A subspace code of size <inline-formula><tex-math id="M1">\begin{document}$ \bf{333} $\end{document}</tex-math></inline-formula> in the setting of a binary <inline-formula><tex-math id="M2">\begin{document}$ \bf{q} $\end{document}</tex-math></inline-formula>-analog of the Fano plane

Abstract

We show that there is a binary subspace code of constant dimension 3 in ambient dimension 7, having minimum distance 4 and cardinality 333, i.e., $333 \le A_2(7,4;3)$, which improves the previous best known lower bound of 329. Moreover, if a code with these parameters has at least 333 elements, its automorphism group is in one of $31$ conjugacy classes. This is achieved by a more general technique for an exhaustive search in a finite group that does not depend on the enumeration of all subgroups.