I have been asked to prove that there exists a (40,13,4)BIBD. I admittedly have no idea where to start with this. Checking some of the necessary conditions for BIBDs shows me that if such a BIBD existed, it would be symmetric (i.e. a (40,40,13,13,4)BIBD). I am absolutely lost on how to move forward. I've been searching for proofs and theorems but have come up empty thus far. Are there any theorems or lemmas that I may have overlooked that could help me here?
2025-01-14 11:29:32.1736854172
Existence of a (40,13,4)BIBD (Balanced Incomplete Block Design)
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The existence of a projective space $\rm PG(3,3)$ implies the existence of this BIBD, as $$(3^3+3^2+3+1,\ 3^2+3+1,\ 3+1)=(40,13,4)$$and in fact it can be constructed using the difference set $$\{1, 2, 3, 5, 6, 9, 14, 15, 18, 20, 25, 27, 35\}$$ by steadily incrementing every element modulo $40$.