Classification of quasi-symmetric 2-(64,24,46) designs of Blokhuis-Haemers type

TL;DR

This paper completes the classification of certain quasi-symmetric 2-(64,24,46) designs of Blokhuis-Haemers type, identifying all 30,264 nonisomorphic designs derived from maximal arcs in affine geometries.

Contribution

It finalizes the classification of these designs supported by the dual code of a specific binary linear code, expanding understanding of their structure and related graphs.

Findings

Exactly 30,264 nonisomorphic designs identified

All designs derived from maximal arcs in affine geometries

Discussion of related strongly regular graphs

Abstract

This paper completes the classification of quasi-symmetric 2-$(64,24,46)$ designs of Blokhuis-Haemers type supported by the dual code $C^{\perp}$ of the binary linear code $C$ spanned by the lines of $AG(3,2^2)$ initiated in \cite{bgr-vdt}. It is shown that $C^{\perp}$ contains exactly 30,264 nonisomorphic quasi-symmetric 2-$(64,24,46)$ designs obtainable from maximal arcs in $AG(2,2^2)$ via the Blokhuis-Haemers construction. The related strongly regular graphs are also discussed.