The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: λ(v − 1) ≡ 0 mod (k − 1) λv(v − 1) ≡ 0 mod k(k − 1) Earlier work has studied k = 9 with λ ∈ {1, 2, 3, 4, 6, 8, 9, 12}. In this article we show that the necessary conditions are sufficient for λ = 9 and every other λ not previously studied.

Some construction methods of the variance balanced block designs with repeated blocks are given. They are based on the incidence matrices of the balanced incomplete block designs with repeated blocks.

A (v, m, m − 1)-BIBD D is said to be near resolvable (NRBIBD) if the blocks of D can be partitioned into classes R1, R2, . . . , Rv such that for each point x of D, there is precisely one class having no block containing x and each class contains precisely v − 1 points of the design. If an (v, m, λ)-NRBIBD has a pair of orthogonal resolutions, it is said to be doubly resolvable and is denoted D...

Stufken (1993) first introduced cyclic polygonal designs (CPDs) (in terms of cyclic balanced sampling plans excluding adjacent units (CBSAs)) for joint distance α ≥ 2. Wei (2002) first suggested the use of Langford sequence for the existence and construction of CPDs with block size k = 3 and λ = 1 for arbitrary α . Zhang and Chang (2005b) used Langford and extended Langford sequence, and constr...

A colouring of a strongly regular graph is an allocation of colours (or treatments) to the vertices of the graph. Such a colouring is balanced if every pair of distinct colours occurs equally often on the ends of an edge. When the graph is the complete regular multipartite graph a balanced colouring is just a balanced incomplete-block design, or 2-design. Some constructions are given. For examp...