On ℤ₂ℤ₂[u]-additive codes

Abstract

In this paper, a new class of additive codes which is referred to as Z 2 Z 2 [u]-additive codes is introduced. This is a generalization towards another direction of recently introduced Z 2 Z 4 -additive codes [J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rifa, and M. Villanueva, Z 2 Z 4 -linear codes: Generator matrices and duality, Designs Codes Cryptogr. 54 (2) (2010), pp. 167-179]. Z 2 Z 4 -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 Z 2 Z 2 [u]-additive codes are presented.

Self-dual cyclic and quantum codes over ℤ2 × (ℤ2 + uℤ2)

“…Most of the concepts about the structure of Z 2 Z 2 [u]-linear codes have been given in [2] with details.…”

“…Recently, inspired by the Z 2 Z 4 -additive codes (introduced in [7]), Z 2 Z 2 [u]-linear codes have been introduced in [2]. Though these code families are similar to each other, Z 2 Z 2 [u]-linear codes have some advantages compared to Z 2 Z 4 -additive codes.…”

<inline-formula><tex-math id="M1">\begin{document}$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$\end{document}</tex-math></inline-formula>-additive cyclic codes

“…Z 2 Z 4 -additive codes are subgroups of Z r 2 Z s 4 , they can be seen as a generalization of binary (when s = 0) and quaternary (when r = 0) linear codes. Since then, there have been extensive study and applications of Z 2 Z 4 -additive codes and other related types of codes (see [2,3,7,11,12,13]). In [4], Borges et al studied the generator matrices and duality of Z 2 Z 4 -additive codes.…”

Linear and cyclic codes over direct product of finite chain rings

“…These codes are defined over the direct product of rings of integers modulo some power of a prime number. Also, we can find the direct product of other finite rings, for example, codes in in Aydogdu et al, codes in in Lu and Zhu, and codes in in Annamalai and Durairajan…”