On Quasi-symmetric 2-(64, 24, 46) Designs Derived from Codes

Abstract

The paper studies quasi-symmetric 2-(64, 24, 46) designs supported by minimum weight codewords in the dual code of the binary code spanned by the lines of AG(3, 2²). We classify up to isomorphism all designs invariant under automorphisms of odd prime order in the full automorphism group G of the code, being of order $\vert G\vert = 2^{13} \cdot 3^{4} \cdot 5 \cdot 7$. We show that there is exactly one isomorphism class of designs invariant under an automorphisms of order 7, 15 isomorphism classes of designs with an automorphism of order 5, and no designs with an automorphism of order 3. Any design in the code that does not admit an automorphism of odd prime order has full group of order 2^m for some m ≤ 13, and there is exactly one isomorphism class of designs with full automorphism group of order 2¹³.