On existence of dual property in symmetric group divisible designs

OF DUAL P

1 1979:teviwl manuscriptreceived1

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ande and J. Stivastava

ition; Semi-regularand RegularGroup Divissb

1. Introduction A binary design with v = Megtreatments arranged in b = ltln blocks eat> of size k is called a symmetrical group divisible design (SGDD) if the in treatments can be arranged into m sets 7’,,Tz, .- ., Tm each having n r2 treatments such that any two treatments belonging to the same set occur together in II blocks and any two treatments belonging to two different sets occur together in A2 blocks. We shail denote such a design by SGDD (v, k,il&,m, n). An SGDD is said 10 possess the dual property if the mn blocks can be arranged into m sets Si, Sz, . . . , Snreach having n 22 blocks such that any twc! blocks belonging to the same set have ,I1treatments ir, common and any two blocks belonging to two different sets have 2 treatments in common. If we denote by A’ the incidence matrix of SGDD possessing the diliral has shown that if a SGDD property then NIV’= N’N. In #;zrecent paper, Bos @J77) x R,, M= (@J) such that every possesses the dual property then there exists a n block of set S’ has exactly rq treatments from the s Tdand every treatment of the set rs exactly rij times in !:he blocks of set Sj. Onthis note we prove that if ihe n be divided into 01 sets of n r 2 blocks each such that the at>we matrix R =((Q)) exists, then the SGDD possesses the dual property. Example of SGDD possessing the above condition bu not satisfying Hoffman’s condition state some known definitions and gAir A.?,m, n) let x=&--A1 +n(A, -L2) = k2 - vA2 and

SGDD is said fo be semi-regl;,lar if k - At >

0

y= n&.

and x = 0 and regular if k - AI > 0 and nto m ark”divi sofI; enote by A$ the n x FJmat;*ix as the

4’fO~!.75@ 1981 Vorth-Holland

incidence matrix of the treatments irr Ts with the blocks in 5”. Treatment *SM7” contains treatmerjts numbered (4:- l)n + 1 to ia and the block set SJ contains blocks numbered u - 1)rr+ I to jn. Then N==(~(Nii))~ i,j = t,2, . .. , sptand NN’ =z(k - A,)&,,+ AfJ,,,n+ [A,- Az)(&,@.J,J where J denotes the square matrix having al): elements unity and Krcmcker product of the matrices,

(1.2) henotes the

2. The results We shall first state i3ose’s result. Theorem 1. i!‘fan SGDD (vBk, A,, 12,m, a) ptxwsses the dual property cored the blsc!i~ sets are denoted l,y &, &, . . . , S,,,, then there exists 4 matrix R = ((Q)) such that every block of Sj has aa-act/’ qj trecltments@om set q and evwy ttwt&nt ,~RXE.x+t77 eccur~ in exact& ro bc’ocksof:wrr S). Theorem 2, Supptwe its a reguk.wSGDD (v, k, A*,Aa,m,n) the mro bloc&s am be divided into m se& SI, $ , a. . , Snr of n r: 2 blocks cock such thot every block sfw 5’ has exactly ru trmtments from the set *Ii, then the negulw SGDQ pas.w~~ the dual property and hence every treatment from set c ocvxrs in aact& tu blucks of S). Proof. Let R,,,x,,,= ((q)), then

.a9(ll;,OJ,)NN’(I,,,QJ,) = n(RR’@J,).

(2.1)

Substituting in (2&1) for NN’ from (1.2’1and simplifying, WC RR’==xl,, +y&. Since -00, we get (RR3”’ and hence

Moreover, from

I(1.1)

and (1.2) we gel

(k-A,)(NN’)-‘==I

?llN -

tk _-‘,)&

-----y--==-

xk

J #lrj

Premulniplying and postmulti we get &/‘N :.z NN’,

f Set

cs in

379

ven in Bose, Shrikhande and lJ,I$ II, l&2422,23) and the 25, we observe that the block fy the condition of theorem 2. Here Hoffman’s . I

i

and j, Box and Co

m,n) the mn Mockscan be & of n;hr2 Mocks ea such that every treatment Sj&then ?he semi+gutar SGDD possesses the dual

t W=N’N-NN’andB=I,@J,.

Then

Now, Tr W=O,

.g. TrW2= - 2(AI - &)Tr (NN’B- NW%). We get

Tr(NN’B)= mn[k + A&t - I)] = nk2 for semi-regular SGDD. if the semi-regular SGDD satisfies the condition stated in the theorem then NB = (kJm)J,,.

A TrIVVVB=nk*. Hence Tr I@= Tr W’W==O. Therefore, W is a null marix s\nd NW= NIV’. Example 2. Consider SR7 iven in Bose, Clatworthy and Shrikhande (1954), It is a lar SODIb (8,4,0,2,4,2) who blocks are I$: (1,2,3,4), B2: (5,&T, B), 881)a &: (6,3,4, S), Bs: (3, 1,6), Be: (X4,5,2), BY: (4,1,6,7& and 44), (Bs,Bd and ( 7&) possess the prwerty tIoffman’s condition is not applicable.

ns with the dual propert:!, J. Statist. ~nat~ri~l properties of group cliivisible incomplete block

[3] Bose, RC, S.S. Shtikhundeand K,N. Bhattru~arya(1953). On &e contluctioj~ of group _ incompleteblock d&u. Ann. [email protected]&Ha 24, 163-195. [4] Bose, RX., WM. Ciatworthyand S.S. Shrikhandt (1954). Tables of parti@ bJanced incomp& block designswith two associateclasses.hkwth Card&r Agticubu~ EkpeiimetW lhd&tih, No. 187. [S] Hoffman, A.9. (i96 I). On the duals of symmetricpartiallybaiancedimcompleteMock &&ns. Ansr. Math. Statist. 34, 528-531.