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| :''For the set of elements in one set but not another, see [[relative complement]]. For the set of differences of pairs of elements, see [[Minkowski difference]].''
| | Oscar is how he's called and he completely loves this title. It's not a typical thing but what she likes doing is foundation jumping and now she is attempting to earn cash with it. Supervising is my profession. Minnesota is exactly where he's been living for years.<br><br>Here is my blog post :: [http://nemoonehfilm.ir/index.php?do=/profile-6876/info/ nemoonehfilm.ir] |
| In [[combinatorics]], a <math>(v,k,\lambda)</math> '''difference set''' is a [[subset]] <math>D</math> of [[cardinality|size]] <math>k</math> of a [[group (mathematics)|group]] <math>G</math> of [[order of a group|order]] <math>v</math> such that every nonidentity element of <math>G</math> can be expressed as a product <math>d_1d_2^{-1}</math> of elements of <math>D</math> in exactly <math>\lambda</math> ways. A difference set <math>D</math> is said to be ''cyclic'', ''abelian'', ''non-abelian'', etc., if the group <math>G</math> has the corresponding property. A difference set with <math>\lambda = 1</math> is sometimes called ''planar'' or ''simple''.<ref>{{harvnb|van Lint|Wilson|1992|loc=p. 331}}</ref> If <math>G</math> is an [[abelian group]] written in additive notation, the defining condition is that every nonzero element of <math>G</math> can be written as a ''difference'' of elements of <math>D</math> in exactly <math>\lambda</math> ways. The term "difference set" arises in this way.
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| ==Basic facts==
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| * A simple counting argument shows that there are exactly <math>k^2-k</math> pairs of elements from <math>D</math> that will yield nonidentity elements, so every difference set must satisfy the equation <math>k^2-k=(v-1)\lambda</math>.
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| * If <math>D</math> is a difference set, and <math>g\in G</math>, then <math>gD=\{gd:d\in D\}</math> is also a difference set, and is called a '''translate''' of <math>D</math> (<math>D + g</math> in additive notation).
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| * The complement of a <math>(v,k,\lambda)</math>-difference set is a <math>(v,v-k,v-2k+\lambda)</math>-difference set.<ref>{{harvnb|Wallis|1988|loc=p. 61 - Theorem 4.5}}</ref>
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| * The set of all translates of a difference set <math>D</math> forms a [[Block design#Symmetric BIBDs|symmetric block design]], called the ''development'' of <math>D</math> and denoted by <math>dev(D)</math>. In such a design there are <math>v</math> ''elements'' (usually called points) and <math>v</math> ''blocks'' (subsets). Each block of the design consists of <math>k</math> points, each point is contained in <math>k</math> blocks. Any two blocks have exactly <math>\lambda</math> elements in common and any two points are simultaneously contained in exactly <math>\lambda</math> blocks. The group <math>G</math> acts as an automorphism group of the design. It is sharply transitive on both points and blocks.<ref>{{harvnb|van Lint|Wilson|1992|loc=p. 331 - Theorem 27.2}}. The theorem only states point transitivity, but block transitivity follows from this by the second corollary on p. 330. </ref>
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| ** In particular, if <math>\lambda=1</math>, then the difference set gives rise to a [[projective plane]]. An example of a (7,3,1) difference set in the group <math>\mathbb{Z}/7\mathbb{Z}</math> is the subset <math>\{1,2,4\}</math>. The translates of this difference set form the [[Fano plane]].
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| * Since every difference set gives a [[Block design#Symmetric BIBDs|symmetric design]], the parameter set must satisfy the [[Bruck–Ryser–Chowla theorem]].<ref>{{harvnb|Colbourn|Dinitz|2007|loc=p. 420 (18.7 Remark 2)}}</ref>
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| * Not every [[Block design#Symmetric BIBDs|symmetric design]] gives a difference set.<ref>{{harvnb|Colbourn|Dinitz|2007|loc=p. 420 (18.7 Remark 1)}}</ref>
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| ==Equivalent and isomorphic difference sets==
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| Two difference sets <math>D_1</math> in group <math>G_1</math> and <math>D_2</math> in group <math>G_2</math> are '''equivalent''' if there is a [[group isomorphism]] <math>\psi</math> between <math>G_1</math> and <math>G_2</math> such that <math>D_1^{\psi} = \{d^{\psi}\colon d \in D_1 \} = g D_2</math> for some <math>g \in G_2</math>. The two difference sets are '''isomorphic''' if the designs <math>dev(D_1)</math> and <math>dev(D_2)</math> are isomorphic as block designs.
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| 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..<ref>{{harvnb|Colbourn|Diniz|2007|loc=p. 420 (Remark 18.9)}}</ref>
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| ==Multipliers==
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| A '''multiplier''' of a difference set <math>D</math> in group <math>G</math> is a [[group automorphism]] <math>\phi</math> of <math>G</math> such that <math>D^{\phi} = gD</math> for some <math>g \in G</math>. If <math>G</math> is abelian and <math>\phi</math> is the automorphism that maps <math>h \mapsto h^t</math>, then <math>t</math> is called a ''numerical'' or ''Hall'' '''multiplier'''.<ref>{{harvnb|van Lint|Wilson|1992|loc=p. 345}}</ref>
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| It has been conjectured that if ''p'' is a prime dividing <math>k-\lambda</math> and not dividing ''v'', then the group automorphism defined by <math>g\mapsto g^p</math> fixes some translate of ''D'' (this is equivalent to being a multiplier). It is known to be true for <math>p>\lambda</math> when <math>G</math> 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 <math>D</math> is a <math>(v,k,\lambda)</math>-difference set in an abelian group <math>G</math> of exponent <math>v^*</math> (the [[least common multiple]] of the orders of every element), let <math>t</math> be an integer coprime to <math>v</math>. If there exists a divisor <math>m>\lambda</math> of <math>k-\lambda</math> such that for every prime ''p'' dividing ''m'', there exists an integer ''i'' with <math>t\equiv p^i\ \pmod{v^*}</math>, then ''t'' is a numerical divisor.<ref>{{harvnb|van Lint|Wilson|1992|loc=p. 349 (Theorem 28.7)}}</ref>
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| For example, 2 is a multiplier of the (7,3,1)-difference set mentioned above.
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| It has been mentioned that a numerical multiplier of a difference set <math>D</math> in an abelian group <math>G</math> fixes a translate of <math>D</math>, but it can also be shown that there is a translate of <math>D</math> which is fixed by all numerical multipliers of <math>D</math>.<ref>{{harvnb|Beth|Jungnickel|Lenz|1986|loc=p. 280 (Theorem 4.6)}}</ref>
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| ==Parameters==
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| The known difference sets or their complements have one of the following parameter sets:{{cn|date=July 2013}}
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| *<math>((q^{n+2}-1)/(q-1), (q^{n+1}-1)/(q-1), (q^n-1)/(q-1))</math>-difference set for some prime power <math>q</math> and some positive integer <math>n</math>.
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| *<math>(4n-1,2n-1,n-1)</math>-difference set for some positive integer <math>n</math>.
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| *<math>(4n^2,2n^2-n,n^2-n)</math>-difference set for some positive integer <math>n</math>.
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| *<math>(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))</math>-difference set for some prime power <math>q</math> and some positive integer <math>n</math>.
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| *<math>(3^{n+1}(3^{n+1}-1)/2,3^n(3^{n+1}+1)/2,3^n(3^n+1)/2)</math>-difference set for some positive integer <math>n</math>.
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| *<math>(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))</math>-difference set for some prime power <math>q</math> and some positive integer <math>n</math>.
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| ==Known difference sets==
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| 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, <math>{\rm GF}(q)</math> is the [[Galois field]] of order <math>q</math>, where <math>q</math> is a prime or prime power. The group under addition is denoted by <math>G = ({\rm GF}(q), +)</math>, while <math>{\rm GF}(q)^*</math> is the multiplicative group of non-zero elements.
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| * Paley <math>(4n-1, 2n-1, n-1)</math>-difference set:
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| ::Let <math>q = 4n -1</math> be a prime power. In the group <math>G = ({\rm GF}(q), +)</math>, let <math>D</math> be the set of all non-zero squares. | |
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| * Singer <math>((q^{n+2}-1)/(q-1), (q^{n+1}-1)/(q-1), (q^n-1)/(q-1))</math>-difference set:
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| ::Let <math>G={\rm GF}(q^{n+2})^*/{\rm GF}(q)^*</math>. Then the set <math>D=\{x\in G~|~{\rm Tr}_{q^{n+2}/q}(x)=0\}</math> is a <math>((q^{n+2}-1)/(q-1), (q^{n+1}-1)/(q-1), (q^n-1)/(q-1))</math>-difference set, where <math>{\rm Tr}_{q^{n+2}/q}:{\rm GF}(q^{n+2})\rightarrow{\rm GF}(q)</math> is the [[trace function]] <math>{\rm Tr}_{q^{n+2}/q}(x)=x+x^q+\cdots+x^{q^{n+1}}</math>.
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| * Twin prime power <math> \left ( q^2 + 2q, \frac{q^2 + 2q -1}{2}, \frac{q^2+2q-3}{4} \right )</math>-difference set when <math>q</math> and <math>q+2</math> are both prime powers:
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| ::In the group <math>G = ({\rm GF}(q), +) \oplus ({\rm GF}(q+2), +)</math>, let <math>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} \}.</math><ref>{{harvnb|Colbourn|Dinitz|2007|loc=p. 425 (Construction 18.49)}}</ref>
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| ==History==
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| 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.<ref>{{citation|first=R.C.|last=Bose|title=On the construction of balanced incomplete block designs|journal=Annals of Eugenics|volume=9|year=1939|pages=353–399}}</ref> However, various examples appeared earlier than this, such as the "Paley Difference Sets" which date back to 1933.<ref>{{harvnb|Wallis|1988|loc=p. 69}}</ref> The generalization of the cyclic difference set concept to more general groups is due to [[Richard Bruck|R.H. Bruck]]<ref>{{citation|first=R.H.|last=Bruck|title=Difference sets in a finite group|journal=Transactions of the American Mathematical Society|volume=78|year=1955|pages=464–481}}</ref> in 1955.<ref>{{harvnb|van Lint|Wilson|1992|loc=p. 340}}</ref> Multipliers were introduced by [[Marshall Hall (mathematician)|Marshall Hall Jr.]]<ref>{{citation|last=Hall Jr.|first=Marshall|title=Cyclic projective planes|journal=Duke Journal of Mathematics|volume=14|year=1947|pages=1079–1090}}</ref> in 1947.<ref>{{harvnb|Beth|Jungnickel|Lenz|1986|loc=p. 275}}</ref>
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| ==Application==
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| It is found by Xia, Zhou and [[Georgios B. Giannakis|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.
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| ==Generalisations==
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| A <math>(v,k,\lambda,s)</math> '''difference family''' is a set of subsets <math>B=\{B_1,...B_s\}</math> of a [[group (mathematics)|group]] <math>G</math> such that the [[order of a group|order]] of <math>G</math> is <math>v</math>, the [[cardinality|size]] of <math>B_i</math> is <math>k</math> for all <math>i</math>, and every nonidentity element of <math>G</math> can be expressed as a product <math>d_1d_2^{-1}</math> of elements of <math>B_i</math> for some <math>i</math> (i.e. both <math>d_1,d_2</math> come from the same <math>B_i</math>) in exactly <math>\lambda</math> ways.
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| A difference set is a difference family with <math>s=1</math>. The parameter equation above generalises to <math>s(k^2-k)=(v-1)\lambda</math>.<ref>{{harvnb|Beth|Jungnickel|Lenz|1986|loc=p. 310 (2.8.a)}}</ref>
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| The development <math>dev (B) = \{B_i+g: i=1,...,s, g \in G\}</math> of a difference family is a [[Block_design|2-design]].
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| Every 2-design with a regular automorphism group is <math>dev (B)</math> for some difference family <math>B</math>.
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| ==See also==
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| *[[Combinatorial design]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{citation|first1=Thomas|last1=Beth|first2=Dieter|last2=Jungnickel|first3=Hanfried|last3=Lenz|title=Design Theory|year=1986|publisher=Cambridge University Press|place=Cambridge|isbn =0521333342}}
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| * {{citation|last1=Colbourn|first1=Charles J.|last2=Dinitz|first2=Jeffrey H.|title=Handbook of Combinatorial Designs|year=2007|publisher=Chapman & Hall/ CRC|location=Boca Raton|isbn=1-58488-506-8|edition=2nd Edition}}
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| * {{cite book | zbl=0157.03301 | last=Storer | first=T. | title=Cyclotomy and difference sets | location=Chicago | publisher=Markham Publishing Company | year=1967 | zbl=0157.03301 }}
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| * {{citation|first1=J.H. |last1=van Lint|first2=R.M. |last2=Wilson|title=A Course in Combinatorics|publisher=Cambridge University Press|place=Cambridge|year=1992|isbn=0-521-42260-4}}
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| * {{cite book | first=W.D. | last=Wallis | title=Combinatorial Designs | publisher=Marcel Dekker |year=1988 |isbn=0-8247-7942-8 | zbl=0637.05004 }}
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| * {{cite book | first=Daniel | last=Zwillinger |title=CRC Standard Mathematical Tables and Formulae |publisher=CRC Press |year=2003 |isbn=1-58488-291-3 |page=246}}
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| * {{cite journal | last1=Xia | first1=Pengfei | last2=Zhou | first2=Shengli | last3=Giannakis | first3=Georgios B. | title=Achieving the Welch Bound with Difference Sets |journal=IEEE Transactions on Information Theory |volume=51 |issue=5 |pages=1900–1907 |year=2005 |doi=10.1109/TIT.2005.846411 |url=http://www.engr.uconn.edu/~shengli/papers/conf05/05icassp.pdf | issn=0018-9448 | zbl=1237.94007}}.
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| :{{ cite journal | last1=Xia | first1=Pengfei | last2=Zhou | first2=Shengli | last3=Giannakis | first3=Georgios B. | title=Correction to ``Achieving the Welch bound with difference sets" | journal=IEEE Trans. Inf. Theory | volume=52 | issue=7 | page=3359 | year=2006 | zbl=1237.94008 }}
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| [[Category:Combinatorics]]
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Oscar is how he's called and he completely loves this title. It's not a typical thing but what she likes doing is foundation jumping and now she is attempting to earn cash with it. Supervising is my profession. Minnesota is exactly where he's been living for years.
Here is my blog post :: nemoonehfilm.ir