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Introduction
Introduction
Since there is some variation between authors of the terminology
employed in design theory, we begin with some definitions. An
incidence structure is a triple D = (P, B, I), where:
- P is a set, the elements of which are called points;
- B is a set, the elements of which are called blocks;
- I is an incidence relation between P and B, so that
I subset P x B. The elements of I are called flags.
Usually, blocks will be subsets of P, so that instead of writing
(p, b) in I, we write p in b. In general, repeated blocks are
allowed so that different blocks may correspond to the same subset
of P. If D has no repeated blocks, then we say that D is
simple.
An incidence structure D is said to be uniform with
blocksize k if D has at least one block and all blocks
contain exactly k points.
A uniform incidence structure is called trivial if each
k-subset of the point set appears as a block (at least once).
Let t >= 0 be an integer. Then
an incidence structure D is said to be t-balanced if there exists
an integer lambda >= 1 such that each t-subset of the point set
is contained in exactly lambda blocks of D.
A near-linear space is an incidence structure in which every
block contains at least two points and any two points lie in at most
one block. A linear space is a near-linear space in which any
two points lie in exactly one block. It is usual, when discussing
near-linear spaces, to use the term line in place of the term
block.
Let v, k, t and lambda be integers with v >= k >= t >= 0
and lambda >= 1. A t-design with v points and
blocksize k is an incidence structure D = (P, B, I) where:
- The cardinality of P is v;
- D is uniform with blocksize k;
- D is simple;
- For each t-subset T of P there are exactly lambda blocks of B incident with all the points of T (so D is t-balanced).
Such a design is usually referred to as a t-(v, k, lambda) design.
The parameter lambda is called the index of the design.
If b denotes the cardinality of B, a t-design with v = b and
t >= 2 is called a symmetric design.
A t-design with lambda = 1 is called a Steiner design.
A design which is trivial is also called a complete design.
Note that a design D must contain at least one block (i.e b > 0).
The category names for the different families of incidence structures
are as follows:
- Incidence structure : Inc
- Near-linear space : IncNsp
- Linear space : IncLsp
- t-design : Dsgn
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