Calderbank, AR, *The application of invariant theory to the existence of quasi-symmetric designs*,
Journal of Combinatorial Theory, Series A, vol. 44 no. 1
(1987),
pp. 94-109 .
**Abstract:**

*Gleason and Mallows and Sloane characterized the weight enumerators of maximal self-orthogonal codes with all weights divisible by 4. We apply these results to obtain a new necessary condition for the existence of 2 - (v, k, λ) designs where the intersection numbers s1...,sn satisfy s1 ≡ s2 ≡ ... ≡ sn (mod 2). Non-existence of quasi-symmetric 2-(21, 18, 14), 2-(21, 9, 12), and 2-(35, 7, 3) designs follows directly from the theorem. We also eliminate quasi-symmetric 2-(33, 9, 6) designs. We prove that the blocks of quasi-symmetric 2-(19, 9, 16), 2-(20, 10, 18), 2-(20,8, 14), and 2-(22, 8, 12) designs are obtained from octads and dodecads in the [24, 12] Golay code. Finally we eliminate quasi-symmetric 2-(19,9, 16) and 2-(22, 8, 12) designs. © 1987.*