 $ \operatorname{dim}_{\mathbb{F}_q}(\mathcal{Z}(\mathbb{F}_q H_k \varepsilon_{\mathcal{C}}^{(k)}(H,K))^{C_k/H_k}) = \frac{\operatorname{dim}_{\mathbb{F}_q}(\mathcal{Z}(\mathbb{F}_q H_k \varepsilon_{\mathcal{C}}^{(k)}(H,K)))