# 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:

**(a)**
$P$ is a set, the elements of which are called *points*;

**(b)**
$B$ is a set, the elements of which are called *blocks*;

**(c)**
$I$ is an incidence relation between $P$ and $B$, so that $I \subset P \times B$. 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 $t \ge 0$ be an integer. Then an incidence structure $D$ is said to be $t$–*balanced* if there exists an integer $\lambda \ge 1$ 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 $v \ge k \ge t \ge 0$ and $\lambda \ge 1$. A $t$–*design* with $v$ points and *blocksize* $k$ is an incidence structure $D = (P, B, I)$ where:

**(a)**
The cardinality of $P$ is $v$;

**(b)**
$D$ is uniform with blocksize $k$;

**(c)**
$D$ is simple;

**(d)**
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 $t \ge 2$ 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`