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This paper studies the compression of a $k$th-order slant Toeplitz operator on the Hardy space $H^2\left({𝕋}^n\right)$ for integers $k\ge 2$ and $n\ge 1$. It also provides a characterization of the compression of a $k$th-order slant Toeplitz operator on $H^2\left({𝕋}^n\right)$. Finally, the paper highlights certain properties, namely isometry, eigenvalues, eigenvectors, spectrum and spectral radius of the compression of $k$th-order slant Toeplitz operator on the Hardy space $H^2\left({𝕋}^n\right)$ of $n$-dimensional torus ${𝕋}^n$.