Block designs with parameters (pⁿ, Δ pⁿ, Δ((pⁿ - 1)/2), (pⁿ - 1)/2, Δ((pⁿ - 3)/4)) are shown to exist whenever pⁿ, a prime power, can be expressed as 2ᵐt + 1 with m a positive integer and t an odd integer $>1$, that is, whenever pⁿ is not one greater than a power of 2; Δ is equal to 2ᵐ⁻¹.
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E. Nemeth (1971) studied this question.
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