Difference set

In combinatorics, a $(v,k,\lambda )$ difference set is a subset $D$ of size $k$ of a group $G$ of order $v$ such that every non-identity element of $G$ can be expressed as a product $d_{1}d_{2}^{-1}$ of elements of $D$ in exactly $\lambda$ ways. A difference set $D$ is said to be cyclic, abelian, non-abelian, etc., if the group $G$ has the corresponding property. A difference set with $\lambda =1$ is sometimes called planar or simple.^[1] If $G$ is an abelian group written in additive notation, the defining condition is that every non-zero element of $G$ can be written as a difference of elements of $D$ in exactly $\lambda$ ways. The term "difference set" arises in this way.

Basic facts

A simple counting argument shows that there are exactly $k^{2}-k$ pairs of elements from $D$ that will yield nonidentity elements, so every difference set must satisfy the equation $k^{2}-k=(v-1)\lambda .$
If $D$ is a difference set and $g\in G,$ then $gD=\{gd:d\in D\}$ is also a difference set, and is called a translate of $D$ ( $D+g$ in additive notation).
The complement of a $(v,k,\lambda )$ -difference set is a $(v,v-k,v-2k+\lambda )$ -difference set.^[2]
The set of all translates of a difference set $D$ $D$ forms a symmetric block design, called the development of $D$ $D$ and denoted by $dev(D).$ $dev(D).$ In such a design there are $v$ $v$ elements (usually called points) and $v$ $v$ blocks (subsets). Each block of the design consists of $k$ $k$ points, each point is contained in $k$ $k$ blocks. Any two blocks have exactly $\lambda$ $\lambda$ elements in common and any two points are simultaneously contained in exactly $\lambda$ $\lambda$ blocks. The group $G$ $G$ acts as an automorphism group of the design. It is sharply transitive on both points and blocks.^[3]
- In particular, if $\lambda =1$ , then the difference set gives rise to a projective plane. An example of a (7,3,1) difference set in the group $\mathbb {Z} /7\mathbb {Z}$ is the subset $\{1,2,4\}$ . The translates of this difference set form the Fano plane.
Since every difference set gives a symmetric design, the parameter set must satisfy the Bruck–Ryser–Chowla theorem.^[4]
Not every symmetric design gives a difference set.^[5]

Equivalent and isomorphic difference sets

Two difference sets $D_{1}$ in group $G_{1}$ and $D_{2}$ in group $G_{2}$ are equivalent if there is a group isomorphism $\psi$ between $G_{1}$ and $G_{2}$ such that $D_{1}^{\psi }=\{d^{\psi }\colon d\in D_{1}\}=gD_{2}$ for some $g\in G_{2}.$ The two difference sets are isomorphic if the designs $dev(D_{1})$ and $dev(D_{2})$ are isomorphic as block designs.

Equivalent difference sets are isomorphic, but there exist examples of isomorphic difference sets which are not equivalent. In the cyclic difference set case, all known isomorphic difference sets are equivalent.^[6]

Multipliers

A multiplier of a difference set $D$ in group $G$ is a group automorphism $\phi$ of $G$ such that $D^{\phi }=gD$ for some $g\in G.$ If $G$ is abelian and $\phi$ is the automorphism that maps $h\mapsto h^{t}$ , then $t$ is called a numerical or Hall multiplier.^[7]

It has been conjectured that if p is a prime dividing $k-\lambda$ and not dividing v, then the group automorphism defined by $g\mapsto g^{p}$ fixes some translate of D (this is equivalent to being a multiplier). It is known to be true for $p>\lambda$ when $G$ is an abelian group, and this is known as the First Multiplier Theorem. A more general known result, the Second Multiplier Theorem, says that if $D$ is a $(v,k,\lambda )$ -difference set in an abelian group $G$ of exponent $v^{*}$ (the least common multiple of the orders of every element), let $t$ be an integer coprime to $v$ . If there exists a divisor $m>\lambda$ of $k-\lambda$ such that for every prime p dividing m, there exists an integer i with $t\equiv p^{i}\ {\pmod {v^{*}}}$ , then t is a numerical divisor.^[8]

For example, 2 is a multiplier of the (7,3,1)-difference set mentioned above.

It has been mentioned that a numerical multiplier of a difference set $D$ in an abelian group $G$ fixes a translate of $D$ , but it can also be shown that there is a translate of $D$ which is fixed by all numerical multipliers of $D.$ ^[9]

Parameters

The known difference sets or their complements have one of the following parameter sets:^[10]

$((q^{n+2}-1)/(q-1),(q^{n+1}-1)/(q-1),(q^{n}-1)/(q-1))$ -difference set for some prime power $q$ and some positive integer $n$ . These are known as the classical parameters and there are many constructions of difference sets having these parameters.
$(4n-1,2n-1,n-1)$ -difference set for some positive integer $n$ . Difference sets with v = 4n − 1 are called Paley-type difference sets.
$(4n^{2},2n^{2}-n,n^{2}-n)$ -difference set for some positive integer $n$ . A difference set with these parameters is a Hadamard difference set.
$(q^{n+1}(1+(q^{n+1}-1)/(q-1)),q^{n}(q^{n+1}-1)/(q-1),q^{n}(q^{n}-1)/(q-1))$ -difference set for some prime power $q$ and some positive integer $n$ . Known as the McFarland parameters.
$(3^{n+1}(3^{n+1}-1)/2,3^{n}(3^{n+1}+1)/2,3^{n}(3^{n}+1)/2)$ -difference set for some positive integer $n$ . Known as the Spence parameters.
$(4q^{2n}(q^{2n}-1)/(q-1),q^{2n-1}(1+2(q^{2n}-1)/(q+1)),q^{2n-1}(q^{2n-1}+1)(q-1)/(q+1))$ -difference set for some prime power $q$ and some positive integer $n$ . Difference sets with these parameters are called Davis-Jedwab-Chen difference sets.

Known difference sets

In many constructions of difference sets, the groups that are used are related to the additive and multiplicative groups of finite fields. The notation used to denote these fields differs according to discipline. In this section, ${\rm {GF}}(q)$ is the Galois field of order $q,$ where $q$ is a prime or prime power. The group under addition is denoted by $G=({\rm {GF}}(q),+)$ , while ${\rm {GF}}(q)^{*}$ is the multiplicative group of non-zero elements.

Paley $(4n-1,2n-1,n-1)$ -difference set:

Let

q=4n-1

be a prime power. In the group

G=({\rm {GF}}(q),+)

, let

D

be the set of all non-zero squares.

Singer $((q^{n+2}-1)/(q-1),(q^{n+1}-1)/(q-1),(q^{n}-1)/(q-1))$ -difference set:

Let

G={\rm {GF}}(q^{n+2})^{*}/{\rm {GF}}(q)^{*}

. Then the set

D=\{x\in G~|~{\rm {Tr}}_{q^{n+2}/q}(x)=0\}

is a

((q^{n+2}-1)/(q-1),(q^{n+1}-1)/(q-1),(q^{n}-1)/(q-1))

-difference set, where

{\rm {Tr}}_{q^{n+2}/q}:{\rm {GF}}(q^{n+2})\rightarrow {\rm {GF}}(q)

is the trace function

{\rm {{Tr}_{q^{n+2}/q}(x)=x+x^{q}+\cdots +x^{q^{n+1}}.}}

Twin prime power $\left(q^{2}+2q,{\frac {q^{2}+2q-1}{2}},{\frac {q^{2}+2q-3}{4}}\right)$ -difference set when $q$ and $q+2$ are both prime powers:

In the group

G=({\rm {GF}}(q),+)\oplus ({\rm {GF}}(q+2),+)

, let

D=\{(x,y)\colon y=0{\text{ or }}x{\text{ and }}y{\text{ are non-zero and both are squares or both are non-squares}}\}.

^[11]

History

The systematic use of cyclic difference sets and methods for the construction of symmetric block designs dates back to R. C. Bose and a seminal paper of his in 1939.^[12] However, various examples appeared earlier than this, such as the "Paley Difference Sets" which date back to 1933.^[13] The generalization of the cyclic difference set concept to more general groups is due to R.H. Bruck^[14] in 1955.^[15] Multipliers were introduced by Marshall Hall Jr.^[16] in 1947.^[17]

Application

It is found by Xia, Zhou and Giannakis that difference sets can be used to construct a complex vector codebook that achieves the difficult Welch bound on maximum cross correlation amplitude. The so-constructed codebook also forms the so-called Grassmannian manifold.

Generalisations

A $(v,k,\lambda ,s)$ difference family is a set of subsets $B=\{B_{1},\ldots ,B_{s}\}$ of a group $G$ such that the order of $G$ is $v$ , the size of $B_{i}$ is $k$ for all $i$ , and every non-identity element of $G$ can be expressed as a product $d_{1}d_{2}^{-1}$ of elements of $B_{i}$ for some $i$ (i.e. both $d_{1},d_{2}$ come from the same $B_{i}$ ) in exactly $\lambda$ ways.

A difference set is a difference family with $s=1.$ The parameter equation above generalises to $s(k^{2}-k)=(v-1)\lambda .$ ^[18] The development $dev(B)=\{B_{i}+g:i=1,\ldots ,s,g\in G\}$ of a difference family is a 2-design. Every 2-design with a regular automorphism group is $dev(B)$ for some difference family $B.$

Notes

^ van Lint & Wilson 1992, p. 331
^ Wallis 1988, p. 61 - Theorem 4.5
^ van Lint & Wilson 1992, p. 331 - Theorem 27.2. The theorem only states point transitivity, but block transitivity follows from this by the second corollary on p. 330.
^ Colbourn & Dinitz 2007, p. 420 (18.7 Remark 2)
^ Colbourn & Dinitz 2007, p. 420 (18.7 Remark 1)
^ Colbourn & Dinitz 2007, p. 420 (Remark 18.9)
^ van Lint & Wilson 1992, p. 345
^ van Lint & Wilson 1992, p. 349 (Theorem 28.7)
^ Beth, Jungnickel & Lenz 1986, p. 280 (Theorem 4.6)
^ Colbourn & Dinitz 2007, pp. 422-425
^ Colbourn & Dinitz 2007, p. 425 (Construction 18.49)
^ Bose, R.C. (1939), "On the construction of balanced incomplete block designs", Annals of Eugenics, 9 (4): 353–399, doi:10.1111/j.1469-1809.1939.tb02219.x, JFM 65.1110.04, Zbl 0023.00102
^ Wallis 1988, p. 69
^ Bruck, R.H. (1955), "Difference sets in a finite group", Transactions of the American Mathematical Society, 78 (2): 464–481, doi:10.2307/1993074, JSTOR 1993074, Zbl 0065.13302
^ van Lint & Wilson 1992, p. 340
^ Hall Jr., Marshall (1947), "Cyclic projective planes", Duke Mathematical Journal, 14 (4): 1079–1090, doi:10.1215/s0012-7094-47-01482-8, S2CID 119846649, Zbl 0029.22502
^ Beth, Jungnickel & Lenz 1986, p. 275
^ Beth, Jungnickel & Lenz 1986, p. 310 (2.8.a)

References

Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986), Design Theory, Cambridge: Cambridge University Press, ISBN 0-521-33334-2, Zbl 0602.05001
Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs, Discrete Mathematics and its Applications (2nd ed.), Boca Raton: Chapman & Hall/ CRC, ISBN 978-1-58488-506-1, Zbl 1101.05001
van Lint, J.H.; Wilson, R.M. (1992), A Course in Combinatorics, Cambridge: Cambridge University Press, ISBN 0-521-42260-4, Zbl 0769.05001
Wallis, W.D. (1988). Combinatorial Designs. Marcel Dekker. ISBN 0-8247-7942-8. Zbl 0637.05004.