Galois self-orthogonal constacyclic codes over finite fields

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.