Partial Spreads and Vector Space Partitions

Abstract

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.

The Lengths of Projective Triply-Even Binary Codes

“…Calling every non-covered point a hole, we can state that the set of holes of a partial k-spread is q k−1 -divisible, see e.g. [7,Theorem 8] containing also a generalization to so-called vector space partitions. 1 So, from the non-existence of q k−1 -divisible sets (or projective q k−1 divisible linear codes, since we have a set of holes in this application) of a suitable effective length n one can conclude the non-existence of partial k-spreads in F v q of a certain cardinality.…”

“…Indeed, all currently known upper bounds for partial k-spreads can be obtained from such non-existence results for divisible codes, see e.g. [7,9].…”

“…For sets of points or projective q r -divisible linear codes the question is more complicated. A partial answer has been given in [7,Theorem 13]:…”

“…upper bounds on the maximum possible cardinality of partial k-spreads. As an example, in [7,Theorem 13], see also [9], it is shown that no projective 2 3 -divisible linear code of length 52 exists, while there are non-projective examples with these parameters. From this non-existence result for projective q r -divisible codes we can conclude that there can be at most 132 solids in F 11 2 with pairwise trivial intersection, which is the tightest currently known upper bound.…”

Subspaces intersecting in at most a point

“…Two k-spaces U , W are said to trivially intersect or to be disjoint if dim(U ∩ W ) = 0, i.e., U and W do not share a common point. Sets of k-spaces that are pairwise disjoint are called partial k-spreads, see [10] for a recent survey on bounds for their maximum possible sizes. In finite projective geometry they are a classical topic.…”