Since we will be using the skew polynomial description of Gabidulin codes, here are a few articles that use that description:
- The following describe some of the algebraic link between the linearized and skew polynomials, and goes on to construct new codes using skew polynomials with derivations:
Boucher, D., and F. Ulmer. n.d. “Linear Codes Using Skew Polynomials with Automorphisms and Derivations.” Designs, Codes and Cryptography, 1–27. doi:10.1007/s10623-012-9704-4. https://hal.archives-ouvertes.fr/hal-00597127/document
- This paper doesn't describe the link but just uses the description of skew polynomials to decode:
Sidorenko, Vladimir, Lan Jiang, and Martin Bossert. 2011.
“Skew-Feedback Shift-Register Synthesis and Decoding Interleaved
Gabidulin Codes.” IEEE Transactions on Information Theory 57 (2):
- A paper I coauthored, where we use the skew polynomial language. We mention the link but do not really spend much time on it: http://jsrn.dk/publications.html#2016-dcc-skew-module
The algorithm in the latter paper, the Mulders-Storjohann for skew polynomials, is quite likely the conceptually easiest way to implement a decoding algorithm. (decoding a standard Gabidulin code is then done by doing a single shift register, where the l=1 in Section 5 of my paper. An error-erasure decoding is also described; it is restricted to only certain "nice" Gabidulin codes).