In this paper, we propose \textit{coded compressive sensing} that recovers an \(n\)-dimensional integer sparse signal vector from a noisy and quantized measurement vector whose dimension \(m\) is far-fewer than \(n\). The core idea of coded compressive sensing is to construct a linear sensing matrix whose columns consist of lattice codes. We present a two-stage decoding method named \textit{compute-and-recover} to detect the sparse signal from the noisy and quantized measurements. In the first stage, we transform such measurements into noiseless finite-field measurements using the linearity of lattice codewords. In the second stage, syndrome decoding is applied over the finite-field to reconstruct the sparse signal vector. A sufficient condition of a perfect recovery is derived. Our theoretical result demonstrates an interplay among the quantization level \(p\), the sparsity level \(k\), the signal dimension \(n\), and the number of measurements \(m\) for the perfect recovery. Considering 1-bit compressive sensing as a special case, we show that the proposed algorithm empirically outperforms an existing greedy recovery algorithm.