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doi:10.1016/0097-3165(87)90062-8 
Copyright © 1987 Published by Elsevier Inc.
The application of invariant theory to the existence of quasi-symmetric designs
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A. R. Calderbank
Received 15 February 1986.
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.
