Algebraic decoding of folded Gabidulin codes

An efficient interpolation-based decoding algorithm for \(h\)-folded Gabidulin codes is presented that can correct rank errors beyond half the minimum rank distance for any code rate \(0\le R\le 1\). The algorithm serves as a list decoder or as a probabilistic unique decoder and improves upon existing schemes, especially for high code rates. A probabilistic unique decoder with adjustable decoding radius is presented. The decoder outputs a unique solution with high probability and requires at most \(\mathcal {O}({s^2n^2})\) operations in \(\mathbb {F}_{q^m}\), where \(1\le s\le h\) is a decoding parameter and \(n\le m\) is the length of the unfolded code over \(\mathbb {F}_{q^m}\). An upper bound on the average list size of folded Gabidulin codes and on the decoding failure probability of the decoder is given. Applying the ideas to a list decoding algorithm by Mahdavifar and Vardy (List-decoding of subspace codes and rank-metric codes up to Singleton bound, ISIT 2012) improves the performance when used as probabilistic unique decoder and gives an upper bound on the failure probability.