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Binary codes and quasi-symmetric designs

✍ Scribed by A.R. Calderbank; P. Frankl

Publisher
Elsevier Science
Year
1990
Tongue
English
Weight
263 KB
Volume
83
Category
Article
ISSN
0012-365X

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✦ Synopsis

obtain a new for the of a -(u, A) design the block intersection st, sZ, . . , s, satisfy sr -sZ-. . . = s, = s(mod 2). This condition eliminates quasi-symmetric 2 -(20,10,18) and 2 -(60,30,58) designs. Quasi-symmetric 2 -(20,8,14) 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:

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The article is concerned with a characterization of quasi-symmetric (QS) designs with intersection numbers 0 and y. It uses the idea of a good block. Such a block G has the property that for any block B with IG n B J = y, every point is on a block containing G n B. It is proved that if a QS design I