A Necessary Condition for Existence of Regular and Symmetrical Experimental Designs of Triangular Type, with Partially Balanced Incomplete Blocks

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“…These results are obtained by using the Grothendieck group, a technique similar to Hasse-Minkowski theory. The results are in a sense generalizations of [63] and [57], respectively, which are only applicable to designs. A general reference for applications of Hasse-Minkowski theory to designs is [60].…”

On the Block Structure of Quartic Designs

“…Further necessary conditions for the existence of quartic designs can be obtained with the help of the Hasse-Minkowski/7-invariant. For a discussion of the properties of the Legendre symbol, the Hilbert norm residue symbol and the Hasse-Minkowski ^-invariant, see Shrikhande and Jain [8] or Ogawa [2].…”

“…For example, Theorem 6.9 may be simplified as follows when 0 2 = O. COROLLARY 6.10. Necessary conditions for the existence of a quartic design with only 6 2 =0, with a=6(s-l) 2 , and with b=s é -6(s-l) 2 , are that (a) if s is odd, that 6 0 be a perfect square, or (b) if s is even, that 6 Q 6± be a perfect square and, if so, that (-1, 0^) p = +1 for all primes p.…”

“…For the quartic association scheme, these parameters have the following values: (2.2) /i! = 4(j-l), n 2 = 6(s-l) 2 3. Characterization of quartic designs.…”

Some results on tactical configurations and non-existence of difference set solutions for certain symmetrical PBIB designs

“…For properties of the Hasse-Minkowski invariant, we refer Ogawa [7] and Shrikhande, Raghavarao and Tharthare [9].…”