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Classification of large partial plane spreads in \(\mathrm{PG}(6,2)\) and related combinatorial objects. (English) Zbl 1407.05042

Summary: The partial plane spreads in \(\mathrm{PG}(6,2)\) of maximum possible size 17 and of size 16 are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: vector space partitions of \(\mathrm{PG}(6,2)\) of type \((3^{16} 4^1)\), binary \(3\times 4\) MRD codes of minimum rank distance 3, and subspace codes with the optimal parameters \((7,17,6)_2\) and \((7,34,5)_2\).

MSC:

05B25 Combinatorial aspects of finite geometries
15A21 Canonical forms, reductions, classification
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
51E14 Finite partial geometries (general), nets, partial spreads
51E20 Combinatorial structures in finite projective spaces
94B60 Other types of codes

Software:

Cliquer; Magma; libexact
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References:

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