Coding for Errors and Erasures in Random Network Coding

Abstract

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 modelled 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. * Supported in part by DARPA ITMANET W911NF-07-I-0029.

Johnson type bounds on constant dimension codes

“…When studying random network coding [3,4], Koetter and Kschischang [1] defined a so-called operator channel and found that an (n, M, ≥ 2δ, l) q constant dimension code C could be employed to correct errors and/or erasures over the operator channel, i.e., the errors and/or erasures could be corrected by a minimum dimension distance decoder if the sum of errors and erasures is less than δ. Some bounds on A q [n, 2δ, l], e.g., the Hamming type upper bound, the Gilbert type lower bound, and the Singleton type upper bound, were derived in [1].…”

“…Some bounds on A q [n, 2δ, l], e.g., the Hamming type upper bound, the Gilbert type lower bound, and the Singleton type upper bound, were derived in [1]. It is known that the Hamming type bound is not very good [1] and there exist no non-trivial perfect codes meeting the Hamming type bound [5,6].…”

“…It is known that the Hamming type bound is not very good [1] and there exist no non-trivial perfect codes meeting the Hamming type bound [5,6]. The Singleton type bound developed in [1] is the following:…”

“…Moreover, Koetter and Kschischang [1] designed a class of Reed-Solomon like constant dimension codes and afforded decoding procedures. They showed that these codes were nearly Singleton-type-bound-achieving.…”

Secure network coding in the presence of eavesdroppers

“…The second attack model is called Byzantine attacks that attackers may modify the coded packets. To address this problem, Koetter [16] and Kschischang [17] proposed the rank metric error-correcting codes, then extended for the scenario in which the channel may supply partial information about erasures and deviations from the sent information flow [18]. At the same time, some network error correction codes have been proposed in [19][20][21].…”