On a generalization of lifted polynomials over finite fields and their applications to DNA codes

^†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, p. 1133, 24–27 June 2013, Almeria, Spain.

Abstract

In this paper, we generalize the lifted polynomials which generate reversible codes over F_q, a finite field with q element. Lifted polynomials are introduced by the authors Oztas and Siap [Lifted polynomials over F₁₆ and their applications to DNA codes, Filomat 27(3) (2013), pp. 459–466] over F₁₆. 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 4^k-lifted polynomials which lead to reversible and reversible complement DNA codes over $F_{4^{2k}}$. Further we construct examples of codes over F₈, F₉, F₁₆ and F₂₅₆ that have the best possible parameters or attain the Griesmer bound, hence they are optimal codes generated by lifted polynomials.

Keywords:

• reversible codes
• linear codes
• lifted polynomials
• DNA codes

2010 Mathematics Subject Classifications:

• 92D10
• 11T71
• 94B05
• 94A45

Acknowledgements

The authors thank to the reviewers for their valuable remarks and suggestions that improved the representation of the paper.

Notes

^† 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, p. 1133, 24–27 June 2013, Almeria, Spain.