Double cyclic codes over the rings Zα2 × Zβ2 and Zα2 × Zβ4

Abstract

Publicació amb motiu del Congreso de la Real Sociedad Matemática Española, Granada, 2 al 6 de Febrero del 2015.

Consider the rings R1 and R2 , such that R1 is an R2-module, and C ⊂ R1α × R2β an additive code. The code C is a double cyclic code if the set of coordinates can be partitioned into two subsets, the set of coordinates in R1 and the set of coordinates in R2 , such that any cyclic shift of the coordinates of both subsets leaves invariant the code. The code can be identified as submodules of the R2[x]-module R1[x]/(x^α - 1) × R2[x]/(x^β - 1). We define two cases. First, when the code C is binary, that is R1 = R2 = Z2 , which is called Z2-double cyclic. The second case is when R1 = Z2 and R2 = Z4 , that is the code is a Z2Z4-additive code, and it is called Z2Z4-cyclic. In both cases, we determine the structure of these double cyclic codes giving their generator polynomials. We also determine the related polynomial representation of its duals in terms of the generator polynomials.