Gabidulin Decoding via Minimal Bases of Linearized Polynomial Modules

Abstract: We show how Gabidulin codes can be decoded via parametrization by using interpolation modules over the ring of linearized polynomials with composition. Our decoding algorithm computes a list of message words that correspond to all closest codewords to a given received word. This involves the computation of a minimal basis for the interpolation module that corresponds to the received word, followed by a search through the parametrization for valid message words. Our module-theoretic approach strengthens the link between Gabidulin decoding and Reed-Solomon decoding. Two subalgorithms are presented to compute the minimal basis, one iterative, the other an extended Euclidean algorithm. Both of these subalgorithms have polynomial time complexity. The complexity order of the overall algorithm, using the parametrization, is then compared to straightforward exhaustive search as well as to chase list decoding.