American Institute of Mathematical Sciences

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November  2010, 4(4): 547-565. doi: 10.3934/amc.2010.4.547

Bounds for binary codes relative to pseudo-distances of $k$ points

Christine Bachoc 1, and  Gilles Zémor 1,

1. 

Université Bordeaux I, Institut de Mathématiques, 351, cours de la Libération, 33405 Talence, France, France

Received  February 2010 Revised  July 2010 Published  November 2010

We apply Schrijver's semidefinite programming method to obtain improved upper bounds on generalized distances and list decoding radii of binary codes.
Keywords: Binary codes, pseudo-distances., semidefinite programming.
Mathematics Subject Classification: 94B65, 90C2.
Citation: Christine Bachoc, Gilles Zémor. Bounds for binary codes relative to pseudo-distances of $k$ points. Advances in Mathematics of Communications, 2010, 4 (4) : 547-565. doi: 10.3934/amc.2010.4.547
References:
[1]

A. Ashikhmin, A. Barg and S. Litsyn, New upper bounds on generalized weights,, IEEE Trans. Inform. Theory, IT-45 (1999), 1258.  doi: 10.1109/18.761280.  Google Scholar

[2]

L. A. Bassalygo, Supports of a code,, in, (1995), 1.   Google Scholar

[3]

C. Bachoc, Semidefinite programming, harmonic analysis and coding theory,, Lecture notes of a CIMPA course, (2009).   Google Scholar

[4]

C. Bachoc, D. Gijswijt, A. Schrijver and F. Vallentin, Invariant semidefinite programs,, preprint, ().   Google Scholar

[5]

V. M. Blinovskii, Bounds for codes in the case of list decoding of finite volume,, Problems of Information Transmission, 22 (1986), 7.   Google Scholar

[6]

V. M. Blinovskii, Generalization of Plotkin bound to the case of multiple packing,, in, (2009).   Google Scholar

[7]

G. Cohen, S. Litsyn and G. Zémor, Upper bounds on generalized Hamming distances,, IEEE Trans. Inform. Theory, 40 (1994), 2090.  doi: 10.1109/18.340487.  Google Scholar

[8]

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups,'', Springer-Verlag, (1988).   Google Scholar

[9]

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

[10]

P. Delsarte, Hahn polynomials, discrete harmonics and $t$-designs,, SIAM J. Appl. Math., 34 (1978), 157.  doi: 10.1137/0134012.  Google Scholar

[11]

V. Guruswami, List decoding from erasures: bounds and code constructions,, IEEE Trans. Inform. Theory, IT-49 (2003), 2826.  doi: 10.1109/TIT.2003.815776.  Google Scholar

[12]

V. I. Levenshtein, Universal bounds for codes and designs,, in, (1998), 499.   Google Scholar

[13]

R. J. McEliece, E. R. Rodemich, H. Rumsey and L. Welch, New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities,, IEEE Trans. Inform. Theory, IT-23 (1977), 157.  doi: 10.1109/TIT.1977.1055688.  Google Scholar

[14]

L. H. Ozarow and A. D. Wyner, Wire-tap channel II,, in, (1985), 33.   Google Scholar

[15]

A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming,, IEEE Trans. Inform. Theory, IT-51 (2005), 2859.  doi: 10.1109/TIT.2005.851748.  Google Scholar

[16]

M. Sudan, Decoding of Reed Solomon codes beyond the error-correction bound,, Journal of Complexity, 13 (1997), 180.  doi: 10.1006/jcom.1997.0439.  Google Scholar

[17]

F. Vallentin, Lecture notes: Semidefinite programs and harmonic analysis,, preprint, ().   Google Scholar

[18]

F. Vallentin, Symmetry in semidefinite programs,, Linear Algebra and Appl., 430 (2009), 360.  doi: 10.1016/j.laa.2008.07.025.  Google Scholar

[19]

V. K. Wei, Generalized Hamming weights for linear codes,, IEEE Trans. Inform. Theory, IT-37 (1991), 1412.  doi: 10.1109/18.133259.  Google Scholar

[20]

G. Zémor, Threshold effects in codes,, in, (1993).   Google Scholar

[21]

G. Zémor and G. Cohen, The threshold probability of a code,, IEEE Trans. Inform. Theory, IT-41 (1995), 469.  doi: 10.1109/18.370148.  Google Scholar

show all references

References:
[1]

A. Ashikhmin, A. Barg and S. Litsyn, New upper bounds on generalized weights,, IEEE Trans. Inform. Theory, IT-45 (1999), 1258.  doi: 10.1109/18.761280.  Google Scholar

[2]

L. A. Bassalygo, Supports of a code,, in, (1995), 1.   Google Scholar

[3]

C. Bachoc, Semidefinite programming, harmonic analysis and coding theory,, Lecture notes of a CIMPA course, (2009).   Google Scholar

[4]

C. Bachoc, D. Gijswijt, A. Schrijver and F. Vallentin, Invariant semidefinite programs,, preprint, ().   Google Scholar

[5]

V. M. Blinovskii, Bounds for codes in the case of list decoding of finite volume,, Problems of Information Transmission, 22 (1986), 7.   Google Scholar

[6]

V. M. Blinovskii, Generalization of Plotkin bound to the case of multiple packing,, in, (2009).   Google Scholar

[7]

G. Cohen, S. Litsyn and G. Zémor, Upper bounds on generalized Hamming distances,, IEEE Trans. Inform. Theory, 40 (1994), 2090.  doi: 10.1109/18.340487.  Google Scholar

[8]

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups,'', Springer-Verlag, (1988).   Google Scholar

[9]

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

[10]

P. Delsarte, Hahn polynomials, discrete harmonics and $t$-designs,, SIAM J. Appl. Math., 34 (1978), 157.  doi: 10.1137/0134012.  Google Scholar

[11]

V. Guruswami, List decoding from erasures: bounds and code constructions,, IEEE Trans. Inform. Theory, IT-49 (2003), 2826.  doi: 10.1109/TIT.2003.815776.  Google Scholar

[12]

V. I. Levenshtein, Universal bounds for codes and designs,, in, (1998), 499.   Google Scholar

[13]

R. J. McEliece, E. R. Rodemich, H. Rumsey and L. Welch, New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities,, IEEE Trans. Inform. Theory, IT-23 (1977), 157.  doi: 10.1109/TIT.1977.1055688.  Google Scholar

[14]

L. H. Ozarow and A. D. Wyner, Wire-tap channel II,, in, (1985), 33.   Google Scholar

[15]

A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming,, IEEE Trans. Inform. Theory, IT-51 (2005), 2859.  doi: 10.1109/TIT.2005.851748.  Google Scholar

[16]

M. Sudan, Decoding of Reed Solomon codes beyond the error-correction bound,, Journal of Complexity, 13 (1997), 180.  doi: 10.1006/jcom.1997.0439.  Google Scholar

[17]

F. Vallentin, Lecture notes: Semidefinite programs and harmonic analysis,, preprint, ().   Google Scholar

[18]

F. Vallentin, Symmetry in semidefinite programs,, Linear Algebra and Appl., 430 (2009), 360.  doi: 10.1016/j.laa.2008.07.025.  Google Scholar

[19]

V. K. Wei, Generalized Hamming weights for linear codes,, IEEE Trans. Inform. Theory, IT-37 (1991), 1412.  doi: 10.1109/18.133259.  Google Scholar

[20]

G. Zémor, Threshold effects in codes,, in, (1993).   Google Scholar

[21]

G. Zémor and G. Cohen, The threshold probability of a code,, IEEE Trans. Inform. Theory, IT-41 (1995), 469.  doi: 10.1109/18.370148.  Google Scholar

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