### Abstract

A code ℤ2ℤ4-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of ℤ2ℤ4-additive codes under an extended Gray map are called ℤ2ℤ4-linear codes. In this paper, the invariants for ℤ2ℤ4-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of ℤ2ℤ4-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a ℤ2ℤ4-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a ℤ2ℤ4-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a ℤ2ℤ4- linear code for each possible pair (r, k) is given. © 2009 Springer Science+Business Media, LLC.

Original language | English |
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Pages (from-to) | 43-59 |

Journal | Designs, Codes, and Cryptography |

Volume | 56 |

DOIs | |

Publication status | Published - 1 Jul 2010 |

### Keywords

- Kernel
- Quaternary linear codes
- Rank
- ℤ -linear codes 4
- ℤ ℤ -linear codes 2 4
- ℤ ℤ -additive codes 2 4

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## Cite this

Fernández-Córdoba, C., Pujol, J., & Villanueva, M. (2010). ℤ<inf>2</inf>ℤ<inf>4</inf>-linear codes: Rank and kernel.

*Designs, Codes, and Cryptography*,*56*, 43-59. https://doi.org/10.1007/s10623-009-9340-9