Binary codes and quasi-symmetric designs

obtain a new for the of a-(u, A) design the block intersection designs are eliminated by an ad hoc coding theoretic argument. A 2-(v, k, A) design 93 is said to be quasi-symmetric if there are two block intersection sizes s1 and s2. The parameters of the complementary design !3* are related to the parameters of 93 as follows: Here Ai denotes the number of blocks through a given i points (and A = A,). Calderbank [l] used Gleason's theorem on self-dual codes to obtain new necessary conditions for the existence of 2-(v, k, A) designs where the block intersection sizes sl, s2,. .. , s, satisfy s1 = s2 = * *-= s,(mod 2). We use (1) to restate these conditions as follows: Theorem 1. Let !I3 be a 2-(v, k, A) design where the block intersection sizes possibly taking complements either (i) k = O(mod 4), v = l(mod S), A, = O(mod S), or (ii) k = O(mod 4), v=-1 (mods), 2&+A,=O(modS). Note that Calderbank [l, Lemma l] proved that k = s(mod 2) and AI = &(mod 2) for designs 93 satisfying the hypotheses of Theorem 1. The next lemma is just proved by simple counting but nevertheless it is very useful.