# American Institute of Mathematical Sciences

November  2011, 5(4): 571-588. doi: 10.3934/amc.2011.5.571

## 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

Received  March 2010 Revised  October 2011 Published  November 2011

The generalized Gray map is defined for codes over $\mathbb{Z}_{2^k}$. We give bounds for the dimension of the kernel and the rank of the image of a code over $\mathbb{Z}_{2^k}$ with a given type and show that there exists such a code for each dimension in the interval for the kernel. We determine when the Gray image of a code over $\mathbb{Z}_{2^k}$ generates a linear self-dual code and give families of codes whose image generate binary self-dual codes. We investigate the Gray image of quaternary self-dual codes and examine when the Gray image of a self-dual code over $\mathbb{Z}_4$ is a binary self-dual code.
Citation: Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571
##### 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.  Google Scholar [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.  Google Scholar [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. Google Scholar [4] J. Borges, C. Fernández and J. Rifà, Propelinear structure of $\mathbbZ$2k-linear codes,, preprint, ().   Google Scholar [5] C. Carlet, $\mathbbZ$2k-linear codes, IEEE Trans. Inform. Theory, 44 (1998), 1543-1547. doi: 10.1109/18.681328.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [12] M. Klemm, Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4, Arch. Math., 53 (1989), 201-207. doi: 10.1007/BF01198572.  Google Scholar [13] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, 1977. Google Scholar [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.  Google Scholar [15] V. S. Pless, W. C. Huffman and R. A. Brualdi, "Handbook of Coding Theory. I,'' North-Holland, 1998.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] Z.-X. Wan, "Quaternary Codes,'' World Scientific, 1997. doi: 10.1142/9789812798121.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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. Google Scholar [4] J. Borges, C. Fernández and J. Rifà, Propelinear structure of $\mathbbZ$2k-linear codes,, preprint, ().   Google Scholar [5] C. Carlet, $\mathbbZ$2k-linear codes, IEEE Trans. Inform. Theory, 44 (1998), 1543-1547. doi: 10.1109/18.681328.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [12] M. Klemm, Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4, Arch. Math., 53 (1989), 201-207. doi: 10.1007/BF01198572.  Google Scholar [13] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, 1977. Google Scholar [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.  Google Scholar [15] V. S. Pless, W. C. Huffman and R. A. Brualdi, "Handbook of Coding Theory. I,'' North-Holland, 1998.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] Z.-X. Wan, "Quaternary Codes,'' World Scientific, 1997. doi: 10.1142/9789812798121.  Google Scholar [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.  Google Scholar
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