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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:

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:

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:

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