomplete Block ◽  
Definition Of ◽  
Image Position ◽  
Combinatorial Configurations

In this paper we establish a connection between rectangular integer matrices and incidence matrices of resolvable balanced incomplete block designs. The definition of these terms has been given in paper I of this series.Our theorem can be stated as follows:THEOREM 2. Let A be a v X b matrix with integer elements such that2.1

On the Construction of Bibd With λ = 1

1973 ◽  
Vol 16 (3) ◽  
pp. 329-335
Author(s):  
Kenneth F. N. Scott
Keyword(s):  
Difference Sets ◽  
Block Designs ◽  
Incomplete Block ◽  
Balanced Incomplete Block Designs ◽  
The Past ◽  
Balanced Incomplete Block ◽  
Fundamental Theorems

In the past three decades the problem of generating (balanced incomplete block) designs by difference sets has received much attention. Bose [2] gave the two "fundamental theorems of the method of differences". Bose, Sprott [9], Lehmer [7], Chowla [4], Takeuchi [10] and others have given specific classes of difference sets.

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