We show that decoding of ℓ-Interleaved Gabidulin codes, as well as
list-ℓ decoding of Mahdavifar--Vardy 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
Mahdavifar--Vardy codes in complexity O(ℓn2), where n is the number of
interpolation constraints.Comment: Accepted for Designs, Codes and Cryptograph