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

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

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.
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]

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[2]

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[4]

J. Borges, C. Fernández, J. Pujol, J. Rifà and M. Villanueva, On $\mathbb Z_2\mathbb Z_4$-linear codes and duality,, in, (2006) , 171.

[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. 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. doi: 10.1109/18.771247.

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R. A. Brualdi and V. S. Pless, Weight enumerators of self-dual codes,, IEEE Trans. Inform. Theory, IT-37 (1991) , 1222. 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. doi: 10.1109/18.59931.

[9]

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

[10]

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

[11]

S. T. Dougherty and P. Solé, Shadows of codes and lattices,, in, (2002) , 139.

[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. 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. 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. doi: 10.1023/A:1025484821641.

[15]

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

[16]

K. T. Phelps and J. Rifà, On binary $1$-perfect additive codes: some structural properties,, IEEE Trans. Inform. Theory, 48 (2002) , 2587. doi: 10.1109/TIT.2002.801474.

[17]

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

[18]

E. Rains and N. J. A. Sloane, Self-dual codes,, in, (1998) , 177.

[19]

H. N. Ward, A restriction on the weight enumerator of a self-dual code,, J. Combin. Theory Ser. A, 21 (1976) , 253. 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.

[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. 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, (2006) , 171.

[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. 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. 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. 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. doi: 10.1109/18.59931.

[9]

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

[10]

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

[11]

S. T. Dougherty and P. Solé, Shadows of codes and lattices,, in, (2002) , 139.

[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. 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. 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. doi: 10.1023/A:1025484821641.

[15]

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

[16]

K. T. Phelps and J. Rifà, On binary $1$-perfect additive codes: some structural properties,, IEEE Trans. Inform. Theory, 48 (2002) , 2587. doi: 10.1109/TIT.2002.801474.

[17]

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

[18]

E. Rains and N. J. A. Sloane, Self-dual codes,, in, (1998) , 177.

[19]

H. N. Ward, A restriction on the weight enumerator of a self-dual code,, J. Combin. Theory Ser. A, 21 (1976) , 253. doi: 10.1016/0097-3165(76)90071-6.

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