Subspace codes and network coding

Summary form only given. In classical coding theory, information transmission is modeled as vector transmission: the transmitter sends a vector, the receiver gathers a vector possibly perturbed by noise, and the coding problem is to design a codebook having a large minimum distance between vectors. In this talk we generalize to the case of network coding and, motivated by the property that linear network coding is vector-space preserving, we model information transmission as vector-space transmission: the transmitter sends a (basis for a) vector space, the receiver gathers a (basis for a) vector space possibly perturbed by noise, and the coding problem is to design a codebook having a large minimum distance between vector spaces. We will show that so-called “lifted” maximum rank distance (MRD) codes such as Gabidulin codes play essentially the same role as that played by maximum distance separable (MDS) codes such as Reed-Solomon codes, both for information transmission in the presence of adversarial errors and for security against a wiretapper. When errors are introduced randomly (rather than chosen by an adversary), we show that a simple matrix-based coding scheme can approach capacity. Finally, we describe how some of these ideas may be useful in the context of lattice-theoretic physical-layer network-coding schemes based on compute-and-forward relaying.