November 2018, 12(4): 773-804. doi: 10.3934/amc.2018046

Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases

1. 

Institute of Communications and Navigation, German Aerospace Center (DLR), D-82234 Oberpfaffenhofen, Germany

2. 

Institute for Communications Engineering, Technical University of Munich (TUM), D-80290 Munich, Germany

Received  February 2018 Published  September 2018

Fund Project: A. Wachter-Zeh's work was supported by the Technical University of Munich|Institute for Advanced Study, funded by the German Excellence Initiative and European Union Seventh Framework Programme under Grant Agreement No. 291763 and the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) unter Grant No. WA3907/1-1

An interpolation-based decoding scheme for $L$-interleaved subspace codes is presented. The scheme can be used as a (not necessarily polynomial-time) list decoder as well as a polynomial-time probabilistic unique decoder. Both interpretations allow to decode interleaved subspace codes beyond half the minimum subspace distance. Both schemes can decode $\gamma $ insertions and $\delta $ deletions up to $\gamma +L\delta \leq L({{n}_{t}}-k)$, where ${{n}_{t}}$ is the dimension of the transmitted subspace and $k$ is the number of data symbols from the field ${{\mathbb{F}}_{{{q}^{m}}}}$. Further, a complementary decoding approach is presented which corrects $\gamma $ insertions and $\delta $ deletions up to $L\gamma +\delta \leq L({{n}_{t}}-k)$. Both schemes use properties of minimal Gröebner bases for the interpolation module that allow predicting the worst-case list size right after the interpolation step. An efficient procedure for constructing the required minimal Gröebner basis using the general Kötter interpolation is presented. A computationally- and memory-efficient root-finding algorithm for the probabilistic unique decoder is proposed. The overall complexity of the decoding algorithm is at most $\mathcal{O}\left( {{L}^{2}}n_{r}^{2} \right)$ operations in ${{\mathbb{F}}_{{{q}^{m}}}}$ where ${{n}_{r}}$ is the dimension of the received subspace and $L$ is the interleaving order. The analysis as well as the efficient algorithms can also be applied for accelerating the decoding of interleaved Gabidulin codes.

Citation: Hannes Bartz, Antonia Wachter-Zeh. Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases. Advances in Mathematics of Communications, 2018, 12 (4) : 773-804. doi: 10.3934/amc.2018046
References:
[1]

W. W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, 3, American Mathematical Soc., 1994. doi: 10.1090/gsm/003.

[2]

C. BachocF. Vallentin and A. Passuello, Bounds for Projective Codes from Semidefinite Programming, Adv. Math. Commun., 7 (2013), 127-145. doi: 10.3934/amc.2013.7.127.

[3]

H. Bartz, Algebraic Decoding of Subspace and Rank-Metric Codes, PhD thesis, Technical University of Munich, 2017.

[4]

H. Bartz, M. Meier and V. Sidorenko, Improved Syndrome Decoding of Interleaved Subspace Codes, in 11th International ITG Conference on Systems, Communications and Coding 2017 (SCC), Hamburg, Germany, 2017.

[5]

H. Bartz and V. Sidorenko, List and probabilistic unique decoding of folded subspace codes in IEEE International Symposium on Information Theory (ISIT), 2015. doi: 10.1109/ISIT.2015.7282407.

[6]

H. Bartz and V. Sidorenko, On list-decoding schemes for punctured reed-solomon, Gabidulin and subspace codes in XV International Symposium "Problems of Redundancy in Information and Control Systems", 2016. doi: 10.1109/RED.2016.7779321.

[7]

H. Bartz and A. Wachter-Zeh, Efficient interpolation-based decoding of interleaved subspace and Gabidulin codes, in 52nd Annual Allerton Conference on Communication, Control, and Computing, 2014, 1349-1356. doi: 10.1109/ALLERTON.2014.7028612.

[8]

D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, vol. 3, Springer, 1992. doi: 10.1007/978-1-4757-2181-2.

[9]

P. Delsarte, Bilinear forms over a finite field with applications to coding theory, J. Combin. Theory, 25 (1978), 226-241. doi: 10.1016/0097-3165(78)90015-8.

[10]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inf. Theory, 55 (2009), 2909-2919. doi: 10.1109/TIT.2009.2021376.

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T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inf. Theory, 57 (2011), 1165-1173. doi: 10.1109/TIT.2010.2095232.

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T. EtzionE. GorlaA. Ravagnani and A. Wachter-Zeh, Optimal Ferrers diagram rank-metric codes, IEEE Trans. Inform. Theory, 62 (2016), 1616-1630. doi: 10.1109/TIT.2016.2522971.

[13]

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

[14]

M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inf. Theory, 56 (2010), 3207-3216. doi: 10.1109/TIT.2010.2048447.

[15]

V. Guruswami and C. Xing, List decoding Reed-Solomon, algebraic-geometric, and Gabidulin subcodes up to the singleton bound, STOC'13??Proceedings of the 2013 ACM Symposium on Theory of Computing, 843-852, ACM, New York, 2013. doi: 10.1145/2488608.2488715.

[16]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.

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A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical Methods in Computer Science, vol. 5393 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2008, 31-42. doi: 10.1007/978-3-540-89994-5_4.

[18]

R. Kötter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inf. Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449.

[19]

M. Kuijper and A. Trautmann, Gröbner bases for linearized polynomials, URL http://arXiv.org/abs/1406.4600.

[20]

W. LiV. Sidorenko and D. Silva, On transform-domain error and erasure correction by Gabidulin codes, Des. Codes Cryptogr., 73 (2014), 571-586. doi: 10.1007/s10623-014-9954-4.

[21]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1997.

[22]

P. Loidreau and R. Overbeck, Decoding rank errors beyond the error correcting capability, in Int. Workshop Alg. Combin. Coding Theory (ACCT), 2006,186-190.

[23]

H. Mahdavifar and A. Vardy, Algebraic list-decoding on the operator channel, in IEEE Int. Symp. Inf. Theory (ISIT), 2010, 1193-1197. doi: 10.1109/ISIT.2010.5513656.

[24]

H. Mahdavifar and A. Vardy, List-Decoding of Subspace Codes and Rank-Metric Codes up to Singleton Bound, in IEEE Int. Symp. Inf. Theory (ISIT), 2012, 1488-1492.

[25]

J. N. Nielsen, List Decoding of Algebraic Codes, PhD thesis, 2013.

[26]

∅. Ore, On a special class of polynomials, Trans. Amer. Math. Soc., 35 (1933), 559-584. doi: 10.1090/S0002-9947-1933-1501703-0.

[27]

S. Puchinger, J. S. R. Nielsen, W. Li and V. Sidorenko, Row reduction applied to decoding of rank metric and subspace codes, Designs, Codes and Cryptography, 82 (2017), 389-409, URL http://arXiv.org/abs/1510.04728v2. doi: 10.1007/s10623-016-0257-9.

[28]

N. Raviv and A. Wachter-Zeh, Some Gabidulin codes cannot be list decoded efficiently at any radius, IEEE Trans. Inform. Theory, 62 (2016), 1605-1615. doi: 10.1109/TIT.2016.2532343.

[29]

R. M. Roth, Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. Inf. Theory, 37 (1991), 328-336. doi: 10.1109/18.75248.

[30]

V. R. Sidorenko and M. Bossert, Decoding interleaved Gabidulin codes and multisequence linearized shift-register synthesis, in IEEE Int. Symp. Inf. Theory (ISIT), 2010, 1148-1152. doi: 10.1109/ISIT.2010.5513676.

[31]

V. R. SidorenkoL. Jiang and M. Bossert, Skew-feedback shift-register synthesis and decoding interleaved Gabidulin codes, IEEE Trans. Inf. Theory, 57 (2011), 621-632. doi: 10.1109/TIT.2010.2096032.

[32]

D. Silva, Error Control for Network Coding, PhD thesis, University of Toronto, Toronto, Canada, 2009.

[33]

D. SilvaF. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding, IEEE Trans. Inf. Theory, 54 (2008), 3951-3967. doi: 10.1109/TIT.2008.928291.

[34]

V. Skachek, Recursive code construction for random networks, IEEE Trans. Inf. Theory, 56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163.

[35]

A. L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - A new concept in the area of network coding in IEEE Information Theory Workshop 2019 (ITW 2012), 2010. doi: 10.1109/CIG.2010.5592788.

[36]

A.-L. Trautmann and M. Kuijper, Gabidulin decoding via minimal bases of linearized polynomial modules, preprint, arXiv: 1408.2303.

[37]

A.-L. Trautmann, N. Silberstein and J. Rosenthal, List decoding of lifted Gabidulin codes via the Plücker embedding, in Int. Workshop Coding Cryptogr. (WCC), 2013.

[38]

A. Wachter-Zeh, Bounds on list decoding of rank-metric codes, IEEE Trans. Inf. Theory, 59 (2013), 7268-7277. doi: 10.1109/TIT.2013.2274653.

[39]

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

[40]

B. Wang, R. J. McEliece and K. Watanabe, Kötter interpolation over free modules, in Proc. 43rd Annu. Allerton Conf. Comm., Control, and Comp., 2005.

[41]

S. Xia and F. Fu, Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172. doi: 10.1007/s10623-008-9221-7.

[42]

H. Xie, Z. Yan and B. W. Suter, General linearized polynomial interpolation and its applications, in IEEE Int. Symp. Network Coding (Netcod), 2011, 1-4. doi: 10.1109/ISNETCOD.2011.5978942.

show all references

References:
[1]

W. W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, 3, American Mathematical Soc., 1994. doi: 10.1090/gsm/003.

[2]

C. BachocF. Vallentin and A. Passuello, Bounds for Projective Codes from Semidefinite Programming, Adv. Math. Commun., 7 (2013), 127-145. doi: 10.3934/amc.2013.7.127.

[3]

H. Bartz, Algebraic Decoding of Subspace and Rank-Metric Codes, PhD thesis, Technical University of Munich, 2017.

[4]

H. Bartz, M. Meier and V. Sidorenko, Improved Syndrome Decoding of Interleaved Subspace Codes, in 11th International ITG Conference on Systems, Communications and Coding 2017 (SCC), Hamburg, Germany, 2017.

[5]

H. Bartz and V. Sidorenko, List and probabilistic unique decoding of folded subspace codes in IEEE International Symposium on Information Theory (ISIT), 2015. doi: 10.1109/ISIT.2015.7282407.

[6]

H. Bartz and V. Sidorenko, On list-decoding schemes for punctured reed-solomon, Gabidulin and subspace codes in XV International Symposium "Problems of Redundancy in Information and Control Systems", 2016. doi: 10.1109/RED.2016.7779321.

[7]

H. Bartz and A. Wachter-Zeh, Efficient interpolation-based decoding of interleaved subspace and Gabidulin codes, in 52nd Annual Allerton Conference on Communication, Control, and Computing, 2014, 1349-1356. doi: 10.1109/ALLERTON.2014.7028612.

[8]

D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, vol. 3, Springer, 1992. doi: 10.1007/978-1-4757-2181-2.

[9]

P. Delsarte, Bilinear forms over a finite field with applications to coding theory, J. Combin. Theory, 25 (1978), 226-241. doi: 10.1016/0097-3165(78)90015-8.

[10]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inf. Theory, 55 (2009), 2909-2919. doi: 10.1109/TIT.2009.2021376.

[11]

T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inf. Theory, 57 (2011), 1165-1173. doi: 10.1109/TIT.2010.2095232.

[12]

T. EtzionE. GorlaA. Ravagnani and A. Wachter-Zeh, Optimal Ferrers diagram rank-metric codes, IEEE Trans. Inform. Theory, 62 (2016), 1616-1630. doi: 10.1109/TIT.2016.2522971.

[13]

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

[14]

M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inf. Theory, 56 (2010), 3207-3216. doi: 10.1109/TIT.2010.2048447.

[15]

V. Guruswami and C. Xing, List decoding Reed-Solomon, algebraic-geometric, and Gabidulin subcodes up to the singleton bound, STOC'13??Proceedings of the 2013 ACM Symposium on Theory of Computing, 843-852, ACM, New York, 2013. doi: 10.1145/2488608.2488715.

[16]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.

[17]

A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical Methods in Computer Science, vol. 5393 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2008, 31-42. doi: 10.1007/978-3-540-89994-5_4.

[18]

R. Kötter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inf. Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449.

[19]

M. Kuijper and A. Trautmann, Gröbner bases for linearized polynomials, URL http://arXiv.org/abs/1406.4600.

[20]

W. LiV. Sidorenko and D. Silva, On transform-domain error and erasure correction by Gabidulin codes, Des. Codes Cryptogr., 73 (2014), 571-586. doi: 10.1007/s10623-014-9954-4.

[21]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1997.

[22]

P. Loidreau and R. Overbeck, Decoding rank errors beyond the error correcting capability, in Int. Workshop Alg. Combin. Coding Theory (ACCT), 2006,186-190.

[23]

H. Mahdavifar and A. Vardy, Algebraic list-decoding on the operator channel, in IEEE Int. Symp. Inf. Theory (ISIT), 2010, 1193-1197. doi: 10.1109/ISIT.2010.5513656.

[24]

H. Mahdavifar and A. Vardy, List-Decoding of Subspace Codes and Rank-Metric Codes up to Singleton Bound, in IEEE Int. Symp. Inf. Theory (ISIT), 2012, 1488-1492.

[25]

J. N. Nielsen, List Decoding of Algebraic Codes, PhD thesis, 2013.

[26]

∅. Ore, On a special class of polynomials, Trans. Amer. Math. Soc., 35 (1933), 559-584. doi: 10.1090/S0002-9947-1933-1501703-0.

[27]

S. Puchinger, J. S. R. Nielsen, W. Li and V. Sidorenko, Row reduction applied to decoding of rank metric and subspace codes, Designs, Codes and Cryptography, 82 (2017), 389-409, URL http://arXiv.org/abs/1510.04728v2. doi: 10.1007/s10623-016-0257-9.

[28]

N. Raviv and A. Wachter-Zeh, Some Gabidulin codes cannot be list decoded efficiently at any radius, IEEE Trans. Inform. Theory, 62 (2016), 1605-1615. doi: 10.1109/TIT.2016.2532343.

[29]

R. M. Roth, Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. Inf. Theory, 37 (1991), 328-336. doi: 10.1109/18.75248.

[30]

V. R. Sidorenko and M. Bossert, Decoding interleaved Gabidulin codes and multisequence linearized shift-register synthesis, in IEEE Int. Symp. Inf. Theory (ISIT), 2010, 1148-1152. doi: 10.1109/ISIT.2010.5513676.

[31]

V. R. SidorenkoL. Jiang and M. Bossert, Skew-feedback shift-register synthesis and decoding interleaved Gabidulin codes, IEEE Trans. Inf. Theory, 57 (2011), 621-632. doi: 10.1109/TIT.2010.2096032.

[32]

D. Silva, Error Control for Network Coding, PhD thesis, University of Toronto, Toronto, Canada, 2009.

[33]

D. SilvaF. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding, IEEE Trans. Inf. Theory, 54 (2008), 3951-3967. doi: 10.1109/TIT.2008.928291.

[34]

V. Skachek, Recursive code construction for random networks, IEEE Trans. Inf. Theory, 56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163.

[35]

A. L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - A new concept in the area of network coding in IEEE Information Theory Workshop 2019 (ITW 2012), 2010. doi: 10.1109/CIG.2010.5592788.

[36]

A.-L. Trautmann and M. Kuijper, Gabidulin decoding via minimal bases of linearized polynomial modules, preprint, arXiv: 1408.2303.

[37]

A.-L. Trautmann, N. Silberstein and J. Rosenthal, List decoding of lifted Gabidulin codes via the Plücker embedding, in Int. Workshop Coding Cryptogr. (WCC), 2013.

[38]

A. Wachter-Zeh, Bounds on list decoding of rank-metric codes, IEEE Trans. Inf. Theory, 59 (2013), 7268-7277. doi: 10.1109/TIT.2013.2274653.

[39]

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

[40]

B. Wang, R. J. McEliece and K. Watanabe, Kötter interpolation over free modules, in Proc. 43rd Annu. Allerton Conf. Comm., Control, and Comp., 2005.

[41]

S. Xia and F. Fu, Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172. doi: 10.1007/s10623-008-9221-7.

[42]

H. Xie, Z. Yan and B. W. Suter, General linearized polynomial interpolation and its applications, in IEEE Int. Symp. Network Coding (Netcod), 2011, 1-4. doi: 10.1109/ISNETCOD.2011.5978942.

Figure 1.  Decoding region for the Kötter-Kschischang [18] decoder $(L = 1)$ and for list decoding a homogeneous interleaved KK subspace code with $(L = 4)$. Both codes have minimum subspace distance ${{d}_{s}} = 8$
Figure 2.  Decoding region for Kötter-Kschischang [18] codes $(L = 1)$ and for probabilistic unique decoding of $(L = 4)$-interleaved KK subspace codes. Both codes have minimum subspace distance ${{d}_{s}} = 8$. The decoding region for insertions increases with the interleaving order $L$
Figure 3.  Simulation results and upper bounds on $P_f$ of the probabilistic-unique decoder with parameters $m = {{n}_{t}} = 6, k = 2, L = 2$
Figure 4.  Multiplications vs. the number of insertions for $L = 4$
Figure 5.  Number of multiplications for root-finding step for different interleaving orders L. The complexity of the recursive GE is independent of $\gamma $
Figure 6.  Memory requirements of the root-finding step for ${{n}_{t}} = 80, k = 60$
Table 1.  Simulation results of the probabilistic-unique decoder with parameters $m = {{n}_{t}} = 6, k = 2$ and $L = 2$
$\gamma + L\delta $ $\gamma $ $\delta $TransmissionsObserved dec. failuresSimulated $P_{f}$
880 $1.2\cdot10^{6}$18749 $1.56\cdot10^{-2}$
61 $4.0\cdot10^{5}$6202 $1.55\cdot10^{-2}$
770 $4.4\cdot10^{6}$1025 $2.33\cdot10^{-4}$
51 $6.0\cdot10^{5}$144 $2.40\cdot10^{-4}$
660 $9.0\cdot10^{6}$21 $2.33\cdot10^{-6}$
41 $3.4\cdot10^{6}$17 $5.00\cdot10^{-6}$
550 $3.2\cdot10^{6}$750 $2.33\cdot10^{-4}$
31 $8.0\cdot10^{5}$184 $2.30\cdot10^{-4}$
440 $4.4\cdot10^{6}$21 $4.77\cdot10^{-6}$
21 $1.6\cdot10^{6}$0 $0$
$\gamma + L\delta $ $\gamma $ $\delta $TransmissionsObserved dec. failuresSimulated $P_{f}$
880 $1.2\cdot10^{6}$18749 $1.56\cdot10^{-2}$
61 $4.0\cdot10^{5}$6202 $1.55\cdot10^{-2}$
770 $4.4\cdot10^{6}$1025 $2.33\cdot10^{-4}$
51 $6.0\cdot10^{5}$144 $2.40\cdot10^{-4}$
660 $9.0\cdot10^{6}$21 $2.33\cdot10^{-6}$
41 $3.4\cdot10^{6}$17 $5.00\cdot10^{-6}$
550 $3.2\cdot10^{6}$750 $2.33\cdot10^{-4}$
31 $8.0\cdot10^{5}$184 $2.30\cdot10^{-4}$
440 $4.4\cdot10^{6}$21 $4.77\cdot10^{-6}$
21 $1.6\cdot10^{6}$0 $0$
Table 2.  Comparison of different decoding schemes for interleaved KK subspace codes
Decoding schemeDecoding regionOp. in ${{\mathbb{F}}_{{{q}^{m}}}}$
Li-Sidorenko-Silva [20,31] $(L+1)t+L\varkappa+L\rho\leq L({{n}_{t}}-k)$ $ \mathcal{O}\left( Ln_{t}^{2} \right) \phantom{{}^{2}}$
Wachter-Zeh-Zeh [39] $(L+1)t+L\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{3}}n_{t}^{2} \right)$
Guruswami-Xing [15] $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{6}}n_{t}^{2} \right)$
Bartz-Meier-Sidorenko [4] $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{3}}n_{t}^{3} \right)$
This contribution $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{4}}n_{t}^{2} \right)$
Decoding schemeDecoding regionOp. in ${{\mathbb{F}}_{{{q}^{m}}}}$
Li-Sidorenko-Silva [20,31] $(L+1)t+L\varkappa+L\rho\leq L({{n}_{t}}-k)$ $ \mathcal{O}\left( Ln_{t}^{2} \right) \phantom{{}^{2}}$
Wachter-Zeh-Zeh [39] $(L+1)t+L\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{3}}n_{t}^{2} \right)$
Guruswami-Xing [15] $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{6}}n_{t}^{2} \right)$
Bartz-Meier-Sidorenko [4] $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{3}}n_{t}^{3} \right)$
This contribution $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{4}}n_{t}^{2} \right)$
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