Characterization and constructions of self-dual codes over \mathbb Z_2\times \mathbb Z_4

Previous Article
Secondary constructions of bent functions and their enforcement
AMC Home
This Issue
Next Article
Wet paper codes and the dual distance in steganography

August 2012, 6(3): 287-303. doi: 10.3934/amc.2012.6.287

Characterization and constructions of self-dual codes over $\mathbb Z_2\times \mathbb Z_4$

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

1.	Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain
2.	Department of Mathematics, University of Scranton, Scranton, PA 18510, United States

Received March 2011 Revised March 2012 Published August 2012

Abstract
Related Papers

Self-dual codes over $\mathbb Z_2\times\mathbb Z_4$ are subgroups of $\mathbb Z_2^\alpha\times\mathbb Z_4^\beta$ that are equal to their orthogonal under an inner-product that relates these codes to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible values $\alpha,\beta$ such that there exist a self-dual code $\mathcal C\subseteq \mathbb Z_2^\alpha \times\mathbb Z_4^\beta$ are established. Moreover, the construction of such a code for each type and possible pair $(\alpha,\beta)$ is given. The standard techniques of invariant theory are applied to describe the weight enumerators for each type. Finally, we give a construction of self-dual codes from existing self-dual codes.

Keywords: Type I codes, $\mathbb Z_2\mathbb Z_4$-additive codes., Type II codes, Self-dual codes.

Mathematics Subject Classification: Primary: 94B60; Secondary: 94B2.

Citation: 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

References:

[1]	C. Bachoc and P. Gaborit, On extremal additive $\mathbb F_4$ codes of length $10$ to $18$, J. Théorie Nombres Bordeaux, 12 (2000), 255-271.
[2]	J. Bierbrauer, "Introduction to Coding Theory,'' Chapman & Hall/CRC, 2005.
[3]	A. Blokhuis and A. E. Brouwer, Small additive quaternary codes, European J. Combin., 25 (2004), 161-167. doi: 10.1016/S0195-6698(03)00096-9.
[4]	J. Borges, C. Fernández, J. Pujol, J. Rifà and M. Villanueva, On $\mathbb Z_2\mathbb Z_4$-linear codes and duality, in "Fifth Conference on Discrete Mathematics and Computer Science (Spanish),'' (2006), 171-177.
[5]	J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rifà and M. Villanueva, $\mathbb Z_2\mathbb Z_4$-linear codes: generator matrices and duality, Des. Codes Crypt., 54 (2010), 167-179. doi: 10.1007/s10623-009-9316-9.
[6]	J. Borges and J. Rifà, A characterization of 1-perfect additive codes, IEEE Trans. Inform. Theory, 45 (1999), 1688-1697. doi: 10.1109/18.771247.
[7]	R. A. Brualdi and V. S. Pless, Weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, IT-37 (1991), 1222-1225. doi: 10.1109/18.86979.
[8]	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.
[9]	P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., 10 (1973), 97 pp.
[10]	P. Delsarte and V. Levenshtein, Association schemes and coding theory, IEEE Trans. Inform. Theory, 44 (1998), 2477-2504. doi: 10.1109/18.720545.
[11]	S. T. Dougherty and P. Solé, Shadows of codes and lattices, in "Proceedings of the Third Asian Mathematical Conference, 2000 (Diliman),'' World Sci. Publ., (2002), 139-152.
[12]	C. Fernández, J. Pujol and M. Villanueva, $\mathbb Z_2\mathbb Z_4$-linear codes: rank and kernel, Des. Codes Crypt., 56 (2010), 43-59. doi: 10.1007/s10623-009-9340-9.
[13]	A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of kerdock, preparata, goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.
[14]	J.-L. Kim and V. Pless, Designs in additive codes over GF(4), Des. Codes Crypt., 30 (2003), 187-199. doi: 10.1023/A:1025484821641.
[15]	F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland Publishing Co., Amsterdam, 1977.
[16]	K. T. Phelps and J. Rifà, On binary $1$-perfect additive codes: some structural properties, IEEE Trans. Inform. Theory, 48 (2002), 2587-2592. doi: 10.1109/TIT.2002.801474.
[17]	J. Pujol and J. Rifà, Translation invariant propelinear codes, IEEE Trans. Inform. Theory, 43 (1997), 590-598. doi: 10.1109/18.556115.
[18]	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.
[19]	H. N. Ward, A restriction on the weight enumerator of a self-dual code, J. Combin. Theory Ser. A, 21 (1976), 253-255. doi: 10.1016/0097-3165(76)90071-6.

show all references

References:

[1]	C. Bachoc and P. Gaborit, On extremal additive $\mathbb F_4$ codes of length $10$ to $18$, J. Théorie Nombres Bordeaux, 12 (2000), 255-271.
[2]	J. Bierbrauer, "Introduction to Coding Theory,'' Chapman & Hall/CRC, 2005.
[3]	A. Blokhuis and A. E. Brouwer, Small additive quaternary codes, European J. Combin., 25 (2004), 161-167. doi: 10.1016/S0195-6698(03)00096-9.
[4]	J. Borges, C. Fernández, J. Pujol, J. Rifà and M. Villanueva, On $\mathbb Z_2\mathbb Z_4$-linear codes and duality, in "Fifth Conference on Discrete Mathematics and Computer Science (Spanish),'' (2006), 171-177.
[5]	J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rifà and M. Villanueva, $\mathbb Z_2\mathbb Z_4$-linear codes: generator matrices and duality, Des. Codes Crypt., 54 (2010), 167-179. doi: 10.1007/s10623-009-9316-9.
[6]	J. Borges and J. Rifà, A characterization of 1-perfect additive codes, IEEE Trans. Inform. Theory, 45 (1999), 1688-1697. doi: 10.1109/18.771247.
[7]	R. A. Brualdi and V. S. Pless, Weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, IT-37 (1991), 1222-1225. doi: 10.1109/18.86979.
[8]	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.
[9]	P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., 10 (1973), 97 pp.
[10]	P. Delsarte and V. Levenshtein, Association schemes and coding theory, IEEE Trans. Inform. Theory, 44 (1998), 2477-2504. doi: 10.1109/18.720545.
[11]	S. T. Dougherty and P. Solé, Shadows of codes and lattices, in "Proceedings of the Third Asian Mathematical Conference, 2000 (Diliman),'' World Sci. Publ., (2002), 139-152.
[12]	C. Fernández, J. Pujol and M. Villanueva, $\mathbb Z_2\mathbb Z_4$-linear codes: rank and kernel, Des. Codes Crypt., 56 (2010), 43-59. doi: 10.1007/s10623-009-9340-9.
[13]	A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of kerdock, preparata, goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.
[14]	J.-L. Kim and V. Pless, Designs in additive codes over GF(4), Des. Codes Crypt., 30 (2003), 187-199. doi: 10.1023/A:1025484821641.
[15]	F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland Publishing Co., Amsterdam, 1977.
[16]	K. T. Phelps and J. Rifà, On binary $1$-perfect additive codes: some structural properties, IEEE Trans. Inform. Theory, 48 (2002), 2587-2592. doi: 10.1109/TIT.2002.801474.
[17]	J. Pujol and J. Rifà, Translation invariant propelinear codes, IEEE Trans. Inform. Theory, 43 (1997), 590-598. doi: 10.1109/18.556115.
[18]	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.
[19]	H. N. Ward, A restriction on the weight enumerator of a self-dual code, J. Combin. Theory Ser. A, 21 (1976), 253-255. doi: 10.1016/0097-3165(76)90071-6.

[1]	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
[2]	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
[3]	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
[4]	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
[5]	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
[6]	Padmapani Seneviratne, Martianus Frederic Ezerman. New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2021073
[7]	Habibul Islam, Om Prakash, Patrick Solé. $ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes. Advances in Mathematics of Communications, 2021, 15 (4) : 737-755. doi: 10.3934/amc.2020094
[8]	Om Prakash, Shikha Yadav, Habibul Islam, Patrick Solé. On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022017
[9]	Ken Saito. Self-dual additive $ \mathbb{F}_4 $-codes of lengths up to 40 represented by circulant graphs. Advances in Mathematics of Communications, 2019, 13 (2) : 213-220. doi: 10.3934/amc.2019014
[10]	W. Cary Huffman. Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order. Advances in Mathematics of Communications, 2007, 1 (3) : 357-398. doi: 10.3934/amc.2007.1.357
[11]	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
[12]	Jie Geng, Huazhang Wu, Patrick Solé. On one-lee weight and two-lee weight $ \mathbb{Z}_2\mathbb{Z}_4[u] $ additive codes and their constructions. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021046
[13]	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
[14]	Raj Kumar, Maheshanand Bhaintwal. Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2020135
[15]	Hengming Zhao, Rongcun Qin, Dianhua Wu. Balanced ($\mathbb{Z} _{2u}\times \mathbb{Z}_{38v}$, {3, 4, 5}, 1) difference packings and related codes. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022008
[16]	Lingyu Diao, Jian Gao, Jiyong Lu. Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Advances in Mathematics of Communications, 2020, 14 (4) : 555-572. doi: 10.3934/amc.2020029
[17]	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
[18]	Adrian Korban, Serap Sahinkaya, Deniz Ustun. New type I binary $[72, 36, 12]$ self-dual codes from $M_6(\mathbb{F}_2)G$ - Group matrix rings by a hybrid search technique based on a neighbourhood-virus optimisation algorithm. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022032
[19]	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
[20]	Yuan Cao, Yonglin Cao, Hai Q. Dinh, Ramakrishna Bandi, Fang-Wei Fu. An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021, 15 (2) : 291-309. doi: 10.3934/amc.2020067

2020 Impact Factor: 0.935

Tools

Metrics

Other articles
by authors

on AIMS
Joaquim Borges Steven T. Dougherty Cristina Fernández-Córdoba
on Google Scholar
Joaquim Borges Steven T. Dougherty Cristina Fernández-Córdoba

[Back to Top]