Optimal binary subspace codes of length 6, constant dimension 3 and minimum subspace distance 4
Thomas Honold, Michael Kiermaier, Sascha Kurz · Contemporary mathematics - American Mathematical Society · 2015
It is shown that the maximum size of a binary subspace code of packet length v = 6 v=6 , minimum subspace distance d = 4 d=4 , and constant dimension k = 3 k=3 is M = 77 M=77 ; in Finite Geometry terms, the maximum number of planes in P G ( 5 , 2 ) \mathrm {PG}(5,2) mutually intersecting in at most a point is 77 77 . Optimal binary ( v , M , d ; k ) = ( 6 , 77 , 4 ; 3 ) (v,M,d;k)=(6,77,4;3) subspace codes are classified into 5 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 q , yielding a new family of q q -ary ( 6 , q 6 + 2 q 2 + 2 q + 1 , 4 ; 3 ) (6,q^6+2q^2+2q+1,4;3) subspace codes.