Polycyclic codes are a generalization of cyclic and constacyclic codes. Even though they have been known since 1972 and received some attention more recently, there have not been many studies on polycyclic codes. This paper presents an in-depth investigation of polycyclic codes associated with trinomials. Our results include a number of facts about trinomials, some properties of polycyclic codes, and many new quantum codes derived from polycyclic codes. We also state several conjectures about polynomials and polycyclic codes. Hence, we show useful features of polycyclic codes and present some open problems related to them.

翻译：多环法典是全环法典和多环法典的普遍化,尽管自1972年以来人们就已知道这些法典,最近也注意到了这些法典,但关于多环法典的研究并不多,本文对三环法典进行了深入的调查,我们的结果包括关于三环法典、多环法典的一些特性和从多环法典中衍生出的许多新的量法典的一些事实。我们还对多环法典和多环法典作了一些猜测。因此,我们展示了多环法典的有用特征,并提出了一些与它们有关的公开问题。