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In this paper, we propose a new class of low-rate error correction codes called repeat-zigzag-Hadamard (RZH) codes featuring simple encoder and decoder structures, and flexible coding rate. RZH codes are serially concatenated turbo-like codes where the outer code is a repetition code and the inner code is a punctured zigzag-Hadamard (ZH) code. By analyzing the code structure of RZH codes, we prove that both systematic and nonsystematic RZH codes are good codes, in the sense that for an RZH code ensemble, there exists a positive number gamma0 such that for any binary-input memoryless channel whose Bhattacharyya noise parameter is less than , the average block error probability of maximum-likelihood (ML) decoding approaches zero. Two decoding algorithms-serial and parallel decoders for RZH codes-are proposed. We then employ the extrinsic information transfer (EXIT) chart technique to design irregular RZH codes. Results show that the optimized irregular RZH codes exhibit a performance that is very close to capacity in the low-rate regime.