American Institute of Mathematical Sciences

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May  2013, 7(2): 127-145. doi: 10.3934/amc.2013.7.127

Bounds for projective codes from semidefinite programming

Christine Bachoc 1, , Alberto Passuello 1, and  Frank Vallentin 2,

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

Received  May 2012 Published  May 2013

We apply the semidefinite programming method to derive bounds for projective codes over a finite field.
Keywords: semidefinite programming, bounds., Projective codes.
Mathematics Subject Classification: 94B65, 90C2.
Citation: 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
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.  Google Scholar

[2]

C. Bachoc, Applications of semidefinite programming to coding theory, in "IEEE Information Theory Workshop (ITW),'' 2010. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., (1973), 97 pp.  Google Scholar

[7]

P. Delsarte, Hahn polynomials, discrete harmonics and $t$-designs, SIAM J. Appl. Math., 34 (1978), 157-166.  Google Scholar

[8]

C. F. Dunkl, An addition theorem for some $q$-Hahn polynomials, Monatsh. Math., 85 (1977), 5-37.  Google Scholar

[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.  Google Scholar

[10]

T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.  Google Scholar

[11]

P. Frankl and R. M. Wilson, The Erdős-Ko-Rado theorem for vector spaces, J. Combin. Theory Ser. A, 43 (1986), 228-236.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

A. Khaleghi and F. R. Kschischang, Projective space codes for the injection metric, in "Proc. 11th Canadian Workshop Inform. Theory,'' (2009), 9-12. Google Scholar

[15]

R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.  Google Scholar

[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.  Google Scholar

[17]

F. R. Kschischang and D. Silva, On metrics for error correction in network coding, IEEE Trans. Inform. Theory, 55 (2009), 5479-5490.  Google Scholar

[18]

L. Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory, 25 (1979), 1-5.  Google Scholar

[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. 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), 134-152.  Google Scholar

[21]

A. Schrijver, A comparison of the Delsarte and Lovász bound, IEEE Trans. Inform. Theory, 25 (1979), 425-429.  Google Scholar

[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.  Google Scholar

[23]

M. Schwartz and T. Etzion, Codes and anticodes in the Grassmann graph, J. Combin. Theory Ser. A, 97 (2002), 27-42.  Google Scholar

[24]

M. J. Todd, Semidefinite optimization, Acta Numerica, 10 (2001), 515-560.  Google Scholar

[25]

F. Vallentin, Symmetry in semidefinite programs, Linear Algebra Appl., 430 (2009), 360-369.  Google Scholar

[26]

L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), 49-95.  Google Scholar

[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.  Google Scholar

[28]

S. T. Xia and F. W. Fu, Johnson type bounds on constant dimension codes, Des. Codes Crypt., 50 (2009), 163-172.  Google Scholar

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.  Google Scholar

[2]

C. Bachoc, Applications of semidefinite programming to coding theory, in "IEEE Information Theory Workshop (ITW),'' 2010. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., (1973), 97 pp.  Google Scholar

[7]

P. Delsarte, Hahn polynomials, discrete harmonics and $t$-designs, SIAM J. Appl. Math., 34 (1978), 157-166.  Google Scholar

[8]

C. F. Dunkl, An addition theorem for some $q$-Hahn polynomials, Monatsh. Math., 85 (1977), 5-37.  Google Scholar

[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.  Google Scholar

[10]

T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.  Google Scholar

[11]

P. Frankl and R. M. Wilson, The Erdős-Ko-Rado theorem for vector spaces, J. Combin. Theory Ser. A, 43 (1986), 228-236.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

A. Khaleghi and F. R. Kschischang, Projective space codes for the injection metric, in "Proc. 11th Canadian Workshop Inform. Theory,'' (2009), 9-12. Google Scholar

[15]

R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.  Google Scholar

[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.  Google Scholar

[17]

F. R. Kschischang and D. Silva, On metrics for error correction in network coding, IEEE Trans. Inform. Theory, 55 (2009), 5479-5490.  Google Scholar

[18]

L. Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory, 25 (1979), 1-5.  Google Scholar

[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. 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), 134-152.  Google Scholar

[21]

A. Schrijver, A comparison of the Delsarte and Lovász bound, IEEE Trans. Inform. Theory, 25 (1979), 425-429.  Google Scholar

[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.  Google Scholar

[23]

M. Schwartz and T. Etzion, Codes and anticodes in the Grassmann graph, J. Combin. Theory Ser. A, 97 (2002), 27-42.  Google Scholar

[24]

M. J. Todd, Semidefinite optimization, Acta Numerica, 10 (2001), 515-560.  Google Scholar

[25]

F. Vallentin, Symmetry in semidefinite programs, Linear Algebra Appl., 430 (2009), 360-369.  Google Scholar

[26]

L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), 49-95.  Google Scholar

[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.  Google Scholar

[28]

S. T. Xia and F. W. Fu, Johnson type bounds on constant dimension codes, Des. Codes Crypt., 50 (2009), 163-172.  Google Scholar

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