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Isometry and automorphisms of constant dimension codes
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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.
doi: 10.1109/18.850663. |
[2] |
C. Bachoc, Applications of semidefinite programming to coding theory,, in, (2010). Google Scholar |
[3] |
C. Bachoc, D. C. Gijswijt, A. Schrijver and F. Vallentin, Invariant semidefinite programs,, in, (2012), 219.
|
[4] |
C. Bachoc and F. Vallentin, More semidefinite programming bounds (extended abstract),, in, (2007), 129. Google Scholar |
[5] |
C. Bachoc and F. Vallentin, New upper bounds for kissing numbers from semidefinite programming,, J. Amer. Math. Soc., 21 (2008), 909.
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).
|
[7] |
P. Delsarte, Hahn polynomials, discrete harmonics and $t$-designs,, SIAM J. Appl. Math., 34 (1978), 157.
|
[8] |
C. F. Dunkl, An addition theorem for some $q$-Hahn polynomials,, Monatsh. Math., 85 (1977), 5.
|
[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.
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.
|
[11] |
P. Frankl and R. M. Wilson, The Erdős-Ko-Rado theorem for vector spaces,, J. Combin. Theory Ser. A, 43 (1986), 228.
|
[12] |
D. C. Gijswijt, H. D. Mittelmann and A. Schrijver, Semidefinite code bounds based on quadruple distances,, IEEE Trans. Inform. Theory, 58 (2012), 2697.
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, (2003). Google Scholar |
[14] |
A. Khaleghi and F. R. Kschischang, Projective space codes for the injection metric,, in, (2009), 9. Google Scholar |
[15] |
R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding,, IEEE Trans. Inform. Theory, 54 (2008), 3579.
|
[16] |
A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance,, in, (2008), 31.
|
[17] |
F. R. Kschischang and D. Silva, On metrics for error correction in network coding,, IEEE Trans. Inform. Theory, 55 (2009), 5479.
|
[18] |
L. Lovász, On the Shannon capacity of a graph,, IEEE Trans. Inform. Theory, 25 (1979), 1.
|
[19] |
F. Manganiello, E. Gorla and J. Rosenthal, Spread codes and spread decoding in network coding,, in, (2008), 851. Google Scholar |
[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.
|
[21] |
A. Schrijver, A comparison of the Delsarte and Lovász bound,, IEEE Trans. Inform. Theory, 25 (1979), 425.
|
[22] |
A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming,, IEEE Trans. Inform. Theory, 51 (2005), 2859.
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.
|
[24] |
M. J. Todd, Semidefinite optimization,, Acta Numerica, 10 (2001), 515.
|
[25] |
F. Vallentin, Symmetry in semidefinite programs,, Linear Algebra Appl., 430 (2009), 360.
|
[26] |
L. Vandenberghe and S. Boyd, Semidefinite programming,, SIAM Rev., 38 (1996), 49.
|
[27] |
H. Wang, C. Xing and R. Safavi-Naini, Linear authentication codes: bounds and constructions,, IEEE Trans. Inform. Theory, 49 (2003), 866.
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.
|
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.
doi: 10.1109/18.850663. |
[2] |
C. Bachoc, Applications of semidefinite programming to coding theory,, in, (2010). Google Scholar |
[3] |
C. Bachoc, D. C. Gijswijt, A. Schrijver and F. Vallentin, Invariant semidefinite programs,, in, (2012), 219.
|
[4] |
C. Bachoc and F. Vallentin, More semidefinite programming bounds (extended abstract),, in, (2007), 129. Google Scholar |
[5] |
C. Bachoc and F. Vallentin, New upper bounds for kissing numbers from semidefinite programming,, J. Amer. Math. Soc., 21 (2008), 909.
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).
|
[7] |
P. Delsarte, Hahn polynomials, discrete harmonics and $t$-designs,, SIAM J. Appl. Math., 34 (1978), 157.
|
[8] |
C. F. Dunkl, An addition theorem for some $q$-Hahn polynomials,, Monatsh. Math., 85 (1977), 5.
|
[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.
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.
|
[11] |
P. Frankl and R. M. Wilson, The Erdős-Ko-Rado theorem for vector spaces,, J. Combin. Theory Ser. A, 43 (1986), 228.
|
[12] |
D. C. Gijswijt, H. D. Mittelmann and A. Schrijver, Semidefinite code bounds based on quadruple distances,, IEEE Trans. Inform. Theory, 58 (2012), 2697.
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, (2003). Google Scholar |
[14] |
A. Khaleghi and F. R. Kschischang, Projective space codes for the injection metric,, in, (2009), 9. Google Scholar |
[15] |
R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding,, IEEE Trans. Inform. Theory, 54 (2008), 3579.
|
[16] |
A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance,, in, (2008), 31.
|
[17] |
F. R. Kschischang and D. Silva, On metrics for error correction in network coding,, IEEE Trans. Inform. Theory, 55 (2009), 5479.
|
[18] |
L. Lovász, On the Shannon capacity of a graph,, IEEE Trans. Inform. Theory, 25 (1979), 1.
|
[19] |
F. Manganiello, E. Gorla and J. Rosenthal, Spread codes and spread decoding in network coding,, in, (2008), 851. Google Scholar |
[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.
|
[21] |
A. Schrijver, A comparison of the Delsarte and Lovász bound,, IEEE Trans. Inform. Theory, 25 (1979), 425.
|
[22] |
A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming,, IEEE Trans. Inform. Theory, 51 (2005), 2859.
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.
|
[24] |
M. J. Todd, Semidefinite optimization,, Acta Numerica, 10 (2001), 515.
|
[25] |
F. Vallentin, Symmetry in semidefinite programs,, Linear Algebra Appl., 430 (2009), 360.
|
[26] |
L. Vandenberghe and S. Boyd, Semidefinite programming,, SIAM Rev., 38 (1996), 49.
|
[27] |
H. Wang, C. Xing and R. Safavi-Naini, Linear authentication codes: bounds and constructions,, IEEE Trans. Inform. Theory, 49 (2003), 866.
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.
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