# American Institute of Mathematical Sciences

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

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

 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] C. Bachoc and P. Gaborit, On extremal additive $\mathbb F_4$ codes of length $10$ to $18$,, J. Théorie Nombres Bordeaux, 12 (2000), 255.   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.  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, (2006), 171.   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.  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.  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.  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.  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).   Google Scholar [10] P. Delsarte and V. Levenshtein, Association schemes and coding theory,, IEEE Trans. Inform. Theory, 44 (1998), 2477.  doi: 10.1109/18.720545.  Google Scholar [11] S. T. Dougherty and P. Solé, Shadows of codes and lattices,, in, (2002), 139.   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.  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.  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.  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., (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.  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.  doi: 10.1109/18.556115.  Google Scholar [18] E. Rains and N. J. A. Sloane, Self-dual codes,, in, (1998), 177.   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.  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.   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.  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, (2006), 171.   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.  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.  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.  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.  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).   Google Scholar [10] P. Delsarte and V. Levenshtein, Association schemes and coding theory,, IEEE Trans. Inform. Theory, 44 (1998), 2477.  doi: 10.1109/18.720545.  Google Scholar [11] S. T. Dougherty and P. Solé, Shadows of codes and lattices,, in, (2002), 139.   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.  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.  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.  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., (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.  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.  doi: 10.1109/18.556115.  Google Scholar [18] E. Rains and N. J. A. Sloane, Self-dual codes,, in, (1998), 177.   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.  doi: 10.1016/0097-3165(76)90071-6.  Google Scholar
 [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] 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 [7] 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 [8] 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 [9] 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 [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] 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 [12] 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, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020067 [13] 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 [14] 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 [15] Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031 [16] Evangeline P. Bautista, Philippe Gaborit, Jon-Lark Kim, Judy L. Walker. s-extremal additive $\mathbb F_4$ codes. Advances in Mathematics of Communications, 2007, 1 (1) : 111-130. doi: 10.3934/amc.2007.1.111 [17] W. Cary Huffman. Additive cyclic codes over $\mathbb F_4$. Advances in Mathematics of Communications, 2008, 2 (3) : 309-343. doi: 10.3934/amc.2008.2.309 [18] W. Cary Huffman. Additive cyclic codes over $\mathbb F_4$. Advances in Mathematics of Communications, 2007, 1 (4) : 427-459. doi: 10.3934/amc.2007.1.427 [19] T. Aaron Gulliver, Masaaki Harada, Hiroki Miyabayashi. Double circulant and quasi-twisted self-dual codes over $\mathbb F_5$ and $\mathbb F_7$. Advances in Mathematics of Communications, 2007, 1 (2) : 223-238. doi: 10.3934/amc.2007.1.223 [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