A new evaluation method of a block error probability of a block modulation code over an AWGN channel

Let L be a finite set of symbols, and let C be a block code of length n over L. Let h be a positive integer. For u/spl isin/L, let s(u) denote the signal point in /spl Rscr//sup h/ which represents u, where /spl Rscr//sup h/ denotes the set of all h-tuples of real numbers, and for u=(u/sub 1/,u/sub 2/,...,u/sub n/) over L, let s(u) denote the hn-tuple (s(u/sub 1/),s(u/sub 2/),...,s(u/sub n/)). For z and z' in /spl Rscr//sup hn/ let /spl par/z-z'/spl par/ denote the Euclidean distance between z and z'. For u/spl isin/C/{0}, let pe(r,u) denote the probability of an incorrect decoding of s(0) into u under the condition that /spl par/s(0)-z/spl par/=r where r is a non-negative real number. Then, f(r)=/spl Sigma//sub u/spl isin/C/{0}/pe(r,u). For u/spl isin/C{0}, let C/sub u/ be a subcode of C which contains u. Let P(r,C/sub u/) denote the probability that for every v/spl isin/C/sub u/ other than u, a received z satisfies /spl par/s(u)-z/spl par//spl les//spl par/s(v)-z/spl par/ under the condition that /spl par/s(0)-z/spl par/=r. In the examples given, we present a tighter evaluation method of pe(r,u) or the sum of pe(r,u) over all the nearest neighbor us of 0.