A note on λ ‐designs

Let r and r′ , ( r > r′ ) be the two replication numbers of a λ ‐design D . We show that if r − r ′ is a prime power or 33, then D is a design of type‐1. We derive two inequalities, where equality holds if and only if D is a design of type‐1. Let D be a λ ‐design with two block sizes with v = n p + 1 points, where p is prime and 2 ⩽ n ⩽ 22 . We develop a procedure to obtain exceptions for such type‐1 λ ‐designs, refining earlier results.