Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs

摘要

We construct an explicit family of -linear rank-metric codes over any field that enables efficient list-decoding up to a fraction of errors in the rank metric with a rate of , for any desired and . This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an -subspace that evades the structured subspaces over an extension field that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS’13) with subspace-evasive varieties due to Dvir and Lovett (STOC’12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed–Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduc- s to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes.