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Abstract
A code\({{\mathcal C}}\) is\({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -additive if the set of coordinates can be partitioned into two subsetsX andY such that the punctured code of\({{\mathcal C}}\) by deleting the coordinates outsideX (respectively,Y) is a binary linear code (respectively, a quaternary linear code). In this paper\({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -additive codes are studied. Their corresponding binary images, via the Gray map, are\({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity-check matrices are given. In order to do this, the appropriate concept of duality for\({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -additive codes is defined and the parameters of their dual codes are computed.
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Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain
J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rifà & M. Villanueva
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Correspondence toJ. Borges.
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Communicated by T. Helleseth.
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Borges, J., Fernández-Córdoba, C., Pujol, J.et al.\({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -linear codes: generator matrices and duality.Des. Codes Cryptogr.54, 167–179 (2010). https://doi.org/10.1007/s10623-009-9316-9
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Keywords
- Binary linear codes
- Duality
- Quaternary linear codes
- \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -additive codes
- \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -linear codes