Linearity and Complements in Projective Space

The projective space of order $n$ over the finite field $\Fq$, denoted here as $\Ps$, is the set of all subspaces of the vector space $\Fqn$. The projective space can be endowed with distance function $d_S(X,Y) = \dim(X) + \dim(Y) - 2\dim(X\cap Y)$ which turns $\Ps$ into a metric space. With this, \emph{an $(n,M,d)$ code $\C$ in projective space} is a subset of $\Ps$ 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 $\rho$ packet erasures introduced (adversarially) anywhere in the network as long as $2t + 2\rho < d$. This motivates new interest in such codes. In this paper, we examine the two fundamental concepts of \myemph{complements} and \myemph{linear codes} in the context of $\Ps$. 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 $\Ps$ and leaves some interesting problems for further research.