Algebraic list-decoding on the operator channel

The operator channel was introduced by Koetter and Kschischang as a model of errors and erasures for randomized network coding, in the case where network topology is unknown (the noncoherent case). The input and output of the operator channel are vector subspaces of the ambient space; thus error-correcting codes for this channel are collections of such subspaces. Koetter and Kschischang also constructed a remarkable family of codes for the operator channel. The Koetter-Kschischang codes are similar to Reed-Solomon codes in that codewords are obtained by evaluating certain (linearized) polynomials. In this paper, we consider the problem of list-decoding the Koetter-Kschischang codes on the operator channel. In a sense, we are able to achieve for these codes what Sudan was able to achieve for Reed-Solomon codes. In order to do so, we have to modify and generalize the original Koetter-Kschischang construction in many important respects. The end result is this: for any integer L, our list-L decoder guarantess successful recovery of the message subspace provided the normalized dimension of the error is at most L − L2(L + 1) over 2 R where R is the normalized rate of the code. Just as in the case of Sudan's list-decoding algorithm, this exceeds the previously best-known error-correction radius 1 - R, demonstrated by Koetter and Kschischang, for low rates R.