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Designs from maximal subgroups and conjugacy classes of Ree groups
New and updated semidefinite programming bounds for subspace codes
1. | Department of Communications and Networking, Aalto University, Finland |
2. | Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Belgium |
We show that $A_2(7, 4) \leq 388$ and, more generally, $A_q(7, 4) \leq (q^2-q+1) [7] + q^4 - 2q^3 + 3q^2 - 4q + 4$ by semidefinite programming for $q \leq 101$. Furthermore, we extend results by Bachoc et al. on SDP bounds for $A_2(n, d)$, where $d$ is odd and $n$ is small, to $A_q(n, d)$ for small $q$ and small $n$.
References:
[1] |
J. Ai, T. Honold and H. Liu, The expurgation-augmentation method for constructing good plane subspace codes, preprint, arXiv: 1601.01502. |
[2] |
C. Bachoc, A. Passuello and F. Vallentin,
Bounds for projective codes from semidefinite programming, Adv. Math. Commun., 7 (2013), 127-145.
doi: 10.3934/amc.2013.7.127. |
[3] |
A. Beutelspacher,
Partial spreads in finite projective spaces and partial designs, Math. Z., 145 (1975), 211-229.
doi: 10.1007/BF01215286. |
[4] |
M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy and A. Wassermann, Existence of q-analogs of Steiner systems, Forum Math. Pi, 4 (2016), e7, 14pp.
doi: 10.1017/fmp.2016.5. |
[5] |
M. Braun, M. Kiermaier and A. Nakić,
On the automorphism group of a binary q-analog of the Fano plane, European J. Combin., 51 (2016), 443-457.
doi: 10.1016/j.ejc.2015.07.014. |
[6] |
M. Braun and J. Reichelt,
q-analogs of packing designs, J. Combin. Des., 22 (2014), 306-321.
doi: 10.1002/jcd.21376. |
[7] |
A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Results in Mathematics and Related Areas, 18, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-74341-2. |
[8] |
P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., (1973), 97pp. |
[9] |
C. F. Dunkl,
An addition theorem for some q-Hahn polynomials, Monatsh. Math., 85 (1978), 5-37.
doi: 10.1007/BF01300958. |
[10] |
T. Etzion, On the structure of the q-Fano plane, preprint, arXiv: 1508.01839. |
[11] |
T. Etzion, A new approach for examining q-Steiner systems, Electron. J. Combin., 25 (2018), 24pp. |
[12] |
T. Etzion and N. Silberstein,
Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919.
doi: 10.1109/TIT.2009.2021376. |
[13] |
T. Etzion and A. Vardy,
Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.
doi: 10.1109/TIT.2010.2095232. |
[14] |
T. Etzion and A. Vardy,
On q-analogs of Steiner systems and covering designs, Adv. Math. Commun., 5 (2011), 161-176.
doi: 10.3934/amc.2011.5.161. |
[15] |
O. Heden and P. A. Sissokho,
On the existence of a (2, 3)-spread in V(7, 2), Ars Combin., 124 (2016), 161-164.
|
[16] |
D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864. |
[17] |
D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann,
A subspace code of size 333 in the setting of a binary q-analog of the Fano plane, Adv. Math. Commun., 13 (2019), 457-475.
doi: 10.3934/amc.2019029. |
[18] |
D. G. Higman,
Coherent configurations part Ⅰ: Ordinary representation theory, Geometriae Dedicata, 4 (1975), 1-32.
doi: 10.1007/BF00147398. |
[19] |
D. G. Higman,
Coherent configurations part: Ⅱ: Weights, Geometriae Dedicata, 5 (1976), 413-424.
doi: 10.1007/BF00150773. |
[20] |
D. G. Higman,
Coherent algebras, Linear Algebra Appl., 93 (1987), 209-239.
doi: 10.1016/S0024-3795(87)90326-0. |
[21] |
S. A. Hobart,
Bounds on subsets of coherent configurations, Michigan Math. J., 58 (2009), 231-239.
doi: 10.1307/mmj/1242071690. |
[22] |
S. A. Hobart and J. Williford,
Tightness in subset bounds for coherent configurations, J. Algebraic Combin., 39 (2014), 647-658.
doi: 10.1007/s10801-013-0459-4. |
[23] |
T. Honold and M. Kiermaier, On putative q-analogues of the Fano plane and related combinatorial structures, in Dynamical Systems, Number Theory and Applications, World Sci. Publ., Hackensack, NJ, 2016,141–175. |
[24] |
T. Honold, M. Kiermaier and S. Kurz,
Constructions and bounds for mixed-dimension subspace codes, Adv. Math. Commun., 10 (2016), 649-682.
doi: 10.3934/amc.2016033. |
[25] |
T. Honold, M. Kiermaier and S. Kurz, Johnson type bounds for mixed dimension subspace codes, preprint, arXiv: 1808.03580. |
[26] |
M. Kiermaier, S. Kurz and A. Wassermann,
The order of the automorphism group of a binary q-analog of the Fano plane is at most two, Des. Codes Cryptogr., 86 (2018), 239-250.
doi: 10.1007/s10623-017-0360-6. |
[27] |
M. Kiermaier and M. O. Pavčević,
Intersection numbers for subspace designs, J. Combin. Des., 23 (2015), 463-480.
doi: 10.1002/jcd.21403. |
[28] |
A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical Methods in Computer Science, Lecture Notes in Comput. Sci., 5393, Springer, Berlin, 2008, 31–42.
doi: 10.1007/978-3-540-89994-5_4. |
[29] |
R. Kötter and F. R. Kschischang,
Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[30] |
H. Liu and T. Honold, A new approach to the main problem of subspace coding, 9th International Conference on Communications and Networking in China, 2014. Available at arXiv: 1408.1181. |
[31] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, 16, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
[32] |
K. Metsch, Bose-Burton type theorems for finite projective, affine and polar spaces, in
Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., 267, Cambridge Univ. Press,
Cambridge, 1999, 137-166. |
[33] |
M. Miyakawa, A. Munemasa and S. Yoshiara,
On a class of small 2-designs over GF(q), J. Combin. Des., 3 (1995), 61-77.
doi: 10.1002/jcd.3180030108. |
[34] |
A. Schrijver,
New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory, 51 (2005), 2859-2866.
doi: 10.1109/TIT.2005.851748. |
[35] |
S. Thomas,
Designs over finite fields, Geom. Dedicata, 24 (1987), 237-242.
doi: 10.1007/BF00150939. |
[36] |
S. Thomas,
Designs and partial geometries over finite fields, Geom. Dedicata, 63 (1996), 247-253.
doi: 10.1007/BF00181415. |
[37] |
L. Vandenberghe and S. Boyd,
Semidefinite programming, SIAM Rev., 38 (1996), 49-95.
doi: 10.1137/1038003. |
[38] |
Y. Watanabe, An Algebraic Study of Association Schemes and Its Applications, Master's thesis, Tohoku University, 2015. |
[39] |
M. Yamashita, K. Fujisawa, M. Fukuda, K. Kobayashi, K. Nakata and M. Nakata, Latest developments in the SDPA family for solving large-scale SDPs, in Handbook on Semidefinite, Conic and Polynomial Optimization, Internat. Ser. Oper. Res. Management Sci., 166, Springer, New York, 2012,687–713.
doi: 10.1007/978-1-4614-0769-0_24. |
show all references
References:
[1] |
J. Ai, T. Honold and H. Liu, The expurgation-augmentation method for constructing good plane subspace codes, preprint, arXiv: 1601.01502. |
[2] |
C. Bachoc, A. Passuello and F. Vallentin,
Bounds for projective codes from semidefinite programming, Adv. Math. Commun., 7 (2013), 127-145.
doi: 10.3934/amc.2013.7.127. |
[3] |
A. Beutelspacher,
Partial spreads in finite projective spaces and partial designs, Math. Z., 145 (1975), 211-229.
doi: 10.1007/BF01215286. |
[4] |
M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy and A. Wassermann, Existence of q-analogs of Steiner systems, Forum Math. Pi, 4 (2016), e7, 14pp.
doi: 10.1017/fmp.2016.5. |
[5] |
M. Braun, M. Kiermaier and A. Nakić,
On the automorphism group of a binary q-analog of the Fano plane, European J. Combin., 51 (2016), 443-457.
doi: 10.1016/j.ejc.2015.07.014. |
[6] |
M. Braun and J. Reichelt,
q-analogs of packing designs, J. Combin. Des., 22 (2014), 306-321.
doi: 10.1002/jcd.21376. |
[7] |
A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Results in Mathematics and Related Areas, 18, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-74341-2. |
[8] |
P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., (1973), 97pp. |
[9] |
C. F. Dunkl,
An addition theorem for some q-Hahn polynomials, Monatsh. Math., 85 (1978), 5-37.
doi: 10.1007/BF01300958. |
[10] |
T. Etzion, On the structure of the q-Fano plane, preprint, arXiv: 1508.01839. |
[11] |
T. Etzion, A new approach for examining q-Steiner systems, Electron. J. Combin., 25 (2018), 24pp. |
[12] |
T. Etzion and N. Silberstein,
Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919.
doi: 10.1109/TIT.2009.2021376. |
[13] |
T. Etzion and A. Vardy,
Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.
doi: 10.1109/TIT.2010.2095232. |
[14] |
T. Etzion and A. Vardy,
On q-analogs of Steiner systems and covering designs, Adv. Math. Commun., 5 (2011), 161-176.
doi: 10.3934/amc.2011.5.161. |
[15] |
O. Heden and P. A. Sissokho,
On the existence of a (2, 3)-spread in V(7, 2), Ars Combin., 124 (2016), 161-164.
|
[16] |
D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864. |
[17] |
D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann,
A subspace code of size 333 in the setting of a binary q-analog of the Fano plane, Adv. Math. Commun., 13 (2019), 457-475.
doi: 10.3934/amc.2019029. |
[18] |
D. G. Higman,
Coherent configurations part Ⅰ: Ordinary representation theory, Geometriae Dedicata, 4 (1975), 1-32.
doi: 10.1007/BF00147398. |
[19] |
D. G. Higman,
Coherent configurations part: Ⅱ: Weights, Geometriae Dedicata, 5 (1976), 413-424.
doi: 10.1007/BF00150773. |
[20] |
D. G. Higman,
Coherent algebras, Linear Algebra Appl., 93 (1987), 209-239.
doi: 10.1016/S0024-3795(87)90326-0. |
[21] |
S. A. Hobart,
Bounds on subsets of coherent configurations, Michigan Math. J., 58 (2009), 231-239.
doi: 10.1307/mmj/1242071690. |
[22] |
S. A. Hobart and J. Williford,
Tightness in subset bounds for coherent configurations, J. Algebraic Combin., 39 (2014), 647-658.
doi: 10.1007/s10801-013-0459-4. |
[23] |
T. Honold and M. Kiermaier, On putative q-analogues of the Fano plane and related combinatorial structures, in Dynamical Systems, Number Theory and Applications, World Sci. Publ., Hackensack, NJ, 2016,141–175. |
[24] |
T. Honold, M. Kiermaier and S. Kurz,
Constructions and bounds for mixed-dimension subspace codes, Adv. Math. Commun., 10 (2016), 649-682.
doi: 10.3934/amc.2016033. |
[25] |
T. Honold, M. Kiermaier and S. Kurz, Johnson type bounds for mixed dimension subspace codes, preprint, arXiv: 1808.03580. |
[26] |
M. Kiermaier, S. Kurz and A. Wassermann,
The order of the automorphism group of a binary q-analog of the Fano plane is at most two, Des. Codes Cryptogr., 86 (2018), 239-250.
doi: 10.1007/s10623-017-0360-6. |
[27] |
M. Kiermaier and M. O. Pavčević,
Intersection numbers for subspace designs, J. Combin. Des., 23 (2015), 463-480.
doi: 10.1002/jcd.21403. |
[28] |
A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical Methods in Computer Science, Lecture Notes in Comput. Sci., 5393, Springer, Berlin, 2008, 31–42.
doi: 10.1007/978-3-540-89994-5_4. |
[29] |
R. Kötter and F. R. Kschischang,
Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[30] |
H. Liu and T. Honold, A new approach to the main problem of subspace coding, 9th International Conference on Communications and Networking in China, 2014. Available at arXiv: 1408.1181. |
[31] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, 16, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
[32] |
K. Metsch, Bose-Burton type theorems for finite projective, affine and polar spaces, in
Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., 267, Cambridge Univ. Press,
Cambridge, 1999, 137-166. |
[33] |
M. Miyakawa, A. Munemasa and S. Yoshiara,
On a class of small 2-designs over GF(q), J. Combin. Des., 3 (1995), 61-77.
doi: 10.1002/jcd.3180030108. |
[34] |
A. Schrijver,
New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory, 51 (2005), 2859-2866.
doi: 10.1109/TIT.2005.851748. |
[35] |
S. Thomas,
Designs over finite fields, Geom. Dedicata, 24 (1987), 237-242.
doi: 10.1007/BF00150939. |
[36] |
S. Thomas,
Designs and partial geometries over finite fields, Geom. Dedicata, 63 (1996), 247-253.
doi: 10.1007/BF00181415. |
[37] |
L. Vandenberghe and S. Boyd,
Semidefinite programming, SIAM Rev., 38 (1996), 49-95.
doi: 10.1137/1038003. |
[38] |
Y. Watanabe, An Algebraic Study of Association Schemes and Its Applications, Master's thesis, Tohoku University, 2015. |
[39] |
M. Yamashita, K. Fujisawa, M. Fukuda, K. Kobayashi, K. Nakata and M. Nakata, Latest developments in the SDPA family for solving large-scale SDPs, in Handbook on Semidefinite, Conic and Polynomial Optimization, Internat. Ser. Oper. Res. Management Sci., 166, Springer, New York, 2012,687–713.
doi: 10.1007/978-1-4614-0769-0_24. |
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8 | 9 | 10 | 11 | 12 | 13 | 14 | |
3 | 9191 | 107419 | 2531873 | 57201557 | 2685948795 | 119527379616 | 11215665059647 |
4 | 6479 | 53710 | 1705394 | 28600778 | 1816165540 | 59763689822 | 7496516673358 |
5 | 327 | 2458 | 48255 | 660265 | 26309023 | 688127334 | 54724534275 |
6 | 260 | 1240 | 38455 | 330133 | 21362773 | 344063682 | 43890879895 |
7 | 1219 | 8844 | 314104 | 4678401 | 330331546 | ||
8 | 1090 | 4480 | 279476 | 2343888 | 292988615 | ||
9 | 4483 | 34058 | 2298622 | ||||
10 | 4226 | 17133 | 2164452 | ||||
11 | 259 | 17155 | |||||
12 | 16642 |
8 | 9 | 10 | 11 | 12 | 13 | 14 | |
3 | 9191 | 107419 | 2531873 | 57201557 | 2685948795 | 119527379616 | 11215665059647 |
4 | 6479 | 53710 | 1705394 | 28600778 | 1816165540 | 59763689822 | 7496516673358 |
5 | 327 | 2458 | 48255 | 660265 | 26309023 | 688127334 | 54724534275 |
6 | 260 | 1240 | 38455 | 330133 | 21362773 | 344063682 | 43890879895 |
7 | 1219 | 8844 | 314104 | 4678401 | 330331546 | ||
8 | 1090 | 4480 | 279476 | 2343888 | 292988615 | ||
9 | 4483 | 34058 | 2298622 | ||||
10 | 4226 | 17133 | 2164452 | ||||
11 | 259 | 17155 | |||||
12 | 16642 |
6 | 7 | 8 | 9 | 10 | 11 | 12 | |
3 | 967 | 15394 | 760254 | 34143770 | 5026344026 | 675225312722 | 298950313257852 |
4 | 788 | 7696 | 627384 | 17071886 | 4112061519 | 337612656529 | 244829520433920 |
5 | 166 | 7222 | 123535 | 16008007 | 818518696 | 320387589445 | |
6 | 6727 | 61962 | 14893814 | 409259348 | 298571221318 | ||
7 | 490 | 61002 | 1076052 | 400831735 | |||
8 | 59539 | 539351 | 391178436 | ||||
9 | 1462 | 537278 | |||||
10 | 532903 |
6 | 7 | 8 | 9 | 10 | 11 | 12 | |
3 | 967 | 15394 | 760254 | 34143770 | 5026344026 | 675225312722 | 298950313257852 |
4 | 788 | 7696 | 627384 | 17071886 | 4112061519 | 337612656529 | 244829520433920 |
5 | 166 | 7222 | 123535 | 16008007 | 818518696 | 320387589445 | |
6 | 6727 | 61962 | 14893814 | 409259348 | 298571221318 | ||
7 | 490 | 61002 | 1076052 | 400831735 | |||
8 | 59539 | 539351 | 391178436 | ||||
9 | 1462 | 537278 | |||||
10 | 532903 |
6 | 7 | 8 | 9 | 10 | 11 | |
3 | 4772 | 142313 | 20482322 | 2341621613 | 1343547758223 | 614496020025690 |
4 | 4231 | 71156 | 18245203 | 1170810807 | 1194101275238 | 307248010015067 |
5 | 516 | 68117 | 2132181 | 1122729102 | 140323867490 | |
6 | 66054 | 1067796 | 1088550221 | 70161933745 | ||
7 | 2052 | 1058831 | 33669242 | |||
8 | 1050630 | 16847095 | ||||
9 | 8196 |
6 | 7 | 8 | 9 | 10 | 11 | |
3 | 4772 | 142313 | 20482322 | 2341621613 | 1343547758223 | 614496020025690 |
4 | 4231 | 71156 | 18245203 | 1170810807 | 1194101275238 | 307248010015067 |
5 | 516 | 68117 | 2132181 | 1122729102 | 140323867490 | |
6 | 66054 | 1067796 | 1088550221 | 70161933745 | ||
7 | 2052 | 1058831 | 33669242 | |||
8 | 1050630 | 16847095 | ||||
9 | 8196 |
6 | 7 | 8 | 9 | 10 | |
3 | 17179 | 821170 | 277100135 | 64262978412 | 108238287449582 |
4 | 15883 | 410585 | 256754528 | 32131489207 | 100215014898311 |
5 | 1254 | 398154 | 19675409 | 31196584033 | |
6 | 391883 | 9847885 | 30703887393 | ||
7 | 6254 | 9803150 | |||
8 | 9771883 |
6 | 7 | 8 | 9 | 10 | |
3 | 17179 | 821170 | 277100135 | 64262978412 | 108238287449582 |
4 | 15883 | 410585 | 256754528 | 32131489207 | 100215014898311 |
5 | 1254 | 398154 | 19675409 | 31196584033 | |
6 | 391883 | 9847885 | 30703887393 | ||
7 | 6254 | 9803150 | |||
8 | 9771883 |
6 | 7 | 8 | 9 | 10 | |
3 | 123239 | 11807778 | 14753449680 | 9728400942608 | 85039309360944189 |
4 | 118347 | 5903889 | 14176726504 | 4864200471305 | 81703574152063079 |
5 | 4806 | 5803270 | 566262547 | 4784663914039 | |
6 | 5769615 | 283240686 | 4756893963688 | ||
7 | 33618 | 282744208 | |||
8 | 282508875 |
6 | 7 | 8 | 9 | 10 | |
3 | 123239 | 11807778 | 14753449680 | 9728400942608 | 85039309360944189 |
4 | 118347 | 5903889 | 14176726504 | 4864200471305 | 81703574152063079 |
5 | 4806 | 5803270 | 566262547 | 4784663914039 | |
6 | 5769615 | 283240686 | 4756893963688 | ||
7 | 33618 | 282744208 | |||
8 | 282508875 |
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Christine Bachoc, Alberto Passuello, Frank Vallentin. Bounds for projective codes from semidefinite programming. Advances in Mathematics of Communications, 2013, 7 (2) : 127-145. doi: 10.3934/amc.2013.7.127 |
[2] |
Shouhong Yang. Semidefinite programming via image space analysis. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1187-1197. doi: 10.3934/jimo.2016.12.1187 |
[3] |
Antonio Cossidente, Sascha Kurz, Giuseppe Marino, Francesco Pavese. Combining subspace codes. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021007 |
[4] |
Michael Kiermaier, Reinhard Laue. Derived and residual subspace designs. Advances in Mathematics of Communications, 2015, 9 (1) : 105-115. doi: 10.3934/amc.2015.9.105 |
[5] |
Daniel Heinlein, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. A subspace code of size $ \bf{333} $ in the setting of a binary $ \bf{q} $-analog of the Fano plane. Advances in Mathematics of Communications, 2019, 13 (3) : 457-475. doi: 10.3934/amc.2019029 |
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Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565 |
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Thomas Honold, Michael Kiermaier, Sascha Kurz. Constructions and bounds for mixed-dimension subspace codes. Advances in Mathematics of Communications, 2016, 10 (3) : 649-682. doi: 10.3934/amc.2016033 |
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Ghislain Fourier, Gabriele Nebe. Degenerate flag varieties in network coding. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021027 |
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Lori Badea, Marius Cocou. Approximation results and subspace correction algorithms for implicit variational inequalities. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1507-1524. doi: 10.3934/dcdss.2013.6.1507 |
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