Linearity and Complements in Projective Space

Abstract The projective space of order n over the finite field F q , denoted here as P q ( n ) , is the set of all subspaces of the vector space F q n . The projective space can be endowed with distance function d S ( X , Y ) = dim ( X ) + dim ( Y ) - 2 dim ( X ∩ Y ) which turns P q ( n ) into a metric space. With this, an ( n , M , d ) code C in projective space is a subset of P q ( 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 2 t + 2 ρ 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 P q ( 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 results reveal a number of surprising phenomena pertaining to complements and linearity in P q ( n ) and gives rise to several interesting problems.