The design and evaluation of real-time implementable coding schemes for Gaussian multisource-multichannel communication systems are considered. The problem is to transmit the output from ann-dimensional Gaussian source over a memorylessm-dimensional Gaussian channel with power constraint and a mean-squared error distortion measure. Shannon performance bounds are obtained for this system through the optimum performance theoretically attainable (OPTA) relationship between the source distortion levelbetaand the channel input power levelalpha. The optimum encoder is determined within the class of linear memoryless transformations, in which case the optimum decoder transformation becomes a least-mean-square estimator. The optimum linear encoder-decoder pair either coincides with or is very close to the Shannon bound in a number of significant cases.