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What is block design? What is its mysterious role in mathematics?
In mathematical combinatorial design, block design is an incidencia structure that includes a set and its subsets (called blocks). The selection of these subsets meets certain conditions in order to make the entire block set behave Show symmetry and balance. Block design has a wide range of applications, including experimental design, finite geometry, physical chemistry, software testing, cryptography, algebraic geometry and other fields. Generally speaking, the block design mentioned usually refers to the balanced incomplete block design (BIBD), which is a special 2 design that has historically been the most intensively studied type and is mainly used in experimental designs.
Block design shows the combination and arrangement of elements, opening up many mysterious aspects of mathematics.
Basic concepts of block design
Mathematically, if a design is balanced (up to t), it means that all t subsets of the original set occur in an equal number of blocks. When t is not specified, it is usually assumed that t=2, which means that each pair of elements occurs in the same number of blocks and the design is pairwise balanced. For t=1, then each element appears in the same number of blocks (this is called the number of repetitions), and this design is called a regular design. Furthermore, a design in which all blocks are of the same size is said to be uniform or correct. The designs discussed in this article are all uniform, and the basis of the block design is not uniform, so they are called pairwise balanced designs (PBDs).
Regular uniform design
The most basic "balanced" design (t=1) is called a tactical configuration or 1 design. In geometry, the corresponding incidencia structures are called configurations. This design is both uniform and regular: each block contains k elements, and each element is contained in r blocks. There is a relationship between the number v of elements in the design and the number of blocks b, the total number of occurrences of the element, as bk = vr. Every binary matrix with constant row and column sums is an incidencia matrix of regular uniform block design.
Paired Balanced Uniform Design (2 Design or BIBD)
Given a finite set block. In this design, any x in X is contained in r blocks, and any two distinct points x and y in X are also contained in λ blocks. The condition here means that it is unnecessary for any x to be contained in r blocks in X, as can be seen from the previous derivation. We can call this design a (v, k, λ)-design or a (v, b, r, k, λ)-design.
Because of the existence of imperfect balance, block design shows the mystery and beauty of combinatorial mathematics.
Symmetry 2 Design (SBIBD)
In all 2 designs, when the number of blocks and points is equal, the design is called a symmetric design. This type of design meets the requirements of the other 2 designs with the minimum number of blocks, and in the symmetric design, r=k, and b=v. Among them, any two different blocks intersect at the λ point. Ryser's theorem provides the conditions for symmetric design.
Examples and applications
A unique (6,3,2)-design has 10 blocks and each element is repeated 5 times. Represented using the notation 0-5, these blocks are the following triplet: 012, 013, 024, 035, 045, 125, 134, 145, 234, 235. The corresponding incidencia matrix is a binary matrix with v×b. The examples of block designs are rich and varied, ranging from mathematics to practical applications.
So, can the development and application of block design provide us with new ways of thinking in complex systems?
