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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

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.  Google Scholar

[2]

J. Bierbrauer, "Introduction to Coding Theory,'' Chapman & Hall/CRC, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., 10 (1973), 97 pp.  Google Scholar

[10]

P. Delsarte and V. Levenshtein, Association schemes and coding theory, IEEE Trans. Inform. Theory, 44 (1998), 2477-2504. doi: 10.1109/18.720545.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland Publishing Co., Amsterdam, 1977.  Google Scholar

[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.  Google Scholar

[17]

J. Pujol and J. Rifà, Translation invariant propelinear codes, IEEE Trans. Inform. Theory, 43 (1997), 590-598. doi: 10.1109/18.556115.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. Bierbrauer, "Introduction to Coding Theory,'' Chapman & Hall/CRC, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., 10 (1973), 97 pp.  Google Scholar

[10]

P. Delsarte and V. Levenshtein, Association schemes and coding theory, IEEE Trans. Inform. Theory, 44 (1998), 2477-2504. doi: 10.1109/18.720545.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland Publishing Co., Amsterdam, 1977.  Google Scholar

[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.  Google Scholar

[17]

J. Pujol and J. Rifà, Translation invariant propelinear codes, IEEE Trans. Inform. Theory, 43 (1997), 590-598. doi: 10.1109/18.556115.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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