Exponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankin bound. (English) Zbl 0766.05009
Summary: It is shown that quasi-symmetric designs which are derived or residual designs of nonisomorphic symmetric designs with the symmetric difference property are also nonisomorphic. Combined with a result by W. M. Kantor, this implies that the number of nonisomorphic quasi-symmetric designs with the symmetric difference property grows exponentially. The column spaces of the incidence matrices of these designs provide an exponential number of inequivalent codes meeting the Grey-Rankin bound. A transformation of quasi-symmetric designs by means of maximal arcs is described. In particular, a residual quasi-symmetric design with the symmetric difference property is transformed into a quasi-symmetric design with the same block graph but higher rank over GF(2).
MSC:
| 05B05 | Combinatorial aspects of block designs |
| 94B05 | Linear codes (general theory) |
| 05B30 | Other designs, configurations |
Keywords:
quasi-symmetric designs; residual designs; symmetric designs; symmetric difference property; codes; Grey-Rankin boundReferences:
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