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.
|Journal||Designs, Codes, and Cryptography|
|Publication status||Published - 1 Jul 2010|
- Quaternary linear codes
- ℤ -linear codes 4
- ℤ ℤ -linear codes 2 4
- ℤ ℤ -additive codes 2 4
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