Partial spreads and vector space partitions

Abstract

Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal code size are known. The seminal works of Beutelspacher and Drake \& Freeman on partial spreads date back to 1975, and 1979, respectively. From then until recently, there was almost no progress besides some computer-based constructions and classifications. It turns out that vector space partitions provide the appropriate theoretical framework and can be used to improve the long-standing bounds in quite a few cases. Here, we provide a historic account on partial spreads and an interpretation of the classical results from a modern perspective. To this end, we introduce all required methods from the theory of vector space partitions and Finite Geometry in a tutorial style. We guide the reader to the current frontiers of research in that field, including a detailed description of the recent improvements.

30 pages, 1 table

References in corpus (5)

• Construction of Large Constant Dimension Codes With a Prescribed Minimum Distance
• Tables of subspace codes
• Improved upper bounds for partial spreads
• Upper bounds for partial spreads
• On projective $q^r$-divisible codes

Cited by in corpus (19)

• Tables of subspace codes
• Combining subspace codes
• Projective divisible binary codes
• Classification of large partial plane spreads in $PG(6,2)$ and related combinatorial objects
• Affine vector space partitions
• The lengths of projective triply-even binary codes
• No projective 16-divisible binary linear code of length 131 exists
• A generalization of the cylinder conjecture for divisible codes
• Subspaces intersecting in at most a point
• Heden's bound on the tail of a vector space partition
• New LMRD bounds for constant dimension codes and improved constructions
• On projective $q^r$-divisible codes
• On $\alpha$-points of $q$-analogs of the Fano plane
• Johnson type bounds for mixed dimension subspace codes
• Generalized vector space partitions
• Computer classification of linear codes based on lattice point enumeration and integer linear programming
• Classification of $\Delta$-divisible linear codes spanned by codewords of weight $\Delta$
• Vanishing Flats: A Combinatorial Viewpoint on the Planarity of Functions and Their Application
• Lengths of divisible codes -- the missing cases