Galois self-orthogonal constacyclic codes over finite fields

Abstract

Let ${\mathbb {F}}_{q}$ be a finite field with $q=p^{e}$ elements, where p is a prime and e is a positive integer. In 2017, Fan and Zhang introduced $\ell$-Galois inner products on the n-dimensional vector space ${\mathbb {F}}_{q}^{n}$ for $0\le \ell <e$, which generalized the Euclidean inner product and Hermitian inner product. $\ell$-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes and Hermitian self-orthogonal codes, and can be used to construct entanglement-assisted quantum error-correcting codes. In this paper, we study $\ell$-Galois self-orthogonal constacyclic codes of length n over the finite field ${\mathbb {F}}_{q}$. Sufficient and necessary conditions for constacyclic codes of length n over ${\mathbb {F}}_{q}$ being $\ell$-Galois self-orthogonal and $\ell$-Galois self-dual are characterized. A sufficient and necessary condition for the existence of nonzero $\ell$-Galois self-orthogonal constacyclic codes of length n over ${\mathbb {F}}_{q}$ is obtained. Formulae to enumerate the number of $\ell$-Galois self-orthogonal and $\ell$-Galois self-dual constacyclic codes of length n over ${\mathbb {F}}_{q}$ are found. In particular, formulae to enumerate the number of Hermitian self-orthogonal and Hermitian self-dual constacyclic codes of length n over ${\mathbb {F}}_{q}$ are obtained. Weight distributions of two classes of $\ell$-Galois self-orthogonal constacyclic codes are calculated. A family of MDS $\ell$-Galois self-orthogonal constacyclic codes over ${\mathbb {F}}_{q}$ is constructed.