We present an estimator-based, or soft, vector quantizer decoder for communication over a noisy channel. The decoder is optimal according to the mean-square error criterion, and Hadamard-based in the sense that a Hadamard transform representation of the vector quantizer is utilized in the implementation of the decoder. An efficient algorithm for optimal decoding is derived. We furthermore investigate suboptimal versions of the decoder, providing good performance at lower complexity. The issue of joint encoder-decoder design is considered both for optimal and suboptimal decoding. Results regarding the channel distortion and the structure of a channel robust code are also provided. Through numerical simulations, soft decoding is demonstrated to outperform hard decoding in several aspects.
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