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.

Highlights

  • Let Fq be the finite field with q elements, where q > 1 is a prime power

  • Before we go into the details we present another construction – the so-called Echelon-Ferrers construction for general subspace codes, see [18]

  • For parameters excluded by Lemma 15 this small linear program is infeasible and the infeasibility can be seen at a certain basis solution, i.e., a choice of linear inequalities that are satisfied with equality

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Summary

Introduction

By Fvq we denote the standard vector space of dimension v ≥ 1 over Fq. The set of all subspaces of Fnq , ordered by the incidence relation ⊆, is called (v − 1)-dimensional projective geometry over Fq and denoted by PG(v −1, Fq). An important problem is the determination of the maximum possible cardinality ASq (v, d; k) of a constant dimension code with minimum subspace distance d in Fvq , where all codewords have dimension k. The maximum possible minimum distance of a constant dimension code with codewords of dimension k is 2k Partial k-spreads are just a special case of vector space partitions, where the elements all have a dimension of either k or 1.

Bounds and constructions for partial spreads
One way2 to write down the underlying relation are the so-called Mac Williams identities : n−ν n − j ν n−j
Constructions for qr-divisible sets
More non-existence results for qr-divisible sets
Open research problems
Better constructions for partial spreads
Existence and classification results for qr-divisible sets
Vector space partitions

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