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 moderate lengths and $q \in \{2,3,4\}$.

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