In this paper, we derive an efficient iterative algorithm for the recovery of block-sparse signals given the finite data alphabet and the non-zero block probability. The non-zero block number is supposed to be far smaller than the total block number (block-sparse). The key principle is the separation of the unknown signal vector into an unknown support vector s and an unknown data symbol vector a. Both number (‖s‖0) and positions (s ∈ {0, 1}) of non-zero blocks are unknown. The proposed algorithms use an iterative two-stage LASSO procedure consisting in optimizing the recovery problem alternatively with respect to a and with respect to s. The first algorithm resorts on ℓ1-norm of the support vector and the second one applies reweighted ℓ1-norm, which further improves the recovery performance. Performance of proposed algorithms is illustrated in the context of sporadic multiuser communications. Simulations show that the reweighted-ℓ1 algorithm performs close to its lower bound (perfect knowledge of the support vector).