.I@rul ofscie ce: Pb-sical Sciences,15,17-25(2009)
:IVALENCEOF SQUAREDESIGNS,DIFFERENCESETSAND COMPLETE
CRAPIIS
A.A.I. PERERA
Deparunen t of Mathe latics, Uniwrsitf of Peradenifa, Peradeniya.
ABSTRACT
DiffereDcc scts can be corstncted by differenr tec|niques ln rbis work. we construct
J i m l ' r . . c l . u , i i . 4 / , d , o , n t , , ' d i \ d n da t h i ' ,
The equivaldrc bctwcen (v,f,Z) - a;nerencesets in a -qroup G and a symmetric Gquare)
(v,&,2)-design
with a regular auromorphismgroup C rs well krcwn. we prove the
equivalenceof graphs, squarc dcsigns and diffcrcncc scts using complete graphs and the
equivalenceis illustatcd by an exanrplc.GcncElization of this result for a regulaf g|apl is
I. INTRODUCTION
A combinatorial
designis a way of choosing,ftom a givenfinite set; a collection of subsets
rith particularpropefiies.The studyof designswerebegunby Euler in 1782.Later were by
Yoolhouse(1844),Kirkman(1E47)andSteiner(1853).
Tb€ firstresulton the theoryofblock designswasdueto Fishcrin 1926.Latcr,Yates(1936),
€ltorvla and Riscr (1950), Mam(1969) and modem conlribution by Shrikhande,Sebery',
Yamadal, Jungicke'], Beth' and Pott1. Th Beth, D- Jungnickcl aDd H. Lenzr havo
ioned that a completeglaph can be parlitioned in to subgraphs-This was basedon my
and provcd the Theoren 3. 1
G be a groupof order v rvrittenmultiplicatively.A (r,,ic,Z)-difference set in G is a /.subsel 1) so that the multi set {dd,':d|d.eD,
dr+tlr} containseachnon(i
-,1
=
tity elementof
exactly 2 times. The integer /r /r
is the order of D.
G is an additivc group, then instead of a multi-set we cbnsider the set
dt.,l, D.dt-d)I
t-d1
e thal any grouphaslbur types oft vidl differencesets,nancly, thc cmpty sct, G itsclf,
singletonsubsetof G, and its comple1nellt.h1 general, difference sets occurin seilary palrs.lndeed,if ll is a (r,1,2) -dif'lcrenceset in 6 , then 1)' : c
, isa
'- t-. r'- 2,t + i,) - differencesel in C . Thus to simplily classificationand to climinatethl]
a l t ) ? e sf r o mc o n s i d e r a t i o\ \n, e
, c h o o s c1 < k < r , l 2 ] :
t7
A differencesr:t is called abelian,non-abelianor cyclic accordingas the ulderlying goup is
abelian,non-abelianor cyclic respectively.
The study ofdifference setsis closely connectedwith coding theory becausethe code over a
field F ofthe slanmetricdesignconespondingto a (r,,ft,,l) differenceset may be considered
as the right ideal genemtedby , in the group algebra -F[G] .
Abelian difference sets arise naturally in the solution of many problems of signal design in
digital communication, including sl,nchronization, radar, coded aperture imagilu and optical
imagealignment.
Part of this work is the construction of difference sets using quadratic residues and olbits (or
numerical multipliers)- In the latter part, we use graph theory, specially complete graphs and
regular graphs, to obtain equivalencebetween graphs, designs and difference sets This
equivalencecan be illushated graphicallyand it helps to constructcodes,that may be uscd in
di gital communlcation.
2. METHODOLOGY
2.1 Graphs
A simple graph G is a pair (l/(C),li(G)), wherc /(G) is a finite nonempty set of elements
y'(G) called
calledvertices and ,(G) is a finite set of unorderedpairs of distinct elementsof
u
edges.
The number of vertices of G is calied the o/del of G , and denoteby t. Two verticesthat
6
arl- incident with a common edge are said to be adjacent A set of vefiiccs in which every
dislinct pair is adjacent is called a clique. A k cliqie. (called a complete graph of
is a cliquewilh I \enices.
I venices.;
There arc two types of $aphs that irre ofparticular importancein our work. They are the
complete graphs and the regular graphs. A completc graph K, is a simple graph in which
each pair of distinct vertices is joined by an edge and a regular graph is a graph in which
ev€ry verlex has the samedegree.It can be easily seenthat every completegraph is regular,
3
but the converseis not hrre,in generalr't
2.2 Block Designs
A block desig \s a fnnily of 6 subsetsof a set S of v elements such that, for some fixed *
and 2 , with k <v,),>0,
(i). eachsubsethas i eleme[ts,
(ii).eachpair of elementsof S occur togetherin exactly 2 subsets.
'
The elementsof S are called the tdlielier, and the subsetsof S are called the b/ocft"r.
Example1
and considerthe foliowilg sevensubsetsof S :
Takc S : {0,1,2,3,4,5,611,
{ 0 , 1 , 3 } . { 1 , 2 , 4{}2, , 3 , 5 }{,3 , 4 . 6 }{,4 , 5 , 0 1{ s, , 6 , 1 }{.6 , 0 , 2 } .
18
H e r eb = ' 7 . r r = 7 . k = 3 . 1 - l
simple geometricalrspreseltationof this design is as follows. The el€ments
l, 2, ..., 6 arerepresented
by points,andtheblocksarerepresented
by lines.
Thereis anotheruseful way of represe[tingthe design given in the example_The string of
sevenbitsof0s and I s canbe usedto rcpresent
the first set{0,1,3} as
1101000
Similarly, writing all the block correspondingto the sets,we can obtain tlle following (0,1)matrix which is the lncidence mdtrix ofthe des]sn.
110r
000
0110100
0011010
0001101
1000110
0100011
1010001
The advantageof using an incidencematrix to describea blocL design insteadof listing the
setselementby elementis that tho stucture of lhe designcould be seenmore clearly witlout
any irelevant infomation.
Theorem 1
ln a block designeachelementlies in exactly r blocks, where
r(k
l) = 7(v -l)
a1id bk = vr
Proof: - Givel in'q
When 6 = v, tlrc incidencematrix is a squal-ematrix, and snch a designis called a sE,u.rreor
8''nnetric desigl1.
19
Example 2
to be a (v, f,2) - configurationin which
A rtnite projecth)eplane of order n 1sdef:rned
integen
r >2.Thcscven pointplane
r = n ' 1 + n + 7 , k = r + 1 a n d , 1 = 1 , f o r s o m ep o s i t i v e
corespondsto n = 2. (Exarnple l)
2,3. Difference sets
There arc several tlpes of difference sets such as Plcunr, Hadanurd, Singer, Meno Hadamard, McFarland, Spence, etc. In this scction we discuss constructionsof specitic
53
families ofdifference sctsby using quadraticrcsiduesamdnumericalmultiplier (orbits)
Definition I
The developmentof a differencc set D is the itrcidencestructurc dev, whosepoints are the
elementsof G and whoseblocks are thetrdnsldtes D+ g: \tt + g:d e D\.'
devD=lD+g:geGl
Definition 2
Let , be a differenceset in G. A multiplier for G is an automorphismd of G such that
which maps ll to l.r, then I is
a is thc automorphism
D" =I)+g for some geG.If
5
called,a umerical mr:JliplreL
Delinition 3
d€Z is said tobe a quadraticresiduemodrlo n it (a, )=l
solution.Ifnot jt is saidlobe quadratic an-residue modulo n.
and x'=d(modr)
hasa
E r a m p l e3
Consuuclron
of (1q.9.4J differencesel u.ing quadraticresidues.
Considerx'=a(mod19) ----
'."(l)
Since(l) has solutionswhel1 .t=1,4,5,6.7,9,71.16,17, The set of residuesmodolo 19
is {1,4, 5, 6, 7, 9, 11,16, 17} . Further, differenceof theseelementsmodulo 19 gives each non
zero elementof Zre exactly 4 times.
.'. { 1,4,5,6,7,9,
I l,l 6,17} is a (19,9,4) differenceset.
Example4
Consh'nctionof (21,5,1)-differenceset usi[g multipliers.
2=5 1 = 4 , and 2 is a multiplicr
Here r=f
We may assumethat D consistsof orbitsunder M = 11,2.4, 8, 16, li i. Silce t = 5, the
arethc "smrill"orbits,llanrely,{0}, {7. 14J, {3, 6, 12| ard {9, 18. 15l. Thtts,
only candidates
D hasto be thc union of {7. l4l with oneoftho 3-clcmenlot-bits.
tn fact, bolh possiblechoices 1), and D, wolk.
i.e.
, , = 1 3 , 6 . 7 .1 2 .1 4 ) a n d D r = 1 7 , q .1 4 .1 5 . I S i a r e ( 2 1 . 5 . 1 ) - d r f t c l c n c c s e t s .
_20
3. RISUr_TS
I
I
The equil alenceofsquarc designs,differencesetsand graphsaregivcn by Theoretn3. L
Completesaphs ofordeL r. $here y: rr +n+1; l1>2 hale beenused,and illustratcthis
result by usrngan erampLe.Morcovcr, t]re generalisationof this reslllt is givcn by Theoren
Theorem3.1
Let G be a finitecyclicgroupof ordery. wher-e
r,:n'+n+l;
n>2 andlet , beasubset
of G with (r + 1)elemerts.
Thcnthe followingareequivalentl
(i). Thcrcis a completegraphot'order(n) +n+1).
(ii). Thercisa (/?r+r+l,r+1. 1) squaredesignsuchthat G actsregr adyasan
automorphismgtoup
(iii). D is a differencesct with parameters(r,r + /?+ l, /r + 1, l) in G.
Proof:
{i) = (ii)
Tlere is a completegraphoforder ( r: +lr + 1).
t t ,'
ttrtlrt,
-rl ''
I t i . k r o w nt h a t (
h.rs
e d g e sr./ ] - ,
2
distinct vclticcs arejoined by an edge.
I r \ e n l c e sa. n da n yp a i r o f
Let us partition rf,,, ,,,r info (rr +lt +l ) subgrcphs,ecchrs h",1og
- . 24p
.,1g...
Since the degreeof cach vefiex of K,,, ,,., i. ri(r?+l), without lcissof generalityore can
considerthat each vertex of K,,*, , is in (, + 1)siLbgaphsand cach subgrapltcontributes7?
lo the degrccoleach vertex of K,,,*,,r .
lfeach subgraphhas l,o vertices,then
nrr
n(n +1,) ,. ,,
- r l
)2
That is, eachsubgraphhas l7+ 1 \'erticcsand is regularofdegree z.
eachsubgraphis a conplete graph\\,ith 7?+ I verticesor a (z + 1) clique.
Hence(,,,,, 1 canbc decomposed
into (n' + n +1 ) suchcliques.
Claim: Each edgeis in exactly onc clique.
Supposethal thereexistsan cdgc riri \.\Jlichis common to both cliqucs Q, and Q, rvhere r,
and r, arctr.vodislincL\erlicesin both Q, and p-.
'. r =Q, andr, €?. rhich are coinpletegraph of r+i ve
ices, aDd each of Q,and
I, cont bules /7to the dcgreeof ri - But .1i.r, is an edgecommon both Cr and O, .
- degreeof r, due to qr arld Q. ts 2n
1.
.11
'. vertex ).i is in the other ,?-1 cliques,total contributionto the degreeof t, is,
n ( n - l ) + ( . 2 t -t l ) = n 1+ n - 1 ,
which is a contradiction,since lhe degreeofeach vefiex il n'+n
exactlyone chque.
Hence, each edge is in
I f we deDoteeachvcrtex of a clique by I and other verticesof the graphby 0, we can obtail
a block of 0's and 1's. By doing this for all the cliques' one can obtain all tbc blocks, which
givesa (nI + +7)x(n) -tt +1)squarematrix wilh the following propefiies:
(i)
(iD
Eachblock has (,?+ l) Pojnts
Eachpair of elementsof G occurstogetherin exactlyone block
If we take one block and shift it by one place we can get next block ofthe design Therefore
tbis is the inciclencestructureof (r?r+ +l,tr+1,1)- squaredesignsuch thatG acts regularly
as an automoryhismgroup.
(ii) = (iii)
Supposcthere is a square (v,f,/) - desigll wifh a regular automorphismgroup 6 '
One may thcn selecta "base point" Poandidentify the point set ofthe designwith the group
G as follows: Ifg is the unique elenent of G mapping 10 to P ' then wc identify p wlth g;
in particular, po is identifie<lwith 0 € G (We writc G additively for the time being, even if
G is non-abelian.)Now, choosea "basc block" ,B0,and let D be the coresponding tr-setof
Then D is a
elementsof G (so all blocks now take the form D+h for some leG)
g and 0 are
points
(v,f,/) -differenceset inG. To seethis, onejust notesthat the two distitlct
onablock D+.r if and only if one has the equalionsg=d+jt and 0=d'+r' for some
and ihe fact that
and g-d+r'
But thescequationsare equivalentto g=d-d'
d,d'€D
g and 0 are on cxnctly ,l common blocks now given us exactly 2 different reprcsentatives
g=t|-d'lor
(iii)r
g.
(,)
SupposeD is a difference sct with parameters (ar + a + 1' I + l, 1) Then
dev D = {D + g:g e G} gives the collection of differencesetsof C.Let us constructa graph
vcrtices in the following way. Label all the vcdices of the graph by
of (n'+a+l)
1,2,...,i+n+1 . Takethe differcnceset D andjoin eachpossibledistinctpaitof elements
by an eclge.This correspondsto a complete graph of /t + I vertices(or '?+ I cljque)' Now
conshuct lr + I cliqucs for eachdifferenceset l) + g; .q € G. Then' the resultinggraph 1sthe
completcgraphof zt +t?+l vertices.
Exampleto illustratethe theorem
C o n s i c l et hr c c o n p l e t cg r a p h( K i r ) o f 1 l v e i c e s 1 3 = 3 r + l + 1 ' g i v c s n = S L a b e l i n g
the graphirto
verticesliom0,i.2,- ,l2 whjch are thc clerrentsof(Zr.,+). anddecornpose
l3 cliqoes\\ith 4 vcfiices etrchand eachpair of distinct vefticesarejolned by an edge
22
.
I to eachentrycanconstructall the
O l , 3 . 9 i b ea r e r l e rs c to f . u c ha c l i q u eAtlding
rIqlcs ald cancolorwith 13differentcolors.
110
r 0000
0l
000
00
100
00
010
00
001
00
000
01000110
10
000
00100011
00010001
0t
000
10
100
tl
01
010
101
00
110
00
10
011
0l
I01000
00r
000
10100
r 1010
L 000
0000
r
r
101
000
00000100
10000010
01000001
10100000
001
Fmally, it can be seenthat eachblock correspondsto a (13,4, 1)- differenceser.
This result can be generalizedfor 2 > I . Then, insteadof looking at a complctc graph ,K,,
*-e look at a ,t(v -1) regularmulti-graph of order v, when eachdistinct pair ofvcrticcs arc
i>ined by 2 edgcs.
Theorem3.2
lrt G be a finitecyclic groupoforder r andlet1) beasubsetofG with * clqncnts.Thcn
lhe following slatementsare equivalcnt:
(i)
There is a 2(v - 1)-regulargraph of order v (,i. is the numberofedgcs joincd by any
distinctpair of vqlices).
(ii) There
group.
a (r,*,,1) -squaredesignsuch that Gacts regularlyas an autonoryhism
(iii) D is a differcnccsetwith parameters
(v,k,7) i G .
2_l
Differencesetsconstructedby using quadraticresiduesand multipliers.
vkl
DifferenceSet
Comlnel1t
2
731
! , I, 2 , 4 l
3
13 4
I
{ 0 , 1 , 3 , 9 } ;1 0 , 2 , s , 6 ] i : 1 0 , 14 0, ,1 2 }
Singer/ PIanar/
Hadamard
Singer/ Plarar
3
11 5
2
{ 1 ,3 , 4 , 5 ,9 } ; 1 2 , 6 , 7 , 8 t, o l
Hadamard
1
15
7
3
{ 0 ,1 , 2 , , 1 , 5 , 81,0 } ;{ 0 ,s , 7 , 1 0 ,1 1 ,1 3 ,l 4 }
Singer/ Hadamard
4
21
5I
{ 3 , 6 , 7 , t 2 , 1 4 } ;{ 7 , 9 ,1 4 ,1 5 ,1 8 }
Planar
5
1994
{ 1 ,4 , 5 , 6 , ' . 7 , 9l t, , 1 6 ,l 7 }
{ 2 , 3 , 8 ,1 0 ,1 2 ,1 3 ,1 4 ,1 5 ,1 8 }
Hadamard
5
31
| . t , 5 , ) 1 , 2 4 , 2 5 , 2 7{)2: , 3 ,1 0 ,1 3 ,1 5 ,1 9 }
1 89
, , 1 2 ,1 4 ,2 1, 2 9 1 :1' 4 , 6 , 7 , 2 0 , 2 6 , 3 0 |
Singcr/ Planar
6
23 tl
{ 1 , 2 , 3 , 4 , 6 , 8 , 91, 2 ,1 3 ,1 6 ,1 8 }
{ 5 , 7 ,1 0 ,I l , t 4 , t 5 , 1 ' 11, 9 , 2 0 , 2 1 , 2 2 1
Hadirnard
It, 6,'7,9, 19,38,42,49]l
Singer/ Planar
| t , 2 , 4 , 5 , 7 , 8 , 9 , 1 01, 4 ,1 6 ,1 8 ,1 9 , 2 0 , 2 5 , 2 8 1
l l, 2, 3, 4, 6, 8, 12.15,t6, l7, 23.24,2'1,29,3OJ
Singer/ Hadamard
{,3, 6, 12,19.23,24.3E,46.48|
{ 1 ,2 , 4 . 8 , t 6 , 3 2 ,3 7 , s s ,6 4 ] .
Singer/ Planar
'7
5'7 I
I
,7
8
31 15
8
7391
9
35
t7
{ 0 , t , 3 , 4 , 7 , 9 . t l , 1 2 ,1 3 ,1 4 ,1 6 ,1 7 , 2 t , 2 7 , 2 8 , Hadamard
2 9 , 3 3|
9l
l0
{0, l, 3, 9, 2',l.49.s6,61,77,8l }
Planar
,{. CONCI,USION
Theorem on equivalence of Graphs, Designs and Difference sets have been proved
consideringcomplete graphs,and illustrated by an example.Further this theorem has been
generalizcdfor a regulargraphand is given as Theorem3.2.
A Difference set of higher order has bccn constructcd lusrng Quudrutic Resitlueo^untl
MLitiplicrs. For sornevaiuesofthe parameters,this dcsigocoEespondsto a Hadamarddeslgn
arld the conespondrngHadamardmalrix can be obtained.
24
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2i