Error-Correcting Codes in Projective Space

The projective space of order n over the finite field F_q, denoted P_q(n), is the set of all subspaces of the vector space Fⁿ _q. The distance function d(U,V) = dim U + dim V - 2 dim(UcapV) 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 rho packet erasures introduced (adversarially) anywhere in the network as long as 2t + 2p < d. This motivates our interest in such codes. In this paper, we investigate certain basic aspects of "coding theory in projective space". First, we present several new bounds on the size of codes in P_q(n), which may be thought of as counterparts of the classical bounds in coding theory due to Johnson, Delsarte, and Gilbert-Varshamov. Some of these are stronger than all the previously known bounds, at least for certain code parameters. Next, we examine the fundamental concepts of "linear codes" and "complements" in the context of P_q(n). These turn out to be considerably more involved than their classical counterparts. In particular, we construct linear codes of size 2ⁿ and conjecture that larger linear codes do not exist. We also present several specific constructions of codes and code families in P_q(n). Finally, we prove that nontrivial perfect codes in P_q(n) do not exist.