---
tag: MG1KM4S1BV
kind: concept
name: Difference set
msc: [05B10]
sources:
  - url: https://en.wikipedia.org/wiki/Difference_set
same_as:
  wikidata: Q5275281
  wikipedia: Difference set
---

## Summary

A difference set is a subset $D$ of a finite group with a balance property.
Every non-identity element of the group arises as a "difference" $d_1 d_2^{-1}$ of elements of $D$ the same number $\lambda$ of times.
The translates of a difference set form a symmetric [[block design|4JCSS2Z1BA]] on which the group acts regularly.
Difference sets give many projective planes, Hadamard matrices and sequences with good correlation.
The node [[difference set|D1QA6N9EWJ]] gives the precise definition.

## Explanations

### L1 human

Take the numbers $1, 2, 4$ in arithmetic modulo $7$.
The differences between two of them are $\pm 1$, $\pm 2$ and $\pm 3$, and these are each nonzero residue modulo $7$ exactly once.
So $\set{1, 2, 4}$ "sees" every nonzero distance exactly once.
Shifting it by $0, 1, \dots, 6$ gives seven triples that form a copy of the [[Fano plane|QQ8GW1ASZM]].