${\mathbb {F}}_{q^{2}}$-double cyclic codes with respect to the Hermitian inner product

Abstract

In this paper, we introduce ${\mathbb {F}}_{q^{2}}$-double cyclic codes of length $n=r+s,$ where ${\mathbb {F}}_{q^{2}}$ is the Galois field of $q^{2}$ elements, q is a power of a prime integer p and r, s are positive integers. We determine the generator polynomials for any ${{\mathbb {F}}_{q^{2}} }$-double cyclic code. For any ${\mathbb {F}}_{q^{2}}$-double cyclic code ${\mathcal {C}},$ we will define the Euclidean dual code ${\mathcal {C}}^{\perp }$ based on the Euclidean inner product and the Hermitian dual code ${\mathcal {C}} ^{\perp _{H}}$ based on the Hermitian inner product. We will construct a relationship between ${\mathcal {C}}^{\perp }$ and ${\mathcal {C}}^{\perp _{H}}$ and then find the generator polynomials for the Hermitian dual code ${\mathcal {C}}^{\perp _{H}}.$ As an application of our work, we will present examples of optimal parameter linear codes over the finite field ${\mathbb {F}} _{4}$ and also examples of optimal quantum codes that were derived from ${\mathbb {F}}_{4}$-double cyclic codes using the Hermitian inner product.