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Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes
1. | Department of Mathematics, University of Scranton, Scranton, PA 18510, United States |
2. | Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain |
References:
[1] |
E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices, and invariant rings, IEEE Trans. Inform. Theory, 45 (1999), 1194-1205.
doi: 10.1109/18.761269. |
[2] |
M. Bilal, J. Borges, S. T. Dougherty and C. Fernández-Córdoba, Maximum distance separable codes over $\mathbbZ_4$ and $\mathbbZ_2\times\mathbbZ_4$, Designs Codes Crypt., 61 (2011), 31-40.
doi: 10.1007/s10623-010-9437-1. |
[3] |
J. Borges, C. Fernández and J. Rifà, Every $\mathbbZ$2k-code is a binary propelinear code, in "COMB'01. Electronic Notes in Discrete Mathematics,'' 10 (2001), Elsevier Science. |
[4] |
J. Borges, C. Fernández and J. Rifà, Propelinear structure of $\mathbbZ$2k-linear codes,, preprint, ().
|
[5] |
C. Carlet, $\mathbbZ$2k-linear codes, IEEE Trans. Inform. Theory, 44 (1998), 1543-1547.
doi: 10.1109/18.681328. |
[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] |
S. T. Dougherty, M. Harada and P. Solé, Shadow codes over $Z_4$, Finite Fields Appl., 7 (2001), 507-529.
doi: 10.1006/ffta.2000.0312. |
[8] |
S. T. Dougherty and H. Liu, Independence of vectors in codes over rings, Designs Codes Crypt., 51 (2009), 55-68.
doi: 10.1007/s10623-008-9243-1. |
[9] |
C. Fernández-Córdoba, J. Pujol and M. Villanueva, On rank and kernel of $\mathbbZ_4$-linear codes, in "Code Theory and Applications,'' Springer, (2008), 46-55. |
[10] |
C. Fernández-Córdoba, J. Pujol and M. Villanueva, $\mathbbZ_2\mathbbZ_4$-linear codes: rank and kernel, Designs Codes Crypt., 56 (2010), 43-59.
doi: 10.1007/s10623-009-9340-9. |
[11] |
A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbbZ_4$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[12] |
M. Klemm, Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4, Arch. Math., 53 (1989), 201-207.
doi: 10.1007/BF01198572. |
[13] |
F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, 1977. |
[14] |
Y. H. Park, Modular independence and generator matrices for codes over $Z_m$, Designs Codes Crypt., 50 (2009), 147-162.
doi: 10.1007/s10623-008-9220-8. |
[15] |
V. S. Pless, W. C. Huffman and R. A. Brualdi, "Handbook of Coding Theory. I,'' North-Holland, 1998. |
[16] |
E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory'' (edited by V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 177-294. |
[17] |
K. Shiromoto, A basic exact sequence for the Lee and Euclidean weights of linear codes over $\mathbbZ_l$, Linear Algebra Appl. 295 (1999), 191-200.
doi: 10.1016/S0024-3795(99)00125-1. |
[18] |
H. Tapia-Recillas and G. Vega, On the $\mathbbZ_2^k$-linear and quaternary codes, SIAM J. Discrete Math., 17 (2003), 103-113.
doi: 10.1137/S0895480101397219. |
[19] |
Z.-X. Wan, "Quaternary Codes,'' World Scientific, 1997.
doi: 10.1142/9789812798121. |
[20] |
J. Wood, Duality for modules over finite rings and applications to coding theory, American J. Math., 121 (1999), 555-575.
doi: 10.1353/ajm.1999.0024. |
show all references
References:
[1] |
E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices, and invariant rings, IEEE Trans. Inform. Theory, 45 (1999), 1194-1205.
doi: 10.1109/18.761269. |
[2] |
M. Bilal, J. Borges, S. T. Dougherty and C. Fernández-Córdoba, Maximum distance separable codes over $\mathbbZ_4$ and $\mathbbZ_2\times\mathbbZ_4$, Designs Codes Crypt., 61 (2011), 31-40.
doi: 10.1007/s10623-010-9437-1. |
[3] |
J. Borges, C. Fernández and J. Rifà, Every $\mathbbZ$2k-code is a binary propelinear code, in "COMB'01. Electronic Notes in Discrete Mathematics,'' 10 (2001), Elsevier Science. |
[4] |
J. Borges, C. Fernández and J. Rifà, Propelinear structure of $\mathbbZ$2k-linear codes,, preprint, ().
|
[5] |
C. Carlet, $\mathbbZ$2k-linear codes, IEEE Trans. Inform. Theory, 44 (1998), 1543-1547.
doi: 10.1109/18.681328. |
[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] |
S. T. Dougherty, M. Harada and P. Solé, Shadow codes over $Z_4$, Finite Fields Appl., 7 (2001), 507-529.
doi: 10.1006/ffta.2000.0312. |
[8] |
S. T. Dougherty and H. Liu, Independence of vectors in codes over rings, Designs Codes Crypt., 51 (2009), 55-68.
doi: 10.1007/s10623-008-9243-1. |
[9] |
C. Fernández-Córdoba, J. Pujol and M. Villanueva, On rank and kernel of $\mathbbZ_4$-linear codes, in "Code Theory and Applications,'' Springer, (2008), 46-55. |
[10] |
C. Fernández-Córdoba, J. Pujol and M. Villanueva, $\mathbbZ_2\mathbbZ_4$-linear codes: rank and kernel, Designs Codes Crypt., 56 (2010), 43-59.
doi: 10.1007/s10623-009-9340-9. |
[11] |
A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbbZ_4$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[12] |
M. Klemm, Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4, Arch. Math., 53 (1989), 201-207.
doi: 10.1007/BF01198572. |
[13] |
F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, 1977. |
[14] |
Y. H. Park, Modular independence and generator matrices for codes over $Z_m$, Designs Codes Crypt., 50 (2009), 147-162.
doi: 10.1007/s10623-008-9220-8. |
[15] |
V. S. Pless, W. C. Huffman and R. A. Brualdi, "Handbook of Coding Theory. I,'' North-Holland, 1998. |
[16] |
E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory'' (edited by V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 177-294. |
[17] |
K. Shiromoto, A basic exact sequence for the Lee and Euclidean weights of linear codes over $\mathbbZ_l$, Linear Algebra Appl. 295 (1999), 191-200.
doi: 10.1016/S0024-3795(99)00125-1. |
[18] |
H. Tapia-Recillas and G. Vega, On the $\mathbbZ_2^k$-linear and quaternary codes, SIAM J. Discrete Math., 17 (2003), 103-113.
doi: 10.1137/S0895480101397219. |
[19] |
Z.-X. Wan, "Quaternary Codes,'' World Scientific, 1997.
doi: 10.1142/9789812798121. |
[20] |
J. Wood, Duality for modules over finite rings and applications to coding theory, American J. Math., 121 (1999), 555-575.
doi: 10.1353/ajm.1999.0024. |
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