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.

翻译：$\mathbb{F}_p$上的带状Toeplitz矩阵作为一类众所周知的矩阵，在编码理论与自动机理论领域已被广泛研究。本文发现，$\mathbb{F}_p$上带状Toeplitz矩阵的行列式与逆均呈现周期性。对于带宽为$k$的Toeplitz矩阵，其周期$P(f)$与带上的参数相关且与阶数无关，其上界为$P(f) \le p^{k-1}-1$。我们提出一种算法，可在$O(k^4)$时间内计算任意阶带状Toeplitz矩阵的行列式。其逆矩阵可由三个大小均为$P(f)*P(f)$的子矩阵分别表示在对角线、对角线以上及对角线以下位置。因此，计算逆矩阵的代价是固定的，我们的算法可在$O(k^5)+3kP(f)^2$时间内求解。这是首次对一般$\mathbb{F}_p$上带状Toeplitz矩阵的行列式与逆的周期性进行计算与证明。