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Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights
1. | Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579 |
References:
[1] |
E. F. Assmus, Jr. and V. Pless, On the covering radius of extremal self-dual codes, IEEE Trans. Inform. Theory, 29 (1983), 359-363.
doi: 10.1109/TIT.1983.1056681. |
[2] |
C. Bachoc and P. Gaborit, Designs and self-dual codes with long shadows, J. Combin. Theory Ser. A, 105 (2004), 15-34.
doi: 10.1016/j.jcta.2003.09.003. |
[3] |
A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over $\mathbbZ_4$, IEEE Trans. Inform. Theory, 43 (1997), 969-976.
doi: 10.1109/18.568705. |
[4] |
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[5] |
J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.
doi: 10.1109/18.59931. |
[6] |
J. H. Conway and N. J. A. Sloane, Self-dual codes over the integers modulo $4$, J. Combin. Theory Ser. A, 62 (1993), 30-45.
doi: 10.1016/0097-3165(93)90070-O. |
[7] |
T. A. Gulliver and M. Harada, Certain self-dual codes over $\ZZ_4$ and the odd Leech lattice, in Int. Symp. Appl. Algebra Algebr. Algor. Error-Correcting Codes, Springer, Berlin, 1997, 130-137.
doi: 10.1007/3-540-63163-1_10. |
[8] |
A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\ZZ_4$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[9] |
M. Harada, Extremal type II $\mathbbZ_4$-codes of lengths $56$ and $64$, J. Combin. Theory Ser. A, 117 (2010), 1285-1288.
doi: 10.1016/j.jcta.2009.09.003. |
[10] |
M. Kiermaier, There is no self-dual $\ZZ_4$-linear code whose Gray image has the parameters $(72,2^{36},16)$, IEEE Trans. Inform. Theory, 59 (2013), 3384-3386.
doi: 10.1109/TIT.2013.2246816. |
[11] |
M. Kiermaier and A. Wassermann, Double and bordered $\alpha$-circulant self-dual codes over finite commutative chain rings, in Proc. 7th Int. Workshop Alg. Combin. Coding Theory, Pamporovo, 2008, 144-150. |
[12] |
M. Kiermaier and A. Wassermann, Minimum weights and weight enumerators of $\ZZ_4$-linear quadratic residue codes, IEEE Trans. Inform. Theory, 58 (2012), 4870-4883.
doi: 10.1109/TIT.2012.2191389. |
[13] |
F. J. MacWilliams, N. J. A. Sloane and J. G. Thompson, Good self dual codes exist, Discrete Math., 3 (1972), 153-162. |
[14] |
C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200. |
[15] |
V. Pless, J. Leon and J. Fields, All $\ZZ_4$ codes of Type II and length 16 are known, J. Combin. Theory Ser. A, 78 (1997), 32-50.
doi: 10.1006/jcta.1996.2750. |
[16] |
E. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139.
doi: 10.1109/18.651000. |
[17] |
E. Rains, Optimal self-dual codes over $\ZZ_4$, Discrete Math., 203 (1999), 215-228.
doi: 10.1016/S0012-365X(98)00358-6. |
[18] |
E. Rains, Bounds for self-dual codes over $\ZZ_4$, Finite Fields Appl., 6 (2000), 146-163.
doi: 10.1006/ffta.1999.0258. |
[19] |
E. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, 1998, 177-294. |
[20] |
E. Rains and N. J. A. Sloane, The shadow theory of modular and unimodular lattices, J. Number Theory, 73 (1998), 359-389.
doi: 10.1006/jnth.1998.2306. |
[21] |
S. Zhang, On the nonexistence of extremal self-dual codes, Discrete Appl. Math., 91 (1999), 277-286.
doi: 10.1016/S0166-218X(98)00131-0. |
show all references
References:
[1] |
E. F. Assmus, Jr. and V. Pless, On the covering radius of extremal self-dual codes, IEEE Trans. Inform. Theory, 29 (1983), 359-363.
doi: 10.1109/TIT.1983.1056681. |
[2] |
C. Bachoc and P. Gaborit, Designs and self-dual codes with long shadows, J. Combin. Theory Ser. A, 105 (2004), 15-34.
doi: 10.1016/j.jcta.2003.09.003. |
[3] |
A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over $\mathbbZ_4$, IEEE Trans. Inform. Theory, 43 (1997), 969-976.
doi: 10.1109/18.568705. |
[4] |
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[5] |
J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.
doi: 10.1109/18.59931. |
[6] |
J. H. Conway and N. J. A. Sloane, Self-dual codes over the integers modulo $4$, J. Combin. Theory Ser. A, 62 (1993), 30-45.
doi: 10.1016/0097-3165(93)90070-O. |
[7] |
T. A. Gulliver and M. Harada, Certain self-dual codes over $\ZZ_4$ and the odd Leech lattice, in Int. Symp. Appl. Algebra Algebr. Algor. Error-Correcting Codes, Springer, Berlin, 1997, 130-137.
doi: 10.1007/3-540-63163-1_10. |
[8] |
A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\ZZ_4$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[9] |
M. Harada, Extremal type II $\mathbbZ_4$-codes of lengths $56$ and $64$, J. Combin. Theory Ser. A, 117 (2010), 1285-1288.
doi: 10.1016/j.jcta.2009.09.003. |
[10] |
M. Kiermaier, There is no self-dual $\ZZ_4$-linear code whose Gray image has the parameters $(72,2^{36},16)$, IEEE Trans. Inform. Theory, 59 (2013), 3384-3386.
doi: 10.1109/TIT.2013.2246816. |
[11] |
M. Kiermaier and A. Wassermann, Double and bordered $\alpha$-circulant self-dual codes over finite commutative chain rings, in Proc. 7th Int. Workshop Alg. Combin. Coding Theory, Pamporovo, 2008, 144-150. |
[12] |
M. Kiermaier and A. Wassermann, Minimum weights and weight enumerators of $\ZZ_4$-linear quadratic residue codes, IEEE Trans. Inform. Theory, 58 (2012), 4870-4883.
doi: 10.1109/TIT.2012.2191389. |
[13] |
F. J. MacWilliams, N. J. A. Sloane and J. G. Thompson, Good self dual codes exist, Discrete Math., 3 (1972), 153-162. |
[14] |
C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200. |
[15] |
V. Pless, J. Leon and J. Fields, All $\ZZ_4$ codes of Type II and length 16 are known, J. Combin. Theory Ser. A, 78 (1997), 32-50.
doi: 10.1006/jcta.1996.2750. |
[16] |
E. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139.
doi: 10.1109/18.651000. |
[17] |
E. Rains, Optimal self-dual codes over $\ZZ_4$, Discrete Math., 203 (1999), 215-228.
doi: 10.1016/S0012-365X(98)00358-6. |
[18] |
E. Rains, Bounds for self-dual codes over $\ZZ_4$, Finite Fields Appl., 6 (2000), 146-163.
doi: 10.1006/ffta.1999.0258. |
[19] |
E. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, 1998, 177-294. |
[20] |
E. Rains and N. J. A. Sloane, The shadow theory of modular and unimodular lattices, J. Number Theory, 73 (1998), 359-389.
doi: 10.1006/jnth.1998.2306. |
[21] |
S. Zhang, On the nonexistence of extremal self-dual codes, Discrete Appl. Math., 91 (1999), 277-286.
doi: 10.1016/S0166-218X(98)00131-0. |
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