On ℤ₂ℤ₂[u]-additive codes

Abstract

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.

Keywords:

• ℤ₂ℤ₂[u]-additive codes
• generator matrix
• parity-check matrix
• MacWilliams identity

2000 AMS Subject Classifications:

• 94B05
• 94B60

Acknowledgements

The authors would like to take this opportunity to thank the anonymous reviewers whose suggestions and remarks improved the presentation of the paper.

The preliminary results of this paper are presented in Proceedings of the 2013 International Conference on Computational and Mathematical Methods in Science and Engineering – CMMSE 2013, Almeria, Spain, 24–27 June 2013, p. 169.