Quasi-symmetric designs, codes, quadrics, and hyperplane sections
The author continues the study of designs with the symmetric difference property (SDP-designs), i.e. designs for which the symmetric difference of either three blocks (in the case of a symmetric design) or two blocks (in the case of a nonsymmetric design) is always a block or the complement of a block. Symmetric designs with this property where introduced by W. M. Kantor in his paper ``Symplectic groups, symmetric designs and line ovals'' [J. Algebra 33, 43-58 (1975; Zbl 0298.05016)] and have parameters \(\nu=2^{2m}\), \(k=2^{2m-1}-2^{m-1}\), \(\lambda=2^{2m-2}-2^{m-1}\). Non-symmetric SDP-designs have parameters which correspond to the derived or the residual design of a symmetric SDP-designs. In the paper under review, the author first proves that any non-symmetric SDP-design is actually either the derived or the residual design of a symmetric SDP-design (which is then uniquely determined by a result due to the reviewer and the author, see \textit{D. Jungnickel} and \textit{V. D. Tonchev} [Exponential numbers of quasi-symmetric SDP designs and codes meeting the Grey-Rankin bound, Des. Codes Cryptography, 1, No. 3, 247-253 (1991; Zbl 0766.05009)]. As a second major result, he shows the equivalence of non-symmetric SDP-designs to point sets of elliptic or hyperbolic type in the projective space \(\text{PG}(2m-1,2)\), i.e. point sets which have the same intersection properties as elliptic or hyperbolic quadrics (with respect to hyperplanes). As an application, he obtains the existence of exponentially many projectively non-equivalent sets of both elliptic and hyperbolic type. The case \(m=3\) is then considered in some detail. The proofs make essential use of the codes associated with the designs.
scientific article; zbMATH DE number 493321
scientific article