Some results on ${\mathbb {F}}_4[v]$-double cyclic codes

Abstract

Let $\mathrm {R}={\mathbb {F}}_4+v{\mathbb {F}}_4, v^2=v$. A linear code over $\mathrm {R}$ is a double cyclic code of length (r, s), if the set of its coordinates can be partitioned into two parts of sizes r and s, so that any cyclic shift of coordinates of both parts leave the code invariant. In polynomial representation, these codes can be viewed as $\mathrm {R}[x]$-submodules of $\frac{\mathrm {R}[x]}{\langle x^r-1\rangle }\times \frac{\mathrm {R}[x]}{\langle x^s-1\rangle }$. In this paper, we determine generator polynomials of $\mathrm {R}$-double cyclic codes and their duals for arbitrary values of r and s. We enumerate $\mathrm {R}$-double cyclic codes of length $(2^{e_1},2^{e_2})$ by giving a mass formula, where $e_1$ and $e_2$ are positive integers. Some structural properties of double constacyclic codes over $\mathrm {R}$ are also studied. These results are illustrated with some good examples.