Coding for Errors and Erasures in Random Network Coding

The problem of error-control in random linear network coding is considered. A “noncoherent” or “channel oblivious” model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modeled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U. A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum-distance decoder for this metric achieves correct decoding if the dimension of the space V U is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the Singleton bound are provided for such codes. Finally, a Reed–Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style “list-1” minimum-distance decoding algorithm is provided.