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In this paper, we generalize the lifted polynomials which generate reversible codes overFq, a finite field withqelement. Lifted polynomials are introduced by the authors Oztas and Siap [Lifted polynomials over F16and their applications to DNA codes, Filomat 27(3) (2013), pp. 459–466] overF16. Lifted polynomials have proven to be very advantageous. They are easy to construct and they can be used to construct codes with specific properties such as dimension and the length of codes. We also generalize the 4k-lifted polynomials which lead to reversible and reversible complement DNA codes over. Further we construct examples of codes overF8,F9,F16andF256that have the best possible parameters or attain the Griesmer bound, hence they are optimal codes generated by lifted polynomials. [ABSTRACT FROM PUBLISHER]

In this paper, a new class of additive codes which is referred to as ℤ2ℤ2[u]-additive codes is introduced. This is a generalization towards another direction of recently introduced ℤ2ℤ4-additive codes [J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rif´a, and M. Villanueva, ℤ2ℤ4-linear codes: Generator matrices and duality, Designs Codes Cryptogr. 54(2) (2010), pp. 167–179]. ℤ2ℤ4-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 ℤ2ℤ2[u]-additive codes are presented. [ABSTRACT FROM PUBLISHER]