Optimal Linear Assignment of a Binary Group Code to Integer Vectors for Source-Channel Coding

This paper considers the source-channel coding problem of optimally assigning the codewords of an [n, k] binary linear code to the ^-dimensional vectors produced by an IID, uniformly distributed, integer-valued source with alphabet {0,1,..., 2^k/n-1} when overall distortion is measured by mean squared error (MSE). It finds explicit formulas for the optimal encoding rule, minimum MSE decoding rule, and resulting distortion. This extends to the vector case the 1974 analysis by Wolf and Redinbo [2] of the optimal assignment of binary linear codes to integers. As in [1], [2], the main tool is abstract Fourier analysis for functions defined on groups. The optimal linear assignment found for vectors can be interpreted as applying the optimal assignment found in [2] to the integer in {0,1,..., 2^k-1} formed by multiplexing the binary representations of the nmiddot integers in the vector being encoded in such a way that more significants bits of each integer come before less significant bits of all integers.