Linearity and complements in projective space

The projective space of order n over the finite field Fq, denoted here as Pq(n), is the set of all subspaces of the vector space Fqn. The projective space can be endowed with distance function dS(X,Y)=dim(X)+dim(Y)-2dim(X∩Y) which turns Pq(n) into a metric space. With this, an (n,M,d)code Cin projective space is a subset of Pq(n) of size M such that the distance between any two codewords (subspaces) is at least d. Koetter and Kschischang recently showed that codes in projective space are precisely what is needed for error-correction in networks: an (n,M,d) code can correct t packet errors and ρ packet erasures introduced (adversarially) anywhere in the network as long as 2t+2ρ