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

Recovery of Binary Sparse Signals from Structured Biased Measurements