The least stringent sufficient condition on the optimality of a suboptimally decoded codeword using the most reliable basis

Suppose a binary linear block code C with weight profile W is used for error control over the AWGN channel. A codeword c is mapped into the BPSK sequence x. Let r denote the received noisy vector and z=(z₁ ,...,z_N) denote the corresponding bit-by-bit hard decoded sequence. Let V^N represent the set all binary N-tuples. For u=(u₁,...,u_N) in V^N, we define D₁(u)={i:u_i≠z_i,i⩽N}, n(u)=|D_l(u)| and D₀(u)={1,...,N}-D_l(u). Then the maximum likelihood decoding (MLD) solution is the codeword c _opt which minimize the correlation discrepancy L(c)=Σ _i∈D1(c)|Ti|. Consequently if for c*∈C and α(c*)=min_cεC,c≠c*{L(c)}, L(c*)⩽α(c*), then c*=c_opt. Therefore any lower bound on α(c*) will provide a sufficient condition on the optimality of a candidate codeword