Bounds for codes and designs in complex subspaces

Article

Abstract

We introduce the concepts of complex Grassmannian codes and designs. Let \(\mathcal{G}_{m,n}\) denote the set of m-dimensional subspaces of ℂn: then a code is a finite subset of \(\mathcal{G}_{m,n}\) in which few distances occur, while a design is a finite subset of \(\mathcal{G}_{m,n}\) that polynomially approximates the entire set. Using Delsarte’s linear programming techniques, we find upper bounds for the size of a code and lower bounds for the size of a design, and we show that association schemes can occur when the bounds are tight. These results are motivated by the bounds for real subspaces recently found by Bachoc, Bannai, Coulangeon and Nebe, and the bounds generalize those of Delsarte, Goethals and Seidel for codes and designs on the complex unit sphere.

Keywords

Codes Designs Bounds Grassmannian spaces Complex subspaces Linear programming Delsarte Association schemes 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute for Quantum Information Science & Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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