Density and bounds for Grassmannian codes with chordal distance
Renaud-Alexandre Pitaval, Olav Tirkkonen, Steven D. Blostein · 2011
We investigate the density of codes in the complex Grassmann manifolds Gℂn,pequipped with the chordal distance. The density of a code is defined as the fraction of the Grassmannian covered by `kissing' balls of equal radius centered around the codewords. The kissing radius cannot be determined solely from the minimum distance, nonetheless upper and lower bounds as a function of minimum distance only are provided, along with the corresponding bounds on the density. This leads to a refinement of the Hamming bound for Grassmannian codes. Finally, we provide explicit bounds on code cardinality and minimum distance, notably a generalization of a bound on minimum distance previously proven only for line packing (p = 1).