The loss in quantizing coded symbols in the additive white Gaussian noise (AWGN) channel with binary phase-shift keying (BPSK) or quadrature phase-shift keying (QPSK) modulation is discussed. A quantization scheme and branch metric calculation method are presented. For the uniformly quantized AWGN channel, cutoff rate is used to determine the step size and the smallest number of quantization bits needed for a given bit-signal-to-noise ratio (E/sub b//N/sub 0/) loss. A nine-level quantizer is presented, along with 3-b branch metrics for a rate-1/2 code, which causes an E/sub b//N/sub 0/ loss of only 0.14 dB. These results also apply to soft-decision decoding of block codes. A tight upper bound is derived for the range of path metrics in a Viterbi decoder. The calculations are verified by simulations of several convolutional codes, including the memory-14, rate-1/4 or -1/6 codes used by the big Viterbi decoders at JPL.>

By using coding and combinational techniques, an explicit formula is derived which enumerates the complete weight distribution of decodable words of block codes using partially known weight distributions. Also, an approximation formula for nonbinary block codes is obtained. These results give exact and approximate expressions for the decoder error probability P/sub E/(u) of block codes.>

An explicit formula is derived that enumerates the complete weight distribution of an (n, k, d) linear code using a partially known weight distribution. An approximation formula for the weight distribution of q-ary linear (n, k, d) codes is also derived. It is shown that, for a given q-ary linear (n, k, d) code, the ratio of the number of codewords of weight u to the number of words of weight u approaches the constant Q=q/sup -(n-k)/ as u becomes large. The error term is a decreasing function of the minimum weight of the dual. The results are also valid for nonlinear (n, M, d) codes with the minimum weight of the dual replaced by the dual distance.>

A combinatorial technique similar to the principle of inclusion and exclusion is used to obtain an exact formula for P/sub E/(u), the decoder error probability for Reed-Solomon codes. The P/sub E/(u) for the (255, 223) Reed-Solomon code used by NASA and for the (31, 15) Reed-Solomon code (JTIDS code) are calculated using the exact formula and are observed to approach the Qs of the codes rapidly as u gets large. An upper bound for the expression mod P/sub E/(u)/Q-1 mod is derived and shown to decrease nearly exponentially as u increases.>