Schubert Varieties, Linear Codes and Enumerative Combinatorics
arXiv:math/0409394 · doi:10.1016/j.ffa.2004.09.002
Abstract
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.
12 pages
Cited by in corpus (5)
- Decomposable Subspaces, Linear Sections of Grassmann Varieties, and Higher Weights of Grassmann Codes
- Minimum Distance and the Minimum Weight Codewords of Schubert Codes
- Linear Codes associated to Determinantal Varieties
- The structure of dual Schubert union codes
- Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding