Coding for Errors and Erasures in Random Network Coding

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> 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 <emphasis><formula formulatype="inline"><tex>$V$</tex></formula></emphasis> and the collection by the receiver of a basis for a vector space <emphasis><formula formulatype="inline"> <tex>$U$</tex></formula></emphasis>. 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 <emphasis><formula formulatype="inline"><tex>$V \cap U$</tex></formula></emphasis> 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. </para>