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

- 3.
Skew fields are sometimes known as “division rings”.

- 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.
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.
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.
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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## Additional information

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