Algebraic list-decoding of subspace codes with multiplicities

Abstract

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 wich 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 mostwhere 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 * − 1.

Some Gabidulin Codes Cannot Be List Decoded Efficiently at any Radius

“…List decoding of subspace codes was extensively studied in recent years. In particular, several variants and subcodes of the Kötter and Kschischang codes were shown to be efficiently list decodable (e.g., [7], [15], [16], [22], [23] and references therein), and bounds equivalent to [31] were discussed in [26]. Our results about Gabidulin codes also apply for lifted Gabidulin codes, and thus we get families of subspace codes that cannot be list decoded efficiently at any radius.…”

The combination of sparse learning and list decoding of subspace codes for error correction in random network coding

“…Therefore, it is also inefficient. Based on [9,23], this improves the list model and achieves a much bigger decoding radius C k -1 where C is the max flow min cut and k is the size of information. Guruswami and Sudan [24] can correct C Á ð1−…”

“…Combining list decoding and subspace code, [23] proposes list decoding of subspace codes. For any integers L and r, the list-L decoder with multiplicityr guarantees successful recovery of the message subspace provided that the normalized dimension of error is at most …”

“…Unlike [9] which designs list decoding with a purpose to solve the error-correcting problem in random network coding, [8,23,24,34,35] are not designed for solving the error-correcting problem in random network coding. However, because [8,23,24,34,35] can correct errors beyond d min /2, they may have the potential to solve the propagated errors in random network coding. The fifth is the approach based on the secret channel.…”

Row reduction applied to decoding of rank-metric and subspace codes

“…Their rate unfortunately tend to zero for increasing code lengths. In [26], Mahdavifar and Vardy presented a refined construction which can be decoded "with multiplicities" allowing a better decoding radius and rate; it is future work to adapt our algorithm to this case. The decoding of MV codes is is heavily inspired by the Guruswami-Sudan algorithm for Reed-Solomon codes [16], and our row reduction approach in Section 3.2 is similarly inspired by fast module-based algorithms for realising the Guruswami-Sudan [7,20].Another family of rank-metric codes which can be decoded beyond half the minimum distance are Guruswami-Xing codes [18].…”