Double Toeplitz codes and their average weight enumerators

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 $q\in \{2,3,4\}$.