Upper bounds for partial spreads from divisible codes

Titelangaben

Heinlein, Daniel ; Honold, Thomas ; Kiermaier, Michael ; Kurz, Sascha ; Wassermann, Alfred:
Upper bounds for partial spreads from divisible codes.
2017
Veranstaltung: The 13th International Conference on Finite Fields and their Applications , 04.-10.06.2017 , Gaeta, Italy.
(Veranstaltungsbeitrag: Kongress/Konferenz/Symposium/Tagung , Vortrag )

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Abstract

A partial t-spread in GF(q)^n is a collection of t-dimensional subspaces with trivial intersection such that
each non-zero vector is covered at most once. How many t-dimensional subspaces can be packed into GF(q)^n , i.e., what is the maximum cardinality of a partial t-spread? An upper bound, given by Drake and Freeman, survived almost forty years without any improvement. At the end of 2015, the upper bounds started to crumble. Here, the theoretical foundation is provided by the fact that the uncovered points, called holes in this context, form a projective q^{t-1}-divisible linear block code. This allows to apply
the linear programming method, i.e., to utilize the so-called MacWilliams identities and the positivity of the coefficients of the weight enumerator of the corresponding dual code. In this talk we will exhibit how this well known approach from coding theory can used to obtain analytical bounds on the maximum size of partial $t$-spreads that form the present state-of-the-art.