Soft-decision majority decoding of Reed-Muller codes

We present a new soft-decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of 2^m transmitted symbols are recalculated into the reliabilities of 2^m-r parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft-decision algorithm outperforms its conventional hard-decision counterpart by 10 log₁₀(π/2)≈2 dB at any given output error probability. For fixed code rate R and m→∞, our algorithm increases almost 2^r/2 times the correcting capability of soft-decision bounded distance decoding