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.