Let \(\mathbb F_q\), \(q=2^m\), be a finite field with a primitive element \(\alpha\). For \(\beta\in \mathbb F_q^*\), the \(M\)-sequence is: \[ a_k(\beta )=(-1)^{\text{tr}(\beta\alpha^k)},\qquad 0\leq k < q-1, \] where \(\text{tr}\) is the trace form \(\mathbb F_q\) to \(\mathbb F_2\). Let \(P_q(\beta )\) be the peak-to-mean envelope power ratio of the sequence [see \textit{K. G. Patterson} and \textit{V. Tarokh}, IEEE Trans. Inf. Theory 46, No. 6, 1974--1987 (2000; Zbl 0998.94006)] and let \(P_q\) be the minimum over all primitive \(\alpha\) of \(\max_{\beta\in F_q^*} P_q(\beta )\). \(P_q\) arises in the study of multi-carrier communication systems. Here it is shown that \(P_q\geq (\ln \ln q)^2/(2\pi^2)\), thus showing that \(P_q\) grows with \(q\).