ON THE BLOCK STRUCTURE OF PBIB DESIGNS

Let {θ,θ1,…,θn1} denote the set having any treatment θ and its n1 first associates in a PBIB design with m(≥2) associate classes. A theorem on distribution of the elements of this set among the blocks of the block design is proved and results related to certain well known association schemes of three associate classes are obtained.

of this note is to generalize his theorem for PBIB designs with m(>_2) associate classes.Some connected results for three associate PBIB designs are then obtained in section 5.

ASSOCIATION SCHEMES AND PBIB DESIGNS.
Given v treatments, 1,2,..., v a relation satisfying the following conditions is said to be an association scheme with m classes: (a) Any two treatments are either ist, 2md,..., or m th associates, the relation of association being symmetric, i.e. if treatment is the i th associate of treatment 8, then 8 is the i th associate of treatment () Each treatment has nl, i th associates, the number n i being independent of the treatment taken.
(c) If any two treatments and are i th associates then the number of treatments which are j th associates of and k th associates of 8 is i PJk and is independent of the pair of i th associates and 8.
i The numbers v, n i, Pjk' I <_ i, j, k <_m, are the parameters of the association scheme.If we have an association scheme with m-classes, then we get a PBIB design with r replications, b blocks, and block size k based on the association scheme provided we can arrange the v treatments into b blo@ks such that (i) each block contains k distinct treatments each treatment is contained in r blocks (iii) if two treatments and are i th.associates, then they occur together in %'i blocks (not all %i's equal), the number %i being independent of the particular pair of i th associates and 8 (I < i < m).
If the treatments v v I. v 2 can be arranged in the form of a rectangle of v I rows and v 2 columns, then the first associates of any treat- ment are the other (v2-1) treatments of the same row, the second associates are the other (Vl-i) treatments of the same column and the remaining (vl-i) (v2-1) treatments are the third associates.
The reader may consult the references for examples of schemes and their parameters. 4. DISTRIBUTION OF THE (nl/l) TREATMENTS (VlZ.ANY TREATMENT 8 AND ITS n 1 FIRST ASSOCIATES) AMONG THE b BLOCKS OF THE DESIGN Let 8 be any of the v treatments and let 8 I, 8 2 8nl be the n I first associates of 8. From among these (n I + i) treatments let e. treatments occur in the j th blohk (j 1 2,..., b).We now prove the following: THEOREM.This completes the proof of the theorem.

APPLICATIONS IN PBIB DESIGNS WITH THREE ASSOCIATE CLASSES.
Let N (nij) be the v x b incidence matrix of the design, where nij i or 0 according as the i th treatment occurs in the j th block or not.
(a) Designs based on cubic association scheme.
So, we have Substituting the values of AI and A2 from (5.1) and thereafter some simpli- fications lead to (5.2).An appeal to the Theorem of section 4 completes the proof.

%3
satisfies the requ/rements of the theorem and therefore allows even dletriution of a given treatment and its first associates among the blocks, as can be (b) Designs based on rectangular association scheme.Vartak (1959) as pointed out in Saha (1966) proved that the characteristic roots of NN' of this design are P0 rk, PI r %1 + (Vl -i).

THEOREM 5 . 1 .PROOF
If in any PBIB design based on cubic association scheme _v_ i= 3s(s-l)_ (5s_4)Pl+8(s_l)2P2+(s_l)2(3s_4)-2)k is divisible by s Further every block of the design contains (3s-2)k/s 3 treatments from among the (nl+l) treatments e, el, e2,...,e n 1 then k is divisible by v I. Further every block of the design contains k/v I treatments among the (n I + i) treatments e e 1 ,.., enl I I i PROOF.Here we have n I v 2i, Pll! v 2 2, PI2 P13 0. The Theorem can be proved similar to previous theorem.
parameters.Any treatment with its first associates are evenly distributed among the blocks of the design with parameters b