Determinants and Inverses of banded Toeplitz Matrices over $\mathbb{F}_p$ Are Periodic

Banded Toeplitz matrices over $\mathbb{F}_p$, as a well-known class of matrices, have been extensively studied in the fields of coding theory and automata theory. In this paper, we discover that both determinants and inverses of banded Toeplitz matrices over $\mathbb{F}_p$ exhibit periodicity. For a Toeplitz matrix with bandwidth $k$, The period $P(f)$ is related to the parameters on the band and is independent of the order, with an upper limit of $P(f) \le p^{k-1}-1$. We provide an algorithm which can compute the determinant of any order banded Toeplitz matrix within $O(k^4)$. And its inverse can be represented by three submatrices of size $P(f)*P(f)$ located respectively on the diagonal, above the diagonal, and below the diagonal. Thus, the computational cost for calculating the inverse is fixed, and our algorithm can solve it within $O(k^5)+3kP(f)^2$. This is the first time that the periodicity of determinants and inverses of general banded Toeplitz matrices over $\mathbb{F}_p$ has been computed and proven.