Subspaces intersecting in at most a point

Abstract

We improve on the lower bound of the maximum number of planes in PG(8, q) ∼ = F 9 q pairwise intersecting in at most a point. In terms of constant dimension codes this leads to Aq(9, 4; 3) ≥ q 12 + 2q 8 + 2q 7 + q 6 + 2q 5 + 2q 4 − 2q 2 − 2q + 1. This result is obtained via a more general construction strategy, which also yields other improvements.

Combining subspace codes

“…In [5] the authors combined several (6, , 4; 3) q -CDCs to show A q (9, 4; 3) ≥ q 12 + 2q 8 + 2q 7 + q 6 + q 5 + q 4 + 1, which improves the previously best known lower bound A q (9, 4; 3) ≥ q 12 + 2q 8 + 2q 7 + q 6 + 1 obtained from the improved linkage construction, see [23]. In [34] the bound was further improved to A q (9, 4; 3) ≥…”

“…Here we want to generalize the approach of [5,34] and apply it to a much wider range of parameters. Definition 5.1.…”

“…we assume i < i, so that dim(U ∩ U ) ≤ i. 3 can be used in order to obtain the best known lower bound for A q (9, 4; 3) [5,34] and A q (10, 4; 3), [34]. First let (D 0 , D 1 ) be a (6, 4, 3)-sequence.…”

“…In order to determine a distance-partition (C 0 , C 1 ) of C, we need to find a large partial-spread subcode of C. In [5,Theorem 3.12], it is shown that we can choose C 0 of cardinality q 3 − 1 if we choose C from [6]. However, as shown in [34], the same can be done if we choose C from [26]. This can be made more precise in the language of linearized polynomials.…”

“…In Section 5 we investigate a construction method introduced for specific parameters by the first, third and fourth author in [5] and further developed by the second author in [34]. Here this approach is generalized and applied to a wide range of parameters.…”