**Summary:** The paper establishes a characterization of k-rational singularities for local complete intersection subvarieties of smooth complex varieties. It shows that a local complete intersection subvariety Z of pure codimension r in a smooth variety X has k-rational singularities if and only if the minimal exponent $\widetilde{\alpha}(Z)$ of Z satisfies $\widetilde{\alpha}(Z) > k + r$. This result is proven using the Hodge filtration on the local cohomology sheaf $\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})$ and the Kashiwara-Malgrange V-filtration. The minimal exponent $\widetilde{\alpha}(Z)$ is related to the log canonical threshold of the pair (X, Z), and it is shown that $\widetilde{\alpha}(Z) > r$ if and only if Z has rational singularities. The paper also characterizes k-Du Bois singularities in terms of the minimal exponent, showing that Z has k-Du Bois singularities if and only if $\widetilde{\alpha}(Z) \geq k + r$. Furthermore, it proves that k-rational singularities imply k-Du Bois singularities, and that the dimension of the singular locus of Z is bounded below by $2k + 2$. The paper also discusses the generation level of the Hodge filtration on $\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})$ and its implications for cohomology vanishing. The results are extended to hypersurfaces and provide new insights into the structure of singularities in local complete intersections.**Summary:** The paper establishes a characterization of k-rational singularities for local complete intersection subvarieties of smooth complex varieties. It shows that a local complete intersection subvariety Z of pure codimension r in a smooth variety X has k-rational singularities if and only if the minimal exponent $\widetilde{\alpha}(Z)$ of Z satisfies $\widetilde{\alpha}(Z) > k + r$. This result is proven using the Hodge filtration on the local cohomology sheaf $\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})$ and the Kashiwara-Malgrange V-filtration. The minimal exponent $\widetilde{\alpha}(Z)$ is related to the log canonical threshold of the pair (X, Z), and it is shown that $\widetilde{\alpha}(Z) > r$ if and only if Z has rational singularities. The paper also characterizes k-Du Bois singularities in terms of the minimal exponent, showing that Z has k-Du Bois singularities if and only if $\widetilde{\alpha}(Z) \geq k + r$. Furthermore, it proves that k-rational singularities imply k-Du Bois singularities, and that the dimension of the singular locus of Z is bounded below by $2k + 2$. The paper also discusses the generation level of the Hodge filtration on $\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})$ and its implications for cohomology vanishing. The results are extended to hypersurfaces and provide new insights into the structure of singularities in local complete intersections.