6 research outputs found
Schubert Varieties, Linear Codes and Enumerative Combinatorics
We consider linear error correcting codes associated to higher dimensional
projective varieties defined over a finite field. The problem of determining
the basic parameters of such codes often leads to some interesting and
difficult questions in combinatorics and algebraic geometry. This is
illustrated by codes associated to Schubert varieties in Grassmannians, called
Schubert codes, which have recently been studied. The basic parameters such as
the length, dimension and minimum distance of these codes are known only in
special cases. An upper bound for the minimum distance is known and it is
conjectured that this bound is achieved. We give explicit formulae for the
length and dimension of arbitrary Schubert codes and prove the minimum distance
conjecture in the affirmative for codes associated to Schubert divisors.Comment: 12 page
Decomposable Subspaces, Linear Sections of Grassmann Varieties, and Higher Weights of Grassmann Codes
Given a homogeneous component of an exterior algebra, we characterize those
subspaces in which every nonzero element is decomposable. In geometric terms,
this corresponds to characterizing the projective linear subvarieties of the
Grassmann variety with its Plucker embedding. When the base field is finite, we
consider the more general question of determining the maximum number of points
on sections of Grassmannians by linear subvarieties of a fixed (co)dimension.
This corresponds to a known open problem of determining the complete weight
hierarchy of linear error correcting codes associated to Grassmann varieties.
We recover most of the known results as well as prove some new results. In the
process we obtain, and utilize, a simple generalization of the Griesmer-Wei
bound for arbitrary linear codes.Comment: 16 page