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Higher weights and near-MDR codes over chain rings
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 |
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.
References:
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W. W. Adams and P. Loustaunau,
An Introduction to Gröbner Bases, 3, American Mathematical Soc., 1994.
doi: 10.1090/gsm/003. |
[2] |
C. Bachoc, F. 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. Google Scholar |
[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. Google Scholar |
[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. Etzion, E. Gorla, A. 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. Google Scholar |
[20] |
W. Li, V. 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. Google Scholar |
[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. Google Scholar |
[25] |
J. N. Nielsen, List Decoding of Algebraic Codes, PhD thesis, 2013. Google Scholar |
[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. Sidorenko, L. 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. Google Scholar |
[33] |
D. Silva, F. 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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Bachoc, F. 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. Google Scholar |
[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. Google Scholar |
[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. Etzion, E. Gorla, A. 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. Google Scholar |
[20] |
W. Li, V. 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. Google Scholar |
[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. Google Scholar |
[25] |
J. N. Nielsen, List Decoding of Algebraic Codes, PhD thesis, 2013. Google Scholar |
[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. Sidorenko, L. 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. Google Scholar |
[33] |
D. Silva, F. 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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |




| | Transmissions | Observed dec. failures | Simulated | |
8 | 8 | 0 | | 18749 | |
6 | 1 | | 6202 | | |
7 | 7 | 0 | | 1025 | |
5 | 1 | | 144 | | |
6 | 6 | 0 | | 21 | |
4 | 1 | | 17 | | |
5 | 5 | 0 | | 750 | |
3 | 1 | | 184 | | |
4 | 4 | 0 | | 21 | |
2 | 1 | | 0 | |
| | Transmissions | Observed dec. failures | Simulated | |
8 | 8 | 0 | | 18749 | |
6 | 1 | | 6202 | | |
7 | 7 | 0 | | 1025 | |
5 | 1 | | 144 | | |
6 | 6 | 0 | | 21 | |
4 | 1 | | 17 | | |
5 | 5 | 0 | | 750 | |
3 | 1 | | 184 | | |
4 | 4 | 0 | | 21 | |
2 | 1 | | 0 | |
Decoding scheme | Decoding region | Op. in |
Li-Sidorenko-Silva [20,31] | | |
Wachter-Zeh-Zeh [39] | | |
Guruswami-Xing [15] | | |
Bartz-Meier-Sidorenko [4] | | |
This contribution | | |
Decoding scheme | Decoding region | Op. in |
Li-Sidorenko-Silva [20,31] | | |
Wachter-Zeh-Zeh [39] | | |
Guruswami-Xing [15] | | |
Bartz-Meier-Sidorenko [4] | | |
This contribution | | |
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