Non-Statistical Soft-in, Soft-out Decoding with the Euclidean Metric

Soft-decision decoding of linear error correcting codes, using a quantised or real number soft metric such as Euclidean distance, has been well understood for over 40 years. The output of a minimum soft distance decoder is the codeword (in the case of a block code) or code sequence (for a convolutional code) closest in distance to the soft word or sequence received from the channel and input to the decoder. This is soft-in, hard-out (SIHO) decoding, and it can be implemented optimally by applying the Viterbi algorithm (VA) to the code trellis [1], for example. SIHO decoding minimises the average output block or sequence error rate. Conceptually, but not in practice except in the case of a rather simple code, the codeword or code sequence closest in soft distance to the received word is found by determining the soft distances between the received word and all the words or sequences in the code, and then selecting the closest.