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.

令\({\mathbb {F}}_{q}\)是具有\(q=p^{e}\) 个元素的有限域，其中p是素数，e是正整数。2017 年，范和张在n维向量空间\({\mathbb {F}}_{q}^{n}\) 上为\(0\le \ )引入了\(\ell \) -Galois 内积ell <e\)，它概括了欧几里得内积和厄米内积。\(\ell \) -Galois 自正交码是欧几里得自正交码和厄米自正交码的推广，可用于构造纠缠辅助量子纠错码。在本文中，我们研究\(\ell \)-在有限域\({\mathbb {F}}_{q}\)上长度为n 的伽罗瓦自正交恒环码。长度为n 的恒环码在\({\mathbb {F}}_{q}\)上的充分必要条件是\(\ell \) -Galois 自正交和\(\ell \) -Galois 自对偶被表征。得到了在\({\mathbb {F}}_{q}\)上存在长度为n的非零\(\ell \) -Galois 自正交恒环码的充要条件。枚举\(\ell \) -Galois 自正交和\(\ell \)个数的公式-在\({\mathbb {F}}_{q}\)上找到了长度为n 的Galois 自对偶恒环码。特别地，获得了在\({\mathbb {F}}}_{q}\)上枚举长度为n的Hermitian自正交和Hermitian自对偶恒环码的数量的公式。计算了两类\(\ell \) -Galois 自正交恒环码的权重分布。构造了一系列MDS \(\ell \) -Galois 自正交恒环码在\({\mathbb {F}}_{q}\)上。