Quasi -symmetric designs with good blocks and intersection number one

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We show that a quasi-symmetric design with intersection numbers 1 and y > 1 and a good block belongs to one of three types: (a) it has the same parameters as PG 2(4, q), the design of points and planes in projective 4-space; (b) it is the 2-(23, 7, 21) Witt design; (c) its parameters may be written v = 1 + (( – 1) + 1)(y – 1) and k = 1 + (y – 1), where is an integer and > y 5, and the design induced on a good block is a 2-(k, y, 1) design. No design of type (c) is known; moreover, for large ranges of the parameters, it cannot exist for arithmetic reasons concerning the parameters. We show also that PG 2(4, q) is the only design of type (a) in which all blocks are good.
Original languageEnglish
Pages (from-to)147-162
Number of pages16
JournalDesigns, Codes and Cryptography
Issue number2
Publication statusPublished - Mar 2003


  • quasi-symmetric designs
  • good blocks
  • intersection numbers


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