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arXiv · 1311.0464

Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4

Thomas Honold·Michael Kiermaier·Sascha Kurz
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Abstract

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.

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Thomas Honold, Michael Kiermaier, Sascha Kurz. 2014-11-26. Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4. https://doi.org/10.1090/conm%2F632%2F12627

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