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A method of using reliability information in one-step majority-logic decoders is presented. The idea is, basically, that the received binary digits are corrected in an order such that the least reliable digit is first corrected. With this method the error correction capability of the decoder is extended for large signal-to-noise ratios (SNR's). Different decoding algorithms are used when the number of orthogonal parity-check sums are even and odd, respectively. Computer simulations are presented for some short codes with binary antipodal signals on the additive white Gaussian noise channel.