The \(k\)-th entropy number \(e_k(T)\) of a bounded and linear operator \(T:X \to Y\) between quasi-Banach spaces \(X\) and \(Y\) is defined as the infimum of the \(\delta\)'s such that the image by \(T\) of the unit ball of \(X\) can be covered by \(2^{k-1}\) balls of radius \(\delta\) in \(Y\). For the case when \(T\) is the identity operator between the \(n\)-dimensional Banach spaces \(X=\ell^n_p\) and \(Y=\ell^n_q\) (with \(1 \leq p \leq q \leq \infty\)), it was proved by \textit{C.~Schütt} [J. Approx. Theory 40, 121-128 (1984; Zbl 0522.41017)] that \[ e_k(T) \approx \begin{cases} 1 & \text{if \(1 \leq k \leq \log(n)\)} \\ \left(\dfrac{\log(n/k + 1)}{k}\right)^{1/p-1/q} & \text{if \(\log(n) \leq k \leq n\) } \\ 2^{-(k-1)/n} n^{1/p-1/q} & \text{if \(n \leq k \)., } \end{cases} \] where \(\approx\) means the existence of constants \( c < c'\) such that \(e_k(T)\) is in between \(c\) and \(c'\) times the right hand side. The upper estimates were extended by \textit{D. E. Edmunds} and \textit{H. Triebel} [``Function spaces, entropy numbers and differential operators'', Cambridge Univ. Press (1996; Zbl 0865.46020)] for the quasi-Banach case \(0 < p\leq q \leq \infty\), whereas the lower estimates were extended by \textit{H. Triebel} [``Fractals and spectra related to Fourier analysis and function spaces'', Birkhäuser (1997; Zbl 0898.46030)] again for the quasi-Banach case \(0 < p\leq q \leq \infty\) excluding the medium size case when \(\log(n) \leq k \leq n\). This gap is closed in the present paper, where two different proofs (one using combinatorial arguments, the other one using interpolation) of the lower estimate for \(\log(n) \leq k \leq n\) are given.