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

We show that decoding of ℓ-Interleaved Gabidulin codes, as well as list-ℓ decoding of Mahdavifar–Vardy (MV) codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of F[ x] matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into a certain reduced form. We apply this to solve generalised shift register problems over skew polynomial rings which occur in decoding ℓ-Interleaved Gabidulin codes. We obtain an algorithm with complexity O(ℓμ²) where μ measures the size of the input problem and is proportional to the code length n in the case of decoding. Further, we show how to perform the interpolation step of list-ℓ-decoding MV codes in complexity O(ℓn²) , where n is the number of interpolation constraints.