2011/03/16 by Michael Braun, Tuvi Etzion, Braun, Michael +3
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #graph theory and CDMA systems #math.IT
paper · pdf · doi:10.48550/arxiv.1103.3117
submitted to Linear Algebra and Its Applications
arxiv created 2011/03/16 · openalex publication_date 2011/03/16 · arxiv updated 2015/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 dS(X,Y) = dim(X) + dim(Y) - 2dim(X∩ Y) which turns \Ps into a metric space. With this, 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 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 \myemphcomplements and \myemphlinear 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.