Let [Formula: see text], where [Formula: see text] is a power of a prime number [Formula: see text] and [Formula: see text]. A triple cyclic code of length [Formula: see text] over [Formula: see text] is a set that can be partitioned into three parts that any cyclic shift of the coordinates of the three parts leaves the code invariant. These codes can be viewed as [Formula: see text]-submodules of [Formula: see text]. In this paper, we study the generator polynomials and the minimum generating sets of this kind of codes. Some optimal or almost optimal linear codes are obtained from this family of codes. We present the relationship between the generators of triple cyclic codes and their duals. As a special class of triple cyclic codes, separable codes over [Formula: see text] are discussed briefly in the end.

中文翻译：

---

𝔽q + u𝔽q 上的三重循环码

设[公式：见正文]，其中[公式：见正文]是素数[公式：见正文]和[公式：见正文]的幂。长度为 [公式：见文本] 超过 [公式：见文本] 的三重循环码是一个可以划分为三个部分的集合，这三个部分的坐标的任何循环移位都会使代码保持不变。这些代码可以看作是[公式：见正文]-[公式：见正文]的子模块。在本文中，我们研究了这种代码的生成多项式和最小生成集。一些最优或几乎最优的线性码是从这个码族中获得的。我们提出了三重循环码的生成器与其对偶之间的关系。最后简要讨论[公式：见正文]上的可分码作为一类特殊的三循环码。