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 Fnq . The projective space can be endowed with distance function dS(X, Y ) = dim(X) + dim(Y ) − 2 dim(X ∩ Y ) which turns Pq(n) into a metric space. With this, an (n,M, d) code C in 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 < d. This motivates new interest in such codes. In this paper, we examine the two fundamental concepts of “complements” and “linear codes” in the context of Pq(n). These turn out to be considerably more involved than their classical counterparts. These concepts are examined from two different points of view, coding theory and lattice theory. Our discussion reveals some surprised phenomena of these concepts in Pq(n) and leaves some interesting problems for further research.