  This paper presents explicit formulas and algorithms for the determinants and
inverses of $k$-diagonal Toeplitz matrices. We have achieved a computational
complexity of $O(k^3 \log n+k^5)+kn^2$ for their inverses. By considering the
general case of $k$, we ensure versatility while remaining explicit and
efficient compared to other algorithms. Furthermore, thanks to methodological
innovation, we discovered that the determinants and inverses of $k$-diagonal
Toeplitz matrices exhibit periodicity over finite fields. Over $\mathbb{F}_p$,
the inverses of $n$th-ordered $k$-diagonal Toeplitz matrices can be solved in
$O(k^3 \log c+k^3 \log p+k^5)+3kc^2$ where $c$ is the period. This complexity
is independent of $n$ and therefore makes the computation of $k$-diagonal
Toeplitz matrices over $\mathbb{F}_p$ effortless.
