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

We determine the maximum size $$A_2(8,6;4)$$A2(8,6;4) of a binary subspace code of packet length $$v=8$$v=8, minimum subspace distance $$d=6$$d=6, and constant dimension $$k=4$$k=4 to be 257. There are two isomorphism types of optimal codes. Both of them are extended LMRD codes. In finite geometry terms, the maximum number of solids in $${\text {PG}}(7,2)$$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 result implies that the maximum size $$A_2(8,6)$$A2(8,6) of a binary mixed-dimension subspace code of packet length 8 and minimum subspace distance 6 is 257 as well.

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