Bounds for codes and designs in complex subspaces

J. Algebr. Comb. 31(1), 1-32 (2010)
DOI: 10.1007/s10801-009-0170-7

Summary

Summary: We introduce the concepts of complex Grassmannian codes and designs. Let $G _{ m, n}$ mathcalG_m,n denote the set of $m$-dimensional subspaces of $\Bbb C ^{ n }$: then a $code$ is a finite subset of $G _{ m, n}$ mathcalG_m,n in which few distances occur, while a $design$ is a finite subset of $G _{ m, n}$ mathcalG_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/Phrases

keywords codes, designs, bounds, Grassmannian spaces, complex subspaces, linear programming, delsarte, association schemes