On ℤ2ℤ2[u]-additive codes
İsmail Aydoğdu, Taher Abualrub, İrfan Şiap · International Journal of Computer Mathematics · 2013
In this paper, a new class of additive codes which is referred to as ℤ2 ℤ2[u]-additive codes is introduced. This is a generalization towards another direction of recently introduced ℤ2 ℤ4-additive codes [J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rif´a, and M. Villanueva, ℤ2 ℤ4-linear codes: Generator matrices and duality, Designs Codes Cryptogr. 54(2) (2010), pp. 167–179]. ℤ2 ℤ4-additive codes have shown to provide a promising class of codes with their algebraic structure and applications such as steganography. The standard generator matrices are established and by introducing orthogonality the parity-check matrices are also obtained. A MacWilliams-type identity that relates the weight enumerator of a code with its dual is proved. Furthermore, a Gray map that maps these codes to binary codes is defined and some examples of optimal codes which are the binary Gray images of ℤ2 ℤ2[u]-additive codes are presented.