Channel codes where the redundancy is obtained not from parity symbols, but from expanding the channel signal-set, are addressed. They were initially proposed by G. Ungerboeck (1982) using a convolutional code. Here, a block coding approach is given. Rate m/(m+1) coded 2/sup m+1/-ary phase-shift keying (PSK) is considered. The expanded signal-set is given the structure of a finite field. The code is defined by a square nonsingular circulant generator matrix over the field. Binary data are mapped on a dataword, of the same length as the codewords, over an additive subgroup of the field. The codes using trellises are described, and then the Viterbi algorithm for decoding is applied. The asymptotic coding gain ranges from 1.8 to 6.0 dB for QPSK going from blocklength 3 to 12. For 8-PSK, the gain is from 0.7 to 3.0 dB with blocklength 4 to 8. With only four states in the trellis, codes of any length for QPSK and 8-PSK are constructed, each having an asymptotic coding gain of 3.0 dB. Simulation results are presented. It is found that the bit-error rate performance at moderate signal-to-noise ratios is sensitive to the number of nearest and next-nearest neighbors.
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