A Steiner triple system of order $v$, denoted $STS(v)$ or $S(2,3,v)$, has blocks of size 3 where every pair appears once.

Existence: $STS(v)$ exists if and only if $v \equiv 1$ or $3 \pmod{6}$.

Kirkman's construction: For $v = 3$ (trivial: $\{\{1,2,3\}\}$). For $v = 9$, using $\mathbb{Z}_9$:

Blocks include $\{0,1,2\}, \{3,4,5\}, \{6,7,8\}$ and develop $\{0,1,4\}$ under addition mod 9 to get all $\binom{9}{2}/3 = 12$ blocks.

Application: In round-robin scheduling with bye weeks.

A Hadamard matrix of order $n$ is an $n \times n$ matrix with entries $\pm 1$ such that $HH^T = nI$ (orthogonal rows).

From a Hadamard matrix of order $4m$, construct a $2$-$(4m-1, 2m-1, m-1)$ design:

1. Delete one row and one column
2. Replace $+1$ with 0 and $-1$ with 1
3. Rows are blocks, columns are points

This gives Hadamard designs, used in error-correcting codes and statistical experiments.

A projective plane of order $q$ (denoted $PG(2,q)$) is a $2$-$(q^2+q+1, q+1, 1)$ design with $q^2+q+1$ blocks.

Construction for prime power $q$: Use finite field $\mathbb{F}_q$. Points are equivalence classes $[x:y:z]$ of triples (not all zero) under scalar multiplication. Lines are sets satisfying $ax + by + cz = 0$.

Example: $PG(2,2)$ is the Fano plane ($q=2$: 7 points, 7 lines).

An affine plane of order $q$ (denoted $AG(2,q)$) is a $2$-$(q^2, q, 1)$ design obtained by deleting a line and its points from $PG(2,q)$.

A $(v,k,\lambda)$ difference set in an abelian group $G$ of order $v$ is a $k$-subset $D$ such that every non-identity element of $G$ can be expressed as $d_1 - d_2$ (with $d_1, d_2 \in D$) in exactly $\lambda$ ways.

From $D$, construct a $2$-$(v,k,\lambda)$ design: blocks are $D + g = \{d + g : d \in D\}$ for each $g \in G$.

Example: $D = \{1,2,4\} \subseteq \mathbb{Z}_7$ is a $(7,3,1)$ difference set:

• $1 = 2-1 = 4-1-2 \pmod{7}$ (appears once as $2-1$)
• All non-zero elements of $\mathbb{Z}_7$ appear exactly once as differences

Developing $D$ gives the Fano plane blocks.