Algebraic decoding of folded Gabidulin codes

Abstract

An efficient interpolation-based decoding algorithm for $$h$$h-folded Gabidulin codes is presented that can correct rank errors beyond half the minimum rank distance for any code rate $$0\le R\le 1$$0≤R≤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 $$\mathcal {O}({s^2n^2})$$O(s2n2) operations in $$\mathbb {F}_{q^m}$$Fqm, where $$1\le s\le h$$1≤s≤h is a decoding parameter and $$n\le m$$n≤m is the length of the unfolded code over $$\mathbb {F}_{q^m}$$Fqm. 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.