%0 Journal Article
%T Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4
%A Thomas Honold
%A Michael Kiermaier
%A Sascha Kurz
%J Mathematics
%D 2013
%I arXiv
%R 10.1090/conm/632/12627
%X It is shown that the maximum size of a binary subspace code of packet length $v=6$, minimum subspace distance $d=4$, and constant dimension $k=3$ is $M=77$; in Finite Geometry terms, the maximum number of planes in $\operatorname{PG}(5,2)$ mutually intersecting in at most a point is $77$. Optimal binary $(v,M,d;k)=(6,77,4;3)$ subspace codes are classified into $5$ isomorphism types, and a computer-free construction of one isomorphism type is provided. The construction uses both geometry and finite fields theory and generalizes to any $q$, yielding a new family of $q$-ary $(6,q^6+2q^2+2q+1,4;3)$ subspace codes.
%U http://arxiv.org/abs/1311.0464v2