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This work presents a detailed study of a family of binary message-passing decoding algorithms for low-density parity-check (LDPC) codes, referred to as "majority-based algorithms." Both Gallager's algorithm A (GA) and the standard majority decoding algorithm belong to this family. These algorithms, which are, in fact, the building blocks of Gallager's algorithm B (GB), work based on a generalized majority-decision rule and are particularly attractive for their remarkably simple implementation. We investigate the dynamics of these algorithms using density evolution and compute their (noise) threshold values for regular LDPC codes over the binary symmetric channel. For certain ensembles of codes and certain orders of majority-based algorithms, we show that the threshold value can be characterized as the smallest positive root of a polynomial, and thus can be determined analytically. We also study the convergence properties of majority-based algorithms, including their (convergence) speed. Our analysis shows that the stand-alone version of some of these algorithms provides significantly better performance and/or convergence speed compared with GA. In particular, it is shown that for channel parameters below threshold, while for GA the error probability converges to zero exponentially with iteration number, this convergence for other majority-based algorithms is super-exponential.