American Institute of Mathematical Sciences

Advanced
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

Steven T. Dougherty 1, and  Cristina Fernández-Córdoba 2,

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.
Keywords: $\mathbb{Z}_{2^k}$-codes, self-duality, Gray map..
Mathematics Subject Classification: Primary: 94B25, 94B6.
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.  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.  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, 10 (2001).   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.  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.  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.  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.  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, (2008), 46.   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.  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.  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.  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.  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, (1998), 177.   Google Scholar

[17]

K. Shiromoto, A basic exact sequence for the Lee and Euclidean weights of linear codes over $\mathbbZ_l$,, Linear Algebra Appl. \textbf{295} (1999), 295 (1999), 191.  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.  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.  doi: 10.1353/ajm.1999.0024.  Google Scholar

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.  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.  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, 10 (2001).   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.  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.  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.  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.  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, (2008), 46.   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.  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.  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.  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.  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, (1998), 177.   Google Scholar

[17]

K. Shiromoto, A basic exact sequence for the Lee and Euclidean weights of linear codes over $\mathbbZ_l$,, Linear Algebra Appl. \textbf{295} (1999), 295 (1999), 191.  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.  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.  doi: 10.1353/ajm.1999.0024.  Google Scholar

[1]

Kamil Otal, Ferruh Özbudak, Wolfgang Willems. Self-duality of generalized twisted Gabidulin codes. Advances in Mathematics of Communications, 2018, 12 (4) : 707-721. doi: 10.3934/amc.2018042
[2]

Joaquim Borges, Steven T. Dougherty, Cristina Fernández-Córdoba. Characterization and constructions of self-dual codes over $\mathbb Z_2\times \mathbb Z_4$. Advances in Mathematics of Communications, 2012, 6 (3) : 287-303. doi: 10.3934/amc.2012.6.287
[3]

Helena Rifà-Pous, Josep Rifà, Lorena Ronquillo. $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in Steganography. Advances in Mathematics of Communications, 2011, 5 (3) : 425-433. doi: 10.3934/amc.2011.5.425
[4]

Tingting Wu, Jian Gao, Yun Gao, Fang-Wei Fu. $ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 641-657. doi: 10.3934/amc.2018038
[5]

Masaaki Harada. Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights. Advances in Mathematics of Communications, 2016, 10 (4) : 695-706. doi: 10.3934/amc.2016035
[6]

Makoto Araya, Masaaki Harada, Hiroki Ito, Ken Saito. On the classification of $\mathbb{Z}_4$-codes. Advances in Mathematics of Communications, 2017, 11 (4) : 747-756. doi: 10.3934/amc.2017054
[7]

Thomas Feulner. Canonization of linear codes over $\mathbb Z$4. Advances in Mathematics of Communications, 2011, 5 (2) : 245-266. doi: 10.3934/amc.2011.5.245
[8]

Jean Bourgain. On random Schrödinger operators on $\mathbb Z^2$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 1-15. doi: 10.3934/dcds.2002.8.1
[9]

Delphine Boucher. Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$. Advances in Mathematics of Communications, 2016, 10 (4) : 765-795. doi: 10.3934/amc.2016040
[10]

Joaquim Borges, Cristina Fernández-Córdoba, Roger Ten-Valls. On ${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 169-179. doi: 10.3934/amc.2018011
[11]

Amit Sharma, Maheshanand Bhaintwal. A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation. Advances in Mathematics of Communications, 2018, 12 (4) : 723-739. doi: 10.3934/amc.2018043
[12]

Lingyu Diao, Jian Gao, Jiyong Lu. Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020029
[13]

Anatole Katok, Federico Rodriguez Hertz. Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data. Journal of Modern Dynamics, 2007, 1 (2) : 287-300. doi: 10.3934/jmd.2007.1.287
[14]

Jennifer D. Key, Washiela Fish, Eric Mwambene. Codes from the incidence matrices and line graphs of Hamming graphs $H^k(n,2)$ for $k \geq 2$. Advances in Mathematics of Communications, 2011, 5 (2) : 373-394. doi: 10.3934/amc.2011.5.373
[15]

Dean Crnković, Bernardo Gabriel Rodrigues, Sanja Rukavina, Loredana Simčić. Self-orthogonal codes from orbit matrices of 2-designs. Advances in Mathematics of Communications, 2013, 7 (2) : 161-174. doi: 10.3934/amc.2013.7.161
[16]

Dongchen Li, Dmitry V. Turaev. Existence of heterodimensional cycles near Shilnikov loops in systems with a $\mathbb{Z}_2$ symmetry. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4399-4437. doi: 10.3934/dcds.2017189
[17]

Michael Kiermaier, Matthias Koch, Sascha Kurz. $2$-arcs of maximal size in the affine and the projective Hjelmslev plane over $\mathbb Z$25. Advances in Mathematics of Communications, 2011, 5 (2) : 287-301. doi: 10.3934/amc.2011.5.287
[18]

W. Cary Huffman. Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order. Advances in Mathematics of Communications, 2013, 7 (1) : 57-90. doi: 10.3934/amc.2013.7.57
[19]

Adel Alahmadi, Steven Dougherty, André Leroy, Patrick Solé. On the duality and the direction of polycyclic codes. Advances in Mathematics of Communications, 2016, 10 (4) : 921-929. doi: 10.3934/amc.2016049
[20]

Suat Karadeniz, Bahattin Yildiz. New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes. Advances in Mathematics of Communications, 2013, 7 (2) : 219-229. doi: 10.3934/amc.2013.7.219

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences