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=(z/sub 1/,...,z/sub N/) denote the corresponding bit-by-bit hard decoded sequence. Let V/sup N/ represent the set all binary N-tuples. For u=(u/sub 1/,...,u/sub N/) in V/sup N/, we define D/sub 1/(u)={i:u/sub i//spl ne/z/sub i/,i/spl les/N}, n(u)=|D/sub l/(u)| and D/sub 0/(u)={1,...,N}-D/sub l/(u). Then the maximum likelihood decoding (MLD) solution is the codeword c/sub opt/ which minimize the correlation discrepancy L(c)=/spl Sigma//sub i/spl isin/D1(c)/|Ti|. Consequently if for c*/spl isin/C and /spl alpha/(c*)=min/sub c/spl epsiv/C,c/spl ne/c*/{L(c)}, L(c*)/spl les//spl alpha/(c*), then c*=c/sub opt/. Therefore any lower bound on /spl alpha/(c*) will provide a sufficient condition on the optimality of a candidate codeword.