Spectral approach to linear programming bounds on codes

We give a new asymptotic upper bound on the size of a code in the Grassmannian space. The bound is better than the upper bounds known previously in the entire range of distances except very large values. the problem of estimating the number of planes whose pairwise distances are bounded below by some given value δ, for a suitably defined distance function d(p, q). This problem has attracted attention in the recent years for several reasons. As a coding-theoretic (geometric) problem, it is a natural generalization of the coding problem for the projective space P R n−1 and a closely related case of the sphere in R n , both having long history in coding theory [4]. This problem arises also in engineering applications related to transmission of signals with multiple antennas in wireless environment [1]. Finally, [9] introduced a construction of Grassmannian packings which is closely related to the construction of quantum stabilizer codes, another subject of interest in recent years. There are several possibilities to define a metric on G k,n [5]. We consider the so-called chordal metric (projection 2-norm in the terminology of [5]), which can be defined in two equivalent ways. By a well-known fact [6], given two planes p, q ∈ G k,n one can define k principal angles between them. This is done recursively as follows: take unit vectors x 1 ∈ p, y 1 ∈ q with the maximum possible angular separation and denote this angle by θ 1. with the maximum possible angle between them and denote this angle by θ i. In this way we obtain the set of principal angles 0 ≤ θ k ≤ · · · ≤ θ 1 ≤ π /2; moreover,