The Impossibility of Certain Symmetrical Balanced Incomplete Block Designs

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The Impossibility of Certain Symmetrical Balanced Incomplete Block Designs^∗

Remarkably, the classical paper reproduced here was a part of Shrikhande’s PhD thesis written under the supervision of Raj Chandra Bose. It addresses the existence of a square 2−(v,k, λ) design. Ifvis even, then the necessary condition thatk−λmust be a perfect square is simpler to prove, and was shown already in 1949 by M P Schutzenberger. Ifv is odd, Shrikhande obtained the necessary condition that the equation

(k−λ)x²+(−1)^v−1)² λy²=z²,

must have a solution in three integers x,y,z—not all equal to zero. The proof of this used unexpectedly advanced number-theoretic tools such as the Hilbert symbol from p- adic analysis, and the local-global Hasse–Minkowski principle. For the first time, such sophisticated number theoretic techniques were used in design theory. According to experts, the full strength of this approach is yet to be exploited.

B Sury Email:surybang@gmail.com

Reproduced from Project Euclid, S S Shrikhande, The impossibility of certain symmetrical balanced incomplete block designs,Ann. Math. Statist., 21, No. 1, pp.106–

111, 1950, doi:10.1214/aoms/1177729889.

Euclid URL: https://projecteuclid.org/euclid.aoms/1177729889

∗Vol.26, No.2, DOI: https://doi.org/10.1007/s12045-021-1127-y

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284 RESONANCE | February 2021

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