Abstract
An efficient interpolation-based decoding algorithm for \(h\)-folded Gabidulin codes is presented that can correct rank errors beyond half the minimum rank distance for any code rate \(0\le R\le 1\). 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 \(\mathcal {O}({s^2n^2})\) operations in \(\mathbb {F}_{q^m}\), where \(1\le s\le h\) is a decoding parameter and \(n\le m\) is the length of the unfolded code over \(\mathbb {F}_{q^m}\). 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.
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Acknowledgments
The authors would like to thank Gerhard Kramer, Joschi Brauchle and Johan S. R. Nielsen for fruitful discussions and helpful comments. H. Bartz was supported by the German Ministry of Education and Research in the framework of an Alexander von Humboldt-Professorship.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
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Appendix
Example 2
(Root Space of Interpolation Polynomial) Suppose we transmit a codeword \(\mathbf {c}\) of a folded Gabidulin code with parameters \(N=5\) and \(h=3\) and we receive
where
with \(t={{\mathrm{rk}}}(\mathbf {e})=1\). For decoding parameter \(s=2\) the set of interpolation points from Problem 1 is
By performing \(\mathbb {F}_{q}\)-elementary operations on \(\mathcal {I}_\text {HR}\) we eliminate the errors from the interpolation points for the code locators \(\alpha ^8,\alpha ^9,\dots ,\alpha ^{11}\) and get the noncorrupted interpolation points
In total we have \(t(h+s-1)=4\) interpolation points that are corrupted by errors and \(Nh-(s-1)-t(h+s-1)=10\) noncorrupted interpolation points. The new code locators for the noncorrupted interpolations points are linearly independent over \(\mathbb {F}_{q}\) and thus P(x) must have at least 10 linearly independent roots in \(\mathbb {F}_{q^m}\).
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Bartz, H., Sidorenko, V. Algebraic decoding of folded Gabidulin codes. Des. Codes Cryptogr. 82, 449–467 (2017). https://doi.org/10.1007/s10623-016-0195-6
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DOI: https://doi.org/10.1007/s10623-016-0195-6