On Z2Z4-additive polycyclic codes and their Gray images

In this paper, we first generalize the polycyclic codes over finite fields to polycyclic codes over the mixed alphabet Z(2)Z(4), and we show that these codes can be identified as Z(4)[x]-submodules of R-alpha,R-beta with R-alpha,R-beta = Z(2)[x]/< t(1)(x)> x Z(4)[x]/< t(2)(x)>, where t(1)(x) and t(2)(x) are monic polynomials over Z(2) and Z(4), respectively. Then we provide the generator polynomials and minimal generating sets for this family of codes based on the strong Grobner basis. In particular, under the proper defined inner product, we study the dual of Z(2)Z(4)-additive polycyclic codes. Finally, we focus on the characterization of the Z(2)Z(4)-MDSS and MDSR codes, and as examples, we also present some (almost) optimal binary codes derived from the Z(2)Z(4)-additive polycyclic codes.