Abstract
A code \({\cal C}\) is a quaternary linear code if \({\cal C}\) is a subgroup of ℤ . In this paper, the rank and dimension of the kernel for ℤ4-linear codes, which are the corresponding binary codes of quaternary linear codes, are studied. The possible values of these two parameters for ℤ4-linear codes, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a ℤ4-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a ℤ4-linear code with dimension of the kernel k is given.
This work was supported in part by the Spanish MEC and the European FEDER under Grants MTM2006-03250 and TSI2006-14005-C02-01.
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References
Bauer, H., Ganter, B., Hergert, F.: Algebraic techniques for nonlinear codes. Combinatorica 3, 21–33 (1983)
Borges, J., Rifà, J.: A characterization of 1-perfect additive codes. IEEE Trans. Inform. Theory 45(5), 1688–1697 (1999)
Borges, J., Fernández, C., Phelps, K.T.: Quaternary Reed-Muller codes. IEEE Trans. Inform. Theory 51(7), 2686–2691 (2005)
Borges, J., Phelps, K.T., Rifà, J., Zinoviev, V.A.: On ℤ4-linear Preparata-like and Kerdock-like codes. IEEE Trans. Inform. Theory 49(11), 2834–2843 (2003)
Borges, J., Phelps, K.T., Rifà, J.: The rank and kernel of extended 1-perfect ℤ4-linear and additive non-ℤ4-linear codes. IEEE Trans. Inform. Theory 49(8), 2028–2034 (2003)
Borges, J., Fernández, C., Pujol, J., Rifà, J., Villanueva, M.: ℤ4-linear codes: generator matrices and duality. IEEE Trans. on Inform. Theory (submitted, 2007) arXiv:0710.1149
Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10 (1973)
Hammons, A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Solé, P.: The ℤ4-linearity of kerdock, preparata, goethals and related codes. IEEE Trans. Inform. Theory 40, 301–319 (1994)
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Krotov, D.S.: ℤ4-linear Hadamard and extended perfect codes. In: International Workshop on Coding and Cryptography, Paris, France, January 8-12, 2001, pp. 329–334 (2001)
MacWillams, F.J., Sloane, N.J.A.: The Theory of Error Correcting Codes. North-Holland, Amsterdam (1977)
Phelps, K.T., LeVan, M.: Kernels of nonlinear Hamming codes. Designs, Codes and Cryptography 6(3), 247–257 (1995)
Phelps, K.T., Rifà, J.: On binary 1-perfect additive codes: some structural properties. IEEE Trans. Inform. Theory 48(9), 2587–2592 (2002)
Phelps, K.T., Rifà, J., Villanueva, M.: On the additive ℤ4-linear and non-ℤ4-linear Hadamard codes. Rank and Kernel. IEEE Trans. Inform. Theory 52(1), 316–319 (2005)
Pujol, J., Rifà, J.: Translation invariant propelinear codes. IEEE Trans. Inform. Theory 43, 590–598 (1997)
Wan, Z.-X.: Quaternary Codes. World Scientific, Singapore (1997)
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Fernández-Córdoba, C., Pujol, J., Villanueva, M. (2008). On Rank and Kernel of ℤ4-Linear Codes. In: Barbero, Á. (eds) Coding Theory and Applications. Lecture Notes in Computer Science, vol 5228. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87448-5_6
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DOI: https://doi.org/10.1007/978-3-540-87448-5_6
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