Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6

TL;DR

This paper determines the maximum size of a specific binary subspace code with length 8, constant dimension 4, and minimum distance 6, establishing it as 257 through classification and integer linear programming.

Contribution

It provides the first complete classification of optimal binary subspace codes of this type and size, combining geometric and computational methods.

Findings

Maximum size of the code is 257.

Extended lifted maximum rank distance codes are optimal.

The result applies to both constant and mixed-dimension codes.

Abstract

The maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=6$, and constant dimension $k=4$ is $257$, where the $2$ isomorphism types are extended lifted maximum rank distance codes. In finite geometry terms the maximum number of solids in $\operatorname{PG}(7,2)$, mutually intersecting in at most a point, is $257$. The result was obtained by combining the classification of substructures with integer linear programming techniques. This implies that the maximum size $A_2(8,6)$ of a binary mixed-dimension code of packet length $8$ and minimum subspace distance $6$ is $257$ as well.