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Algebraic decoding of folded Gabidulin codes

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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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References

  1. Bartz H.: List and probabilistic unique decoding of high-rate folded Gabidulin codes. In: International Workshop on Coding and Cryptography (WCC) (2015).

  2. Bartz H., Wachter-Zeh A.: Efficient interpolation-based decoding of interleaved subspace and Gabidulin codes. In: Proceedings of 52nd Annual Allerton Conference on Communication, Control, and Computing (2014).

  3. Cheung K.M.: The weight distribution and randomness of linear codes. TDA Progress Report (42–97), pp. 208–215 (1989).

  4. Delsarte P.: Bilinear forms over a finite field with applications to coding Theory. J. Comb. Theory 25(3), 226–241 (1978).

  5. Gabidulin E.M.: Theory of codes with maximum rank distance. Probl. Inf. Transm. 21(1), 3–16 (1985).

  6. Gadouleau M., Yan Z.: Packing and covering properties of rank metric codes. IEEE Trans. Inf. Theory 54(9), 3873–3883 (2008).

  7. Guruswami V., Rudra A.: Explicit codes achieving list decoding capacity: error-correction with optimal redundancy. IEEE Trans. Inf. Theory 54(1), 135–150 (2008).

  8. Guruswami V., Wang C.: Explicit rank-metric codes list-decodable with optimal redundancy. Electron. Colloq. Comput. Complex. (ECCC) 20, (2013).

  9. Guruswami V., Xing C.: List decoding Reed–Solomon, algebraic–geometric, and Gabidulin subcodes up to the singleton bound. Electron. Colloq. Comput. Complex. 19(146), (2012).

  10. Guruswami V., Narayanan S., Wang C.: List decoding subspace codes from insertions and deletions. In: Proceedings of 3rd Innovations in Theoretical Computer Science Conference, ITCS ’12, New York, pp. 183–189 (2012). doi:10.1145/2090236.2090252.

  11. Lidl R., Niederreiter H.: Finite Fields. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1996).

  12. Loidreau P., Overbeck R.: Decoding rank errors beyond the error correcting capability. In: International Workshop on Algebraic and Combinatorial Coding Theory (ACCT), pp. 186–190 (2006).

  13. Mahdavifar H., Vardy A.: List-decoding of subspace codes and rank-metric codes up to Singleton bound. In: IEEE International Symposium on Information Theory (ISIT), pp. 1488–1492 (2012). doi:10.1109/ISIT.2012.6283511.

  14. McEliece R.J.: On the average list size for the Guruswami–Sudan decoder. In: International Symposium on Communication Theory and Applications (ISCTA) (2003).

  15. Ore Ø.: On a special class of polynomials. Trans. Am. Math. Soc. 35, 559–584 (1933).

  16. Sidorenko V.R., Jiang L., Bossert M.: Skew-feedback shift-register synthesis and decoding interleaved Gabidulin codes. IEEE Trans. Inf. Theory 57(2), 621–632 (2011).

  17. Vadhan S.P.: Pseudorandomness. Found. Trends Theor. Comput. Sci. 7(13), 1–336 (2011).

  18. Wachter-Zeh A.: Bounds on list decoding of rank-metric codes. IEEE Trans. Inf. Theory 59(11), 7268–7277 (2013).

  19. Wachter-Zeh, A., Zeh, A.: List and unique error-erasure decoding of interleaved Gabidulin codes with interpolation techniques. Des. Codes Cryptogr. 73(2), 547–570 (2014). doi:10.1007/s10623-014-9953-5.

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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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Correspondence to Hannes Bartz.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

Appendix

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

$$\begin{aligned} \mathbf {y}=\mathbf {c}+\mathbf {e}= \left( \begin{bmatrix} y_{0} \\ y_{1} \\ y_{2} \end{bmatrix}, \begin{bmatrix} y_{3} \\ y_{4} \\ y_{5} \\ \end{bmatrix}, \begin{bmatrix} y_{6} \\ y_{7} \\ y_{8} \\ \end{bmatrix}, \begin{bmatrix} y_{9} \\ y_{10} \\ y_{11} \\ \end{bmatrix}, \begin{bmatrix} y_{12} \\ y_{13} \\ y_{14} \\ \end{bmatrix} \right) \end{aligned}$$

where

$$\begin{aligned} \mathbf {e}= \left( \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} e_{1} \\ e_{2} \\ e_{3} \\ \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} e_{1} \\ e_{2} \\ e_{3} \\ \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix} \right) \end{aligned}$$

with \(t={{\mathrm{rk}}}(\mathbf {e})=1\). For decoding parameter \(s=2\) the set of interpolation points from Problem 1 is

$$\begin{aligned} \mathcal {I}_\text {HR}=\Big \{&\big (\alpha ^{0},f(\alpha ^{0}),f(\alpha ^{1})\big ),&\big (\alpha ^{1},f(\alpha ^{1}),f(\alpha ^{2})\big ),&\big (\alpha ^{2},f(\alpha ^{2}),f(\alpha ^{3})+e_1\big ),\\&\big (\alpha ^{3},f(\alpha ^{3})+e_1,f(\alpha ^{4})+e_2\big ),&\big (\alpha ^{4},f(\alpha ^{4})+e_2,f(\alpha ^{5})+e_3\big ),&\big (\alpha ^{5},f(\alpha ^{5})+e_3,f(\alpha ^{6})\big ),\\&\big (\alpha ^{6},f(\alpha ^{6}),f(\alpha ^{7})\big ),&\big (\alpha ^{7},f(\alpha ^{7}),f(\alpha ^{8})\big ),&\big (\alpha ^{8},f(\alpha ^{8}),f(\alpha ^{9})+e_1\big ),\\&\big (\alpha ^{9},f(\alpha ^{9})+e_1,f(\alpha ^{10})+e_2\big ),&\big (\alpha ^{10},f(\alpha ^{10})+e_2,f(\alpha ^{11})+e_3\big ),&\big (\alpha ^{11},f(\alpha ^{11})+e_3,f(\alpha ^{12})\big ),\\&\big (\alpha ^{12},f(\alpha ^{12}),f(\alpha ^{13})\big ),&\big (\alpha ^{13},f(\alpha ^{13}),f(\alpha ^{14})\big ) \Big \}. \end{aligned}$$

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

$$\begin{aligned}&\big (\alpha ^{8}-\alpha ^{2},f(\alpha ^{8}-\alpha ^{2}),f(\alpha (\alpha ^{8}-\alpha ^{2}))\big ),&\big (\alpha ^{9}-\alpha ^3,f(\alpha ^{9}-\alpha ^3),f(\alpha (\alpha ^{9}-\alpha ^3))\big ),\\&\big (\alpha ^{10}-\alpha ^4,f(\alpha ^{10}-\alpha ^4),f(\alpha (\alpha ^{10}-\alpha ^4))\big ),&\big (\alpha ^{11}-\alpha ^5,f(\alpha ^{11}-\alpha ^5),f(\alpha (\alpha ^{11}-\alpha ^5))\big ). \end{aligned}$$

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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