Algebraic list-decoding of subspace codes with multiplicities

Koetter and Kschischang introduced subspace codes in order to correct errors and erasures for randomized network coding, in the case where network topology is unknown (the noncoherent case). The codewords are vector subspaces of a fixed ambient space; thus codes for this model are collections of such subspaces. Koetter and Kschischang constructed a remarkable family of codes similar to Reed-Solomon codes in that codewords are obtained by evaluating certain (linearized) polynomials. In a previous work, we introduced a new family of subspace codes based upon Koetter-Kschichang construction which are efficiently list-decodable. By that we could achieve a better decoding radius at low rates than Koetter-Kschischiang codes. In this paper, we consider the problem of list-decoding of subspace codes with multiplicities. We first establish the notion of multiplicity for linearized polynomials in this context. Then we take advantage of enforcing multiple roots for the interpolation polynomial in order to achieve a better decoding radius. We are also able to list-decode at higher rates. The end result is the following: for any integers L and r, our list-L decoder with multiplicity r guarantees successful recovery of the message subspace provided that the normalized dimension of the error is at most 2(L + 1)/r + 1 - 1 -L(L + 1)/r(r + 1)R* where R* is the normalized packet rate. This improves the normalized decoding radius upon the previous results for a wide range of rates. The parameter r is independent of the code construction and can be chosen at the decoder in such a way that the decoding radius is maximized. As L tends to infinity, the decoding radius of our construction with appropriate choice of r approaches 1/R* - .