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=(u ₁,⋯,u_N) in V^N, we define D ₁(u)={i:u_i≠z_i,i≤N}, n(u)=|D ₁(u)| and D_o(u)={1,⋯,N}-D₁(u). Then the maximum likelihood decoding (MLD) solution is the codeword c/sub opt/ which minimize the correlation discrepancy L(c)=Σ_iεD1(c)|Ti|. Consequently if for c*εC and α(c*)=min _εC,c≠c*{L(c)}, L(c*)≤α(c*), then c*=c_opt. Therefore any lower bound on alpha/(c*) will provide a sufficient condition on the optimality of a candidate codeword.