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Bounds for projective codes from semidefinite programming
| 1. | University of Bordeaux, Institut de Mathématiques, 351, cours de la Libération, F-33400 Talence, France, France |
| 2. | Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany |
References:
| [1] |
R. Ahlswede, N. Cai, S.-Y. R. Li and R. W. Yeung, Network information flow, IEEE Trans. Inform. Theory, 46 (2000), 1204-1216.
doi: 10.1109/18.850663. |
| [2] |
C. Bachoc, Applications of semidefinite programming to coding theory, in "IEEE Information Theory Workshop (ITW),'' 2010. |
| [3] |
C. Bachoc, D. C. Gijswijt, A. Schrijver and F. Vallentin, Invariant semidefinite programs, in "Handbook on Semidefinite, Conic and Polynomial Optimization'' (eds. M.F. Anjos and J.B. Lasserre), Springer, (2012), 219-269. |
| [4] |
C. Bachoc and F. Vallentin, More semidefinite programming bounds (extended abstract), in "COE Conference on the Development of Dynamic Mathematics with High Functionality,'' Fukuoka, (2007), 129-132. |
| [5] |
C. Bachoc and F. Vallentin, New upper bounds for kissing numbers from semidefinite programming, J. Amer. Math. Soc., 21 (2008), 909-924.
doi: 10.1090/S0894-0347-07-00589-9. |
| [6] |
P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., (1973), 97 pp. |
| [7] |
P. Delsarte, Hahn polynomials, discrete harmonics and $t$-designs, SIAM J. Appl. Math., 34 (1978), 157-166. |
| [8] |
C. F. Dunkl, An addition theorem for some $q$-Hahn polynomials, Monatsh. Math., 85 (1977), 5-37. |
| [9] |
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. |
| [10] |
T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173. |
| [11] |
P. Frankl and R. M. Wilson, The Erdős-Ko-Rado theorem for vector spaces, J. Combin. Theory Ser. A, 43 (1986), 228-236. |
| [12] |
D. C. Gijswijt, H. D. Mittelmann and A. Schrijver, Semidefinite code bounds based on quadruple distances, IEEE Trans. Inform. Theory, 58 (2012), 2697-2705.
doi: 10.1109/TIT.2012.2184845. |
| [13] |
T. Ho, R. Koetter, M. Médard, D. R. Karger and M. Effros, The benefits of coding over routing in a randomized setting, in "Proc. IEEE ISIT'03,'' 2003. |
| [14] |
A. Khaleghi and F. R. Kschischang, Projective space codes for the injection metric, in "Proc. 11th Canadian Workshop Inform. Theory,'' (2009), 9-12. |
| [15] |
R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591. |
| [16] |
A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in "Mathematical Methods in Computer Science,'' Springer, Berlin, (2008), 31-42. |
| [17] |
F. R. Kschischang and D. Silva, On metrics for error correction in network coding, IEEE Trans. Inform. Theory, 55 (2009), 5479-5490. |
| [18] |
L. Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory, 25 (1979), 1-5. |
| [19] |
F. Manganiello, E. Gorla and J. Rosenthal, Spread codes and spread decoding in network coding, in "Proceedings of the 2008 IEEE International Symposium on Information,'' (2008), 851-855. |
| [20] |
R. J. McEliece, E. R. Rodemich and H. C. Rumsey Jr., The Lovász bound and some generalizations, J. Combin. Inform. Sys. Sci., 3 (1978), 134-152. |
| [21] |
A. Schrijver, A comparison of the Delsarte and Lovász bound, IEEE Trans. Inform. Theory, 25 (1979), 425-429. |
| [22] |
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. |
| [23] |
M. Schwartz and T. Etzion, Codes and anticodes in the Grassmann graph, J. Combin. Theory Ser. A, 97 (2002), 27-42. |
| [24] |
M. J. Todd, Semidefinite optimization, Acta Numerica, 10 (2001), 515-560. |
| [25] |
F. Vallentin, Symmetry in semidefinite programs, Linear Algebra Appl., 430 (2009), 360-369. |
| [26] |
L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), 49-95. |
| [27] |
H. Wang, C. Xing and R. Safavi-Naini, Linear authentication codes: bounds and constructions, IEEE Trans. Inform. Theory, 49 (2003), 866-872.
doi: 10.1109/TIT.2003.809567. |
| [28] |
S. T. Xia and F. W. Fu, Johnson type bounds on constant dimension codes, Des. Codes Crypt., 50 (2009), 163-172. |
show all references
References:
| [1] |
R. Ahlswede, N. Cai, S.-Y. R. Li and R. W. Yeung, Network information flow, IEEE Trans. Inform. Theory, 46 (2000), 1204-1216.
doi: 10.1109/18.850663. |
| [2] |
C. Bachoc, Applications of semidefinite programming to coding theory, in "IEEE Information Theory Workshop (ITW),'' 2010. |
| [3] |
C. Bachoc, D. C. Gijswijt, A. Schrijver and F. Vallentin, Invariant semidefinite programs, in "Handbook on Semidefinite, Conic and Polynomial Optimization'' (eds. M.F. Anjos and J.B. Lasserre), Springer, (2012), 219-269. |
| [4] |
C. Bachoc and F. Vallentin, More semidefinite programming bounds (extended abstract), in "COE Conference on the Development of Dynamic Mathematics with High Functionality,'' Fukuoka, (2007), 129-132. |
| [5] |
C. Bachoc and F. Vallentin, New upper bounds for kissing numbers from semidefinite programming, J. Amer. Math. Soc., 21 (2008), 909-924.
doi: 10.1090/S0894-0347-07-00589-9. |
| [6] |
P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., (1973), 97 pp. |
| [7] |
P. Delsarte, Hahn polynomials, discrete harmonics and $t$-designs, SIAM J. Appl. Math., 34 (1978), 157-166. |
| [8] |
C. F. Dunkl, An addition theorem for some $q$-Hahn polynomials, Monatsh. Math., 85 (1977), 5-37. |
| [9] |
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. |
| [10] |
T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173. |
| [11] |
P. Frankl and R. M. Wilson, The Erdős-Ko-Rado theorem for vector spaces, J. Combin. Theory Ser. A, 43 (1986), 228-236. |
| [12] |
D. C. Gijswijt, H. D. Mittelmann and A. Schrijver, Semidefinite code bounds based on quadruple distances, IEEE Trans. Inform. Theory, 58 (2012), 2697-2705.
doi: 10.1109/TIT.2012.2184845. |
| [13] |
T. Ho, R. Koetter, M. Médard, D. R. Karger and M. Effros, The benefits of coding over routing in a randomized setting, in "Proc. IEEE ISIT'03,'' 2003. |
| [14] |
A. Khaleghi and F. R. Kschischang, Projective space codes for the injection metric, in "Proc. 11th Canadian Workshop Inform. Theory,'' (2009), 9-12. |
| [15] |
R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591. |
| [16] |
A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in "Mathematical Methods in Computer Science,'' Springer, Berlin, (2008), 31-42. |
| [17] |
F. R. Kschischang and D. Silva, On metrics for error correction in network coding, IEEE Trans. Inform. Theory, 55 (2009), 5479-5490. |
| [18] |
L. Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory, 25 (1979), 1-5. |
| [19] |
F. Manganiello, E. Gorla and J. Rosenthal, Spread codes and spread decoding in network coding, in "Proceedings of the 2008 IEEE International Symposium on Information,'' (2008), 851-855. |
| [20] |
R. J. McEliece, E. R. Rodemich and H. C. Rumsey Jr., The Lovász bound and some generalizations, J. Combin. Inform. Sys. Sci., 3 (1978), 134-152. |
| [21] |
A. Schrijver, A comparison of the Delsarte and Lovász bound, IEEE Trans. Inform. Theory, 25 (1979), 425-429. |
| [22] |
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. |
| [23] |
M. Schwartz and T. Etzion, Codes and anticodes in the Grassmann graph, J. Combin. Theory Ser. A, 97 (2002), 27-42. |
| [24] |
M. J. Todd, Semidefinite optimization, Acta Numerica, 10 (2001), 515-560. |
| [25] |
F. Vallentin, Symmetry in semidefinite programs, Linear Algebra Appl., 430 (2009), 360-369. |
| [26] |
L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), 49-95. |
| [27] |
H. Wang, C. Xing and R. Safavi-Naini, Linear authentication codes: bounds and constructions, IEEE Trans. Inform. Theory, 49 (2003), 866-872.
doi: 10.1109/TIT.2003.809567. |
| [28] |
S. T. Xia and F. W. Fu, Johnson type bounds on constant dimension codes, Des. Codes Crypt., 50 (2009), 163-172. |
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