Abstract
Publisher Summary The chapter describes necessary and sufficient conditions for the existence of BIBDs with k = 3, 4, and 5. With a single exception, these designs exist whenever obvious arithmetic conditions are satisfied. The single exception occurs when u = 15, k = 5, λ = 2: the (15, 5, 2)-design does not exist as it would be a residual of a nonexistent symmetric (22, 7, 2)-design. Another highly symmetric solution is given in and as shows in the chapter, this solution, Hall's solution and the second of Hanani's solutions are mutually nonisomorphic. Another 1-rotational design is obtained from the twofold pentagon system of order 15.
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