PROJECT TITLE :

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

ABSTRACT:

We construct an explicit family of -linear rank-metric codes over any field that enables efficient list-decoding up to a fraction of errors within the rank metric with a rate of , for any desired and . This is the first explicit construction of positive rate rank-metric codes for economical list-decoding beyond the distinctive decoding radius. Our codes are specific subcodes of the well-known Gabidulin codes, that encode linearized polynomials of low degree via their values at a collection of linearly freelance 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 styles created by Guruswami and Kopparty (FOCS’thirteen) with subspace-evasive varieties due to Dvir and Lovett (STOC’twelve). We establish an analogous result for subspace codes, that have received much attention recently in the context of network coding. We tend to additionally give express subcodes of folded Reed–Solomon (RS) codes with tiny folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the actual fact that list-decoding RS codes reduc- s to list-decoding such folded RS codes. But, as we solely list-decode a subcode of these codes, the Johnson radius continues to be the most effective known error fraction for list-decoding RS codes.