Let \(q\) be an odd prime power, \(\mathbb{F}_q\) the finite field of order \(q\), \(m\) a positive integer, and \(R= \mathbb{F}_q[X]/(X^{2^m}-1)\). The author shows that the following \(m+1\) elements of \(R\) are idempotents, \[ e_0(X):=2^{-m}\sum_{j=0}^{2^m-1}X^j, \qquad e_i(X):=2^{i-m-1}\left(1+\sum_{k=i+1}^m S_k(X)-S_i(X)\right) \] for \(1\leq i\leq m\), where \(S_i(X)=\sum_{n=1}^{2^{m-i}} X^{2^{i-1}(2n-1)}\), \(1\leq i\leq m\). For \(0\leq i\leq m\) let \(E_i\) be the code of length \(2^m\) with idempotent generator \(e_i(X)\). Then \(E_0\) has dimension \(1\) and minimum distance \(2^m\), and for \(1\leq i\leq m\) the code \(E_i\) has dimension \(2^{i-1}\) and minimum distance \(2^{m-i+1}\).