Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 94B60; Secondary: 94B25.


    \begin{equation} \\ \end{equation}
  • [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.


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


    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.


    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.


    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.


    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.


    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.


    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.


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


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


    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.


    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.


    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.


    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.


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


    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.


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


    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.


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(313) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint