In this paper we study the reconstruction of binary sparse signals from partial random circulant measurements. We show that the reconstruction via the least-squares algorithm is as good as the reconstruction via the usually used program basis pursuit. We further show that we need as many measurements to recover an $s$-sparse signal $x_0\in\mathbb{R}^N$ as we need to recover a dense signal, more-precisely an $N-s$-sparse signal $x_0\in\mathbb{R}^N$. We further establish stability with respect to noisy measurements.