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.
Highlights
Let q be a prime power, Fq be the field with q elements, and V ∼= Fvq a v-dimensional vector space over Fq
By L(V ) we denote the set of all subspaces of V, or flats of the projective geometry PG(V ) ∼= PG(Fvq ) =: PG(v − 1, q)
It forms a metric space with respect to the subspace distance ds(U, W ) := dim(U + W ) − dim(U ∩ W ) = dim(U ) + dim(W ) − 2 dim(U ∩ W ) and may be viewed as a q-analogue of the Hamming space (Fv2, dHam)
Summary
Let q be a prime power, Fq be the field with q elements, and V ∼= Fvq a v-dimensional vector space over Fq. As in [14], we want to apply integer linear programming methods in order to determine the exact maximum size of CDCs with the specified parameters. Since this algorithmic approach suffers from the presence of a large symmetry group, we use the inherent symmetry to prescribe some carefully chosen substructures up to isomorphism.
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