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Block design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as blocks, chosen such that number of occurrences of each element satisfies certain conditions making the collection of blocks exhibit symmetry (balance). Block designs have applications in many areas, including experimental design, finite geometry, physical chemistry, software testing, cryptography, and algebraic geometry.
Without further specifications the term block design usually refers to a balanced incomplete block design (BIBD), specifically (and also synonymously) a 2-design, which has been the most intensely studied type historically due to its application in the design of experiments. Its generalization is known as a t-design.
A design is said to be balanced (up to t) if all t-subsets of the original set occur in equally many (i.e., λ) blocks[clarification needed]. When t is unspecified, it can usually be assumed to be 2, which means that each pair of elements is found in the same number of blocks and the design is pairwise balanced. For t = 1, each element occurs in the same number of blocks (the replication number, denoted r) and the design is said to be regular. A block design in which all the blocks have the same size (usually denoted k) is called uniform or proper. The designs discussed in this article are all uniform. Block designs that are not necessarily uniform have also been studied; for t = 2 they are known in the literature under the general name pairwise balanced designs (PBDs). Any uniform design balanced up to t is also balanced in all lower values of t (though with different λ-values), so for example a pairwise balanced (t = 2) design is also regular (t = 1). When the balancing requirement fails, a design may still be partially balanced if the t-subsets can be divided into n classes, each with its own (different) λ-value. For t = 2 these are known as PBIBD(n) designs, whose classes form an association scheme.
Designs are usually said (or assumed) to be incomplete, meaning that the collection of blocks is not all possible k-subsets, thus ruling out a trivial design.
Block designs may or may not have repeated blocks. Designs without repeated blocks are called simple, in which case the "family" of blocks is a set rather than a multiset.
In statistics, the concept of a block design may be extended to non-binary block designs, in which blocks may contain multiple copies of an element (see blocking (statistics)). There, a design in which each element occurs the same total number of times is called equireplicate, which implies a regular design only when the design is also binary. The incidence matrix of a non-binary design lists the number of times each element is repeated in each block.
The simplest type of "balanced" design (t = 1) is known as a tactical configuration or 1-design. The corresponding incidence structure in geometry is known simply as a configuration, see Configuration (geometry). Such a design is uniform and regular: each block contains k elements and each element is contained in r blocks. The number of set elements v and the number of blocks b are related by , which is the total number of element occurrences.
Every binary matrix with constant row and column sums is the incidence matrix of a regular uniform block design. Also, each configuration has a corresponding biregular bipartite graph known as its incidence or Levi graph.
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Block design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as blocks, chosen such that number of occurrences of each element satisfies certain conditions making the collection of blocks exhibit symmetry (balance). Block designs have applications in many areas, including experimental design, finite geometry, physical chemistry, software testing, cryptography, and algebraic geometry.
Without further specifications the term block design usually refers to a balanced incomplete block design (BIBD), specifically (and also synonymously) a 2-design, which has been the most intensely studied type historically due to its application in the design of experiments. Its generalization is known as a t-design.
A design is said to be balanced (up to t) if all t-subsets of the original set occur in equally many (i.e., λ) blocks[clarification needed]. When t is unspecified, it can usually be assumed to be 2, which means that each pair of elements is found in the same number of blocks and the design is pairwise balanced. For t = 1, each element occurs in the same number of blocks (the replication number, denoted r) and the design is said to be regular. A block design in which all the blocks have the same size (usually denoted k) is called uniform or proper. The designs discussed in this article are all uniform. Block designs that are not necessarily uniform have also been studied; for t = 2 they are known in the literature under the general name pairwise balanced designs (PBDs). Any uniform design balanced up to t is also balanced in all lower values of t (though with different λ-values), so for example a pairwise balanced (t = 2) design is also regular (t = 1). When the balancing requirement fails, a design may still be partially balanced if the t-subsets can be divided into n classes, each with its own (different) λ-value. For t = 2 these are known as PBIBD(n) designs, whose classes form an association scheme.
Designs are usually said (or assumed) to be incomplete, meaning that the collection of blocks is not all possible k-subsets, thus ruling out a trivial design.
Block designs may or may not have repeated blocks. Designs without repeated blocks are called simple, in which case the "family" of blocks is a set rather than a multiset.
In statistics, the concept of a block design may be extended to non-binary block designs, in which blocks may contain multiple copies of an element (see blocking (statistics)). There, a design in which each element occurs the same total number of times is called equireplicate, which implies a regular design only when the design is also binary. The incidence matrix of a non-binary design lists the number of times each element is repeated in each block.
The simplest type of "balanced" design (t = 1) is known as a tactical configuration or 1-design. The corresponding incidence structure in geometry is known simply as a configuration, see Configuration (geometry). Such a design is uniform and regular: each block contains k elements and each element is contained in r blocks. The number of set elements v and the number of blocks b are related by , which is the total number of element occurrences.
Every binary matrix with constant row and column sums is the incidence matrix of a regular uniform block design. Also, each configuration has a corresponding biregular bipartite graph known as its incidence or Levi graph.