On DOUBLE-STRUCK CAPITAL Z2DOUBLE-STRUCK CAPITAL Z2[u]-additive codes

Abstract

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