Definition. Let be a finite abelian group of order . A subset of is said to be a difference set (in ) if there is a positive integer such that every non-zero element of can be expressed as the difference of elements of in exactly ways.
If has elements, then we have the equation
In the equation, we are counting the number of pairs of distinct elements of . On the left hand side, we are counting it by noting that there are pairs of elements of such that their difference is non-zero. On the right hand side, we first count the number of elements in , which is , then subtracted by , since there are pairs of such that .
A difference set with parameters defined above is also called a -difference set. A difference set is said to be non-trivial if . A difference set is said to be planar if .
Difference sets versus square designs. Recall that a square design is a --design (http://planetmath.org/Design) where and the number of points is the same as the number of blocks. In a general design, is related to the other numbers by the equation
So in a square design, the equation reduces to , or
which is identical to the equation above for the difference set. A square design with parameters is called a square -design.
One can show that a subset of an abelian group is an -difference set iff it is a square -design where is the set of points and is the set of blocks.
|Date of creation||2013-03-22 16:50:04|
|Last modified on||2013-03-22 16:50:04|
|Last modified by||CWoo (3771)|
|Defines||non-trivial difference set|
|Defines||planar difference set|