We introduce homothetic-BCH codes. These are a family of $q^2$-ary classical codes $\mathcal{C}$ of length $\lambda n_1$, where $\lambda$ and $n_1$ are suitable positive integers such that the punctured code $\mathcal{B}$ of $\mathcal{C}$ in the last $\lambda n_1 - n_1$ coordinates is a narrow-sense BCH code of length $n_1$. We prove that whenever $\mathcal{B}$ is Hermitian self-orthogonal, so is $\mathcal{C}$. As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths than cannot be reached by BCH codes. With this procedure we get new quantum codes according to Grassl's table. To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.

翻译：本文引入同态BCH码。这是一族$q^2$元经典码$\mathcal{C}$，其长度为$\lambda n_1$，其中$\lambda$和$n_1$为满足特定条件的正整数，使得$\mathcal{C}$在最后$\lambda n_1 - n_1$个坐标上的截短码$\mathcal{B}$成为长度为$n_1$的窄义BCH码。我们证明当$\mathcal{B}$满足厄米特自正交性时，$\mathcal{C}$同样满足该性质。基于此结论，我们提出一种构造量子稳定子码的方法，其可实现的码长超越了传统BCH码的构造范围。通过该方法，我们获得了若干Grassl码表中未收录的新量子码。为证明主要结论，我们针对一大类码长的BCH码，给出了其厄米特自正交性的充分必要条件。