NEW A-D- AND E-OPTIMAL n-ARY BLOCK DESIGNS WITH UNEQUAL REPLICATIONS AND EQUAL BLOCK SIZES

Abstract

We shall restrict our attention to connected n-ary block designs i.e. designs whose C-matrices have 
rank (V-1), and use D (V, B, K) to denote the class of all connected block designs having V treatments arranged 
in B blocks of size K.  For d∈ D (V, B, K) let O = zd0 < zd1 ≤ zd2 ≤ … ≤ zd,V-1 denote the non-zero eigen values of 
Cd. 
A n-ary design d* is said to be φf - optimal in D (V, B, K) if for any other d∈ D (V, B, K) 
( ) ( ) ( ) ( )d f
1V
1 r
di
1V
1 r
di *di f
C Z f Zf C Φ= ≤ = φ ∑ ∑ −
=
−
=
 
where f satisfies the following conditions: 
I. f is continuously differentiable on (O,A) where A = max d∈ D (V,B,K) tr Cd & f’ <0, f’’ > 0 and f’’’ <0 on  
(O,A) 
II. f is continuous at 0 or     ∞=
→
)x(f lim
0 x
    
It is noted that by considering f(x) = 1/x and f(x) = -log(x)