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A new algorithm for the reconstruction of so called block-sparse signals in a compressive sensing framework is presented. The algorithm is based on minimizing an ℓ2/p-norm regularized l2 error. The minimization is carried out by using a sequential conjugate-gradient algorithm where the line search involved is carried out using a technique based on Banach's fixed-point theorem. Simulation results are presented which show that for large-size data the proposed algorithm yields improved reconstruction performance and requires a reduced amount of computation relative to several known algorithms.