Soft decision majority decoding

We present a new soft decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of all received symbols are recalculated into the reliabilities of the parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to the 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 bit error rate ε<1/2