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

Abstract

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.

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Notes

  1. 1.

    We opted for using the term “row reduction” rather than “module minimisation”, as we used in [24], since the former is more common in the literature.

  2. 2.

    \(\mathcal{{R}}\) is also right Euclidean, a right PID and right Noetherian, but we will only need its left module structure.

  3. 3.

    Skew fields are sometimes known as “division rings”.

  4. 4.

    There is a precise notion of “row reduced” [20, p. 384] for \(\mathbb {F}[x]\) matrices. Weak Popov form implies being row reduced, but we will not formally define row reduced in this paper.

  5. 5.

    In [28], the claimed complexity of their root-finding is \(O(\ell ^{O(1)} k)\). However, we have to point out that the complexity analysis of that algorithm has severe issues which are outside the scope of this paper to amend. There are two problems: (1) It is not proven that the recursive calls will not produce many spurious “pseudo-roots” which are sifted away only at the leaf of the recursions; and (2) the cost analysis ignores the cost of computing the shifts \(Q(X, Y^q + \gamma Y)\). Issue 1 is necessary to resolve for assuring polynomial complexity. For the original \(\mathbb {F}[x]\)-algorithm this is proved as [42, Proposition 6.4], and an analogous proof might carry over. Issue 2 is critical since these shifts dominate the complexity: assuming the algorithm makes a total of \(O(\ell k)\) recursive calls to itself, then \(O(\ell k)\) shifts need to be computed, each of which costs \(O(\ell \deg _x Q) \subset O(\ell n)\). If Issue 1 is resolved the algorithm should then have complexity \(O(\ell ^2 k n)\).

  6. 6.

    This is a realistic shift register problem arising in decoding of an Interleaved Gabidulin code with \(n=s=100\), \(k_1 = 58\), \(k_2 = 31\).

  7. 7.

    In the conference version of this paper [24], we erroneously claimed a too strong statement concerning this. However, one can relate the complexity of Algorithm 4 to the number of non-zero monomials of \(g_i\), as long as all but the leading monomial have low degree; however the precise statement becomes cumbersome and is not very relevant for this paper.

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Acknowledgments

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).

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Correspondence to Johan Rosenkilde né Nielsen.

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

Vladimir Sidorenko is on leave from Institute of Information Transmission Problems (IITP), Russian Academy of Sciences.

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Puchinger, S., Rosenkilde né Nielsen, J., Li, W. et al. Row reduction applied to decoding of rank-metric and subspace codes. Des. Codes Cryptogr. 82, 389–409 (2017). https://doi.org/10.1007/s10623-016-0257-9

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Keywords

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

Mathematics Subject Classification

  • 12Y05
  • 12E15
  • 11T71