Fq2-double cyclic codes with respect to the Hermitian inner product

Abstract

In this paper, we introduce F-q2-double cyclic codes of length n = r + s, where F-q2 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 F-q2-double cyclic code. For any F-q2-double cyclic code C, we will define the Euclidean dual code C-perpendicular to based on the Euclidean inner product and the Hermitian dual code C-perpendicular to H based on the Hermitian inner product. We will construct a relationship between C-perpendicular to and C-perpendicular to H and then find the generator polynomials for the Hermitian dual code C-perpendicular to H. As an application of our work, we will present examples of optimal parameter linear codes over the finite field F-4 and also examples of optimal quantum codes that were derived from F-4-double cyclic codes using the Hermitian inner product.