Distance spectra and upper bounds on error probability for trellis codes

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Abstract
The problem of estimating error probability for trellis codes is considered. The set of all squared Euclidean distances between code sequences is presented as a countable set. This representation is used for calculating the generating functions for upper-bounding error probability and bit error probability for trellis codes satisfying some symmetry conditions. The generating functions of squared Euclidean distances (distance spectra) are obtained by inversion of a matrix of order 2ν. It is shown that the generating functions are defined in terms of one formal variable for QAM and uniform AM, and in terms of q/4 formal variables for q-ary PSK, q=2m, where m⩾2 is an integer. For small ν, the generating functions may be found in closed form. For larger ν, a numerical technique for obtaining some initial terms of the power series expansion is proposed. This algorithm is based on the recurrent matrix equations and the Chinese remainder theorem

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