FoldUnfold Table of Contents Paley Difference Sets Paley Difference Sets Recall from the Difference Sets page that if $v$, $k$, and $\lambda$ are positive integers such that $2 \leq k and $(G, +)$ is a group of order $v$ with identity $0$, then a $(v, k, \lambda)$-difference set is a nonempty proper subset $D \subset G$ such that $\mid D \mid = k$ and the multiset $\{ x - … [Read more...]

## The Trivial Difference Sets

FoldUnfold Table of Contents The Trivial Difference Sets The Trivial Difference Sets Recall from the Difference Sets page that if $v$, $k$, and $\lambda$ are positive integers such that $2 \leq k and $(G, +)$ is a finite group of order $v$ then a $(v, k, \lambda)$-difference set in $G$ is a nonempty proper subset $D \subset G$ such that the multiset of differences $\{ x - … [Read more...]

## Partial Difference Sets

FoldUnfold Table of Contents Partial Difference Sets Partial Difference Sets Recall from the Difference Sets page that if $v$, $k$, and $\lambda$ are positive integers such that $2 \leq k and $(G, +)$ is a group of order $v$ with identity element $0$ then a $(v, k, \lambda)$-difference set in this group is a nonempty proper subset $D \subset G$ with the properties that: 1. … [Read more...]

## Difference Sets

FoldUnfold Table of Contents Difference Sets Difference Sets We now turn our attention to a very nice structure known as a difference set on an algebraic group. Definition: Let $v$, $k$, and $\lambda$ be positive integers such that $2 \leq k and let $(G, +)$ be a finite group of order $v$ with identity element $0$. A $(v, k, \lambda)$-Difference Set in $(G, +)$ is a … [Read more...]

## Isomorphisms and Automorphisms of Balanced Incomplete Block Designs

FoldUnfold Table of Contents Isomorphisms and Automorphisms of Balanced Incomplete Block Designs Isomorphisms of Balanced Incomplete Block Designs Automorphisms of Balanced Incomplete Block Designs Isomorphisms and Automorphisms of Balanced Incomplete Block Designs Suppose that we have two block designs $(X, \mathcal A)$ and $(Y, \mathcal B)$ with $\mid X \mid = \mid Y … [Read more...]

## Basic Properties of Incidence Matrices of Balanced Incomplete Block Designs

FoldUnfold Table of Contents Basic Properties of Incidence Matrices of Balanced Incomplete Block Designs Basic Properties of Incidence Matrices of Balanced Incomplete Block Designs Recall from The Incidence Matrix of a Balanced Incomplete Block Design page that if $(X, \mathcal A)$ is a $(v, b, r, k, \lambda)$-BIBD then the incidence matrix of this BIBD is the $v \times b$ … [Read more...]

## The Incidence Matrix of a Balanced Incomplete Block Design

FoldUnfold Table of Contents The Incidence Matrix of a Balanced Incomplete Block Design The Incidence Matrix of a Balanced Incomplete Block Design We will now look at one way to describe the information in a BIBD by putting it in matrix form. Definition: Let $(X, \mathcal A)$ by a $(v, b, r, k, \lambda)$-BIBD where $X = \{ x_1, x_2, ..., x_v \}$ and $\mathcal A = \{ A_1, … [Read more...]

## The Complement Construction of Balanced Incomplete Block Designs

FoldUnfold Table of Contents The Complement Construction of Balanced Incomplete Block Designs The Complement Construction of Balanced Incomplete Block Designs On The Sum Construction of Balanced Incomplete Block Designs page we saw that if $(X, \mathcal A_1)$ is a $(v, k, \lambda_1)$-BIBD and $(X, \mathcal A_2)$ is a $(v, k, \lambda_2)$-BIBD then $(X, \mathcal A)$ where … [Read more...]

## The Sum Construction of Balanced Incomplete Block Designs

FoldUnfold Table of Contents The Sum Construction of Balanced Incomplete Block Designs The Sum Construction of Balanced Incomplete Block Designs Suppose that we have a $(v, k, \lambda_1)$-BIBD and a $(v, k, \lambda_2)$-BIBD on the set $X$. Our goal is to take these two BIBDs and obtain a new BIBD. The following result gives us one method to construct a new BIBD called the … [Read more...]

## The Existence of Balanced Incomplete Block Designs

FoldUnfold Table of Contents The Existence of Balanced Incomplete Block Designs The Existence of Balanced Incomplete Block Designs On The Replication Number of a Balanced Incomplete Block Design page we defined the replication number $r$ of a $(v, k, \lambda)$-BIBD to be the number of blocks containing any arbitrary point and we derived the formula: (1) \begin{align} \quad … [Read more...]