Randomly Chosen Index Assignments Are Asymptotically Bad for Uniform Sources

It is known that among all redundancy-free codes (or index assignments), the natural binary code minimizes the mean-squared error (MSE) of the uniform source and uniform quantizer on a binary symmetric channel. We derive a code which maximizes the MSE and demonstrate that the code is linear and its distortion is asymptotically equivalent, as the blocklength grows, to the expected distortion of an index assignment chosen uniformly at random.