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Input-state-output representations and constructions of finite support 2D convolutional codes
Bounds for binary codes relative to pseudo-distances of $k$ points
1. | Université Bordeaux I, Institut de Mathématiques, 351, cours de la Libération, 33405 Talence, France, France |
References:
[1] |
A. Ashikhmin, A. Barg and S. Litsyn, New upper bounds on generalized weights, IEEE Trans. Inform. Theory, IT-45 (1999), 1258-1263.
doi: 10.1109/18.761280. |
[2] |
L. A. Bassalygo, Supports of a code, in "Proc. AAECC 11,'' (1995), 1-3. |
[3] |
C. Bachoc, Semidefinite programming, harmonic analysis and coding theory, Lecture notes of a CIMPA course, 2009, arXiv:0909.4767 |
[4] |
C. Bachoc, D. Gijswijt, A. Schrijver and F. Vallentin, Invariant semidefinite programs,, preprint, ().
|
[5] |
V. M. Blinovskii, Bounds for codes in the case of list decoding of finite volume, Problems of Information Transmission, 22 (1986), 7-19. |
[6] |
V. M. Blinovskii, Generalization of Plotkin bound to the case of multiple packing, in "ISIT 2009''. |
[7] |
G. Cohen, S. Litsyn and G. Zémor, Upper bounds on generalized Hamming distances, IEEE Trans. Inform. Theory, 40 (1994), 2090-2092.
doi: 10.1109/18.340487. |
[8] |
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups,'' Springer-Verlag, 1988. |
[9] |
P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., (1973), vi+97. |
[10] |
P. Delsarte, Hahn polynomials, discrete harmonics and $t$-designs, SIAM J. Appl. Math., 34 (1978), 157-166.
doi: 10.1137/0134012. |
[11] |
V. Guruswami, List decoding from erasures: bounds and code constructions, IEEE Trans. Inform. Theory, IT-49 (2003), 2826-2833.
doi: 10.1109/TIT.2003.815776. |
[12] |
V. I. Levenshtein, Universal bounds for codes and designs, in "Handbook of Coding Theory'' (eds. V. Pless and W.C. Huffmann), North-Holland, Amsterdam, (1998), 499-648. |
[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), 157-166.
doi: 10.1109/TIT.1977.1055688. |
[14] |
L. H. Ozarow and A. D. Wyner, Wire-tap channel II, in "Advances in cryptology (Paris, 1984),'' Springer, Berlin, (1985), 33-50. |
[15] |
A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory, IT-51 (2005), 2859-2866.
doi: 10.1109/TIT.2005.851748. |
[16] |
M. Sudan, Decoding of Reed Solomon codes beyond the error-correction bound, Journal of Complexity, 13 (1997), 180-193.
doi: 10.1006/jcom.1997.0439. |
[17] |
F. Vallentin, Lecture notes: Semidefinite programs and harmonic analysis,, preprint, ().
|
[18] |
F. Vallentin, Symmetry in semidefinite programs, Linear Algebra and Appl., 430 (2009), 360-369.
doi: 10.1016/j.laa.2008.07.025. |
[19] |
V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, IT-37 (1991), 1412-1418.
doi: 10.1109/18.133259. |
[20] |
G. Zémor, Threshold effects in codes, in "Algebraic coding (Paris, 1993),'' |
[21] |
G. Zémor and G. Cohen, The threshold probability of a code, IEEE Trans. Inform. Theory, IT-41 (1995), 469-477.
doi: 10.1109/18.370148. |
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), 1258-1263.
doi: 10.1109/18.761280. |
[2] |
L. A. Bassalygo, Supports of a code, in "Proc. AAECC 11,'' (1995), 1-3. |
[3] |
C. Bachoc, Semidefinite programming, harmonic analysis and coding theory, Lecture notes of a CIMPA course, 2009, arXiv:0909.4767 |
[4] |
C. Bachoc, D. Gijswijt, A. Schrijver and F. Vallentin, Invariant semidefinite programs,, preprint, ().
|
[5] |
V. M. Blinovskii, Bounds for codes in the case of list decoding of finite volume, Problems of Information Transmission, 22 (1986), 7-19. |
[6] |
V. M. Blinovskii, Generalization of Plotkin bound to the case of multiple packing, in "ISIT 2009''. |
[7] |
G. Cohen, S. Litsyn and G. Zémor, Upper bounds on generalized Hamming distances, IEEE Trans. Inform. Theory, 40 (1994), 2090-2092.
doi: 10.1109/18.340487. |
[8] |
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups,'' Springer-Verlag, 1988. |
[9] |
P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., (1973), vi+97. |
[10] |
P. Delsarte, Hahn polynomials, discrete harmonics and $t$-designs, SIAM J. Appl. Math., 34 (1978), 157-166.
doi: 10.1137/0134012. |
[11] |
V. Guruswami, List decoding from erasures: bounds and code constructions, IEEE Trans. Inform. Theory, IT-49 (2003), 2826-2833.
doi: 10.1109/TIT.2003.815776. |
[12] |
V. I. Levenshtein, Universal bounds for codes and designs, in "Handbook of Coding Theory'' (eds. V. Pless and W.C. Huffmann), North-Holland, Amsterdam, (1998), 499-648. |
[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), 157-166.
doi: 10.1109/TIT.1977.1055688. |
[14] |
L. H. Ozarow and A. D. Wyner, Wire-tap channel II, in "Advances in cryptology (Paris, 1984),'' Springer, Berlin, (1985), 33-50. |
[15] |
A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory, IT-51 (2005), 2859-2866.
doi: 10.1109/TIT.2005.851748. |
[16] |
M. Sudan, Decoding of Reed Solomon codes beyond the error-correction bound, Journal of Complexity, 13 (1997), 180-193.
doi: 10.1006/jcom.1997.0439. |
[17] |
F. Vallentin, Lecture notes: Semidefinite programs and harmonic analysis,, preprint, ().
|
[18] |
F. Vallentin, Symmetry in semidefinite programs, Linear Algebra and Appl., 430 (2009), 360-369.
doi: 10.1016/j.laa.2008.07.025. |
[19] |
V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, IT-37 (1991), 1412-1418.
doi: 10.1109/18.133259. |
[20] |
G. Zémor, Threshold effects in codes, in "Algebraic coding (Paris, 1993),'' |
[21] |
G. Zémor and G. Cohen, The threshold probability of a code, IEEE Trans. Inform. Theory, IT-41 (1995), 469-477.
doi: 10.1109/18.370148. |
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