Recently, double Toeplitz codes have been introduced as a generalization of double circulant codes. In this paper, we study the average weight enumerators of double Toeplitz codes. As an application, we consider the existence of double Toeplitz codes over $\mathbb{F}_q$ with some specified minimum weights for $q \in \{2,3,4\}$. We also give a classification of double Toeplitz codes over $\mathbb{F}_q$ with the largest minimum weights for modest lengths and $q \in \{2,3,4\}$.

翻译：近期，双托普利茨码作为双循环码的推广被引入。本文研究了双托普利茨码的平均重量枚举器。作为应用，我们考虑了在$\mathbb{F}_q$上存在某些指定最小权重的双托普利茨码的情形，其中$q \in \{2,3,4\}$。我们还对于较小长度和$q \in \{2,3,4\}$的情况，给出了$\mathbb{F}_q$上具有最大最小权重的双托普利茨码的分类。