Designs, Codes and Cryptography

, Volume 82, Issue 1–2, pp 389–409 | Cite as

Row reduction applied to decoding of rank-metric and subspace codes

  • Sven Puchinger
  • Johan Rosenkilde né Nielsen
  • Wenhui Li
  • Vladimir Sidorenko


We show that decoding of \(\ell \)-Interleaved Gabidulin codes, as well as list-\(\ell \) decoding of Mahdavifar–Vardy (MV) codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of \(\mathbb {F}[x]\) 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 \(\ell \)-Interleaved Gabidulin codes. We obtain an algorithm with complexity \(O(\ell \mu ^2)\) where \(\mu \) 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-\(\ell \)-decoding MV codes in complexity \(O(\ell n^2)\), where n is the number of interpolation constraints.


Skew polynomials Row reduction Module minimisation Gabidulin codes Shift register synthesis Mahdavifar–Vardy codes 

Mathematics Subject Classification

12Y05 12E15 11T71 



The authors would like to thank the anonymous reviewers for suggestions that have substantially improved the clarity of the paper. Sven Puchinger (Grant BO 867/29-3), Wenhui Li and Vladimir Sidorenko (Grant BO 867/34-1) were supported by the German Research Foundation “Deutsche Forschungsgemeinschaft” (DFG).


  1. 1.
    Abramov S.A., Bronstein M.: On solutions of linear functional systems. In: Proceedings of ISSAC, pp. 1–6 (2001).Google Scholar
  2. 2.
    Alekhnovich M.: Linear Diophantine equations over polynomials and soft decoding of Reed–Solomon codes. IEEE Trans. Inf. Theory 51(7), 2257–2265 (2005).Google Scholar
  3. 3.
    Augot D., Loidreau P., Robert G.: Rank metric and Gabidulin codes in characteristic zero. In: ISIT (2013).Google Scholar
  4. 4.
    Baker G., Graves-Morris P.: Padé Approximants, vol. 59. Cambridge University Press, Cambridge (1996).Google Scholar
  5. 5.
    Beckermann B., Labahn G.: A uniform approach for the fast computation of matrix-type Padé approximants. SIAM J. Matrix Anal. Appl. 15(3), 804–823 (1994).Google Scholar
  6. 6.
    Beckermann B., Cheng H., Labahn G.: Fraction-free row reduction of matrices of Ore polynomials. J. Symb. Comput. 41(5), 513–543 (2006).Google Scholar
  7. 7.
    Beelen P., Brander K.: Key equations for list decoding of Reed–Solomon codes and how to solve them. J. Symb. Comput. 45(7), 773–786 (2010).Google Scholar
  8. 8.
    Boucher D., Ulmer F.: Linear codes using skew polynomials with automorphisms and derivations. Des. Codes Cryptogr. 70(3), 405–431 (2014).Google Scholar
  9. 9.
    Cohn H., Heninger N.: Ideal forms of Coppersmith’s theorem and Guruswami–Sudan list decoding (2010). arXiv:1008.1284.
  10. 10.
    Clark P.L.: Non-commutative algebra. University of Georgia. (2012).
  11. 11.
    Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory 25(3), 226–241 (1978).Google Scholar
  12. 12.
    Dieudonné J.: Les déterminants sur un corps non commutatif. Bull. Soc. Math. France 71, 27–45 (1943).Google Scholar
  13. 13.
    Draxl P.K.: Skew Fields, vol. 81. Cambridge University Press, Cambridge (1983).Google Scholar
  14. 14.
    Feng G.L., Tzeng K.K.: A generalization of the Berlekamp–Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes. IEEE Trans. Inf. Theory 37(5), 1274–1287 (1991).Google Scholar
  15. 15.
    Gabidulin E.M.: Theory of codes with maximum rank distance. Problemy Peredachi Informatsii 21(1), 3–16 (1985).Google Scholar
  16. 16.
    Giorgi P., Jeannerod C., Villard G.: On the complexity of polynomial matrix computations. In: Proceedings of ISSAC, pp. 135–142 (2003).Google Scholar
  17. 17.
    Guruswami V., Sudan M.: Improved decoding of Reed–Solomon codes and algebraic-geometric codes. IEEE Trans. Inf. Theory 45(6), 1757–1767 (1999).Google Scholar
  18. 18.
    Guruswami V., Wang C.: Explicit rank-metric codes list-decodable with optimal redundancy. In: Proceedings of RANDOM (2014). arXiv:1311.7084.
  19. 19.
    Guruswami V., Xing C.: List decoding Reed-solomon, algebraic-geometric, and Gabidulin subcodes up to the singleton bound. In: Proceedings of STOC, pp. 843–852. ACM, New York (2013)Google Scholar
  20. 20.
    Kailath T.: Linear Systems. Prentice-Hall, Upper Saddle River (1980).Google Scholar
  21. 21.
    Lee K., O’Sullivan M.E.: List decoding of Reed–Solomon codes from a Gröbner basis perspective. J. Symb. Comput. 43(9), 645–658 (2008).Google Scholar
  22. 22.
    Lenstra A.: Factoring multivariate polynomials over finite fields. J. Comput. Syst. Sci. 30(2), 235–246 (1985).Google Scholar
  23. 23.
    Li W., Sidorenko V., Silva D.: On transform-domain error and erasure correction by Gabidulin codes. Des. Codes Cryptogr. 73(2), 571–586 (2014).Google Scholar
  24. 24.
    Li W., Nielsen J.S.R., Puchinger S., Sidorenko V.: Solving shift register problems over skew polynomial rings using module minimisation. In: Proceedings of WCC (2015).Google Scholar
  25. 25.
    Loidreau P., Overbeck R.: Decoding rank errors beyond the error correcting capability. In: Proceedings of ACCT, pp. 186–190 (2006).Google Scholar
  26. 26.
    Mahdavifar H.: List decoding of subspace codes and rank-metric codes. Ph.D. thesis, University of California, San Diego (2012).Google Scholar
  27. 27.
    Mahdavifar H., Vardy A.: Algebraic list-decoding of subspace codes with multiplicities. In: Allerton Conference on Communication, Control, and Computing, pp. 1430–1437 (2011).Google Scholar
  28. 28.
    Mahdavifar H., Vardy A.: Algebraic list-decoding of subspace codes. IEEE Trans. Inf. Theory 59(12), 7814–7828 (2013).Google Scholar
  29. 29.
    Middeke J.: A computational view on normal forms of matrices of Ore polynomials. Ph.D. thesis, Research Institute for Symbolic Computation (RISC) (2011).Google Scholar
  30. 30.
    Müelich S., Puchinger S., Mödinger D., Bossert M.: An alternative decoding method for Gabidulin codes in characteristic zero. In: Proceedings of IEEE ISIT (2016). arXiv:1601.05205.
  31. 31.
    Mulders T., Storjohann A.: On lattice reduction for polynomial matrices. J. Symb. Comput. 35(4), 377–401 (2003).CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Nielsen J.S.R.: Generalised multi-sequence shift-register synthesis using module minimisation. In: Proceedings of IEEE ISIT, pp. 882–886 (2013).Google Scholar
  33. 33.
    Nielsen J.S.R.: Power decoding Reed–Solomon codes up to the Johnson radius. In: Proceedings of ACCT (2014).Google Scholar
  34. 34.
    Nielsen J.S.R., Beelen P.: Sub-quadratic decoding of one-point Hermitian codes. IEEE Trans. Inf. Theory 61(6), 3225–3240 (2015).Google Scholar
  35. 35.
    Nielsen R.R., Høholdt T.: Decoding Reed–Solomon codes beyond half the minimum distance. In: Coding Theory, Cryptography and Related Areas, pp. 221–236. Springer, Berlin (1998).Google Scholar
  36. 36.
    Olesh Z., Storjohann A.: The vector rational function reconstruction problem. In: Proceedings of WWCA, pp. 137–149 (2006).Google Scholar
  37. 37.
    Ore O.: On a special class of polynomials. Trans. Am. Math. Soc. 35(3), 559–584 (1933).Google Scholar
  38. 38.
    Ore O.: Theory of non-commutative polynomials. Ann. Math. 34(3), 480–508 (1933).Google Scholar
  39. 39.
    Puchinger S., Wachter-Zeh A.: Fast operations on linearized polynomials and their applications in coding theory. J. Symb. Comput. (2015). arXiv:1512.06520.
  40. 40.
    Puchinger S., Müelich S., Mödinger D., Nielsen J.S.R., Bossert M.: Decoding interleaved Gabidulin codes using Alekhnovich’s algorithm. In: Proceedings of ACCT (2016). arXiv:1604.04397.
  41. 41.
    Roth R.M.: Maximum-rank array codes and their application to crisscross error correction. IEEE Trans. Inf. Theory 37(2), 328–336 (1991).Google Scholar
  42. 42.
    Roth R., Ruckenstein G.: Efficient decoding of Reed–Solomon codes beyond half the minimum distance. IEEE Trans. Inf. Theory 46(1), 246–257 (2000).Google Scholar
  43. 43.
    Sidorenko V., Bossert M.: Fast skew-feedback shift-register synthesis. Des. Codes Cryptogr. 70(1–2), 55–67 (2014).Google Scholar
  44. 44.
    Sidorenko V., Jiang L., Bossert M.: Skew-feedback shift-register synthesis and decoding interleaved Gabidulin codes. IEEE Trans. Inf. Theory 57(2), 621–632 (2011).Google Scholar
  45. 45.
    Sidorenko V., Schmidt G.: A linear algebraic approach to multisequence shift-register synthesis. Probl. Inf. Transm. 47(2), 149–165 (2011).Google Scholar
  46. 46.
    Taelman L.: Dieudonné determinants for skew polynomial rings. J. Algebra Appl. 5(1), 89–93 (2006).Google Scholar
  47. 47.
    Wachter-Zeh A.: Decoding of block and convolutional codes in rank metric. Ph.D. thesis, Universität Ulm (2013).Google Scholar
  48. 48.
    Xie H., Lin J., Yan Z., Suter B.W.: Linearized polynomial interpolation and its applications. IEEE Trans. Signal Process. 61(1), 206–217 (2013).Google Scholar
  49. 49.
    Zeh A., Gentner C., Augot D.: An interpolation procedure for list decoding Reed–Solomon codes based on generalized key equations. IEEE Trans. Inf. Theory 57(9), 5946–5959 (2011).Google Scholar
  50. 50.
    Zhou W., Labahn G.: Efficient algorithms for order basis computation. J. Symb. Comput. 47(7), 793–819 (2012).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sven Puchinger
    • 1
  • Johan Rosenkilde né Nielsen
    • 2
  • Wenhui Li
    • 1
  • Vladimir Sidorenko
    • 3
  1. 1.Institute of Communications EngineeringUlm UniversityUlmGermany
  2. 2.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKongens LyngbyDenmark
  3. 3.Institute for Communications EngineeringTU MünchenMunichGermany

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