Perfect compression looks like random noise

True random noise is incompressible: each bit is a fair coin flip, independent of the rest. An optimal compressed stream should be indistinguishable from that noise to the receiver .

Fix a message length of $n$ bits. If all $2^n$ strings of that length are equally likely, each underlying message had probability $2^{-n}$. Picture binary strings organized in layers by length: moving one message to a shorter codeword forces another message upward, costing extra bits globally .

Saving one bit locally can cost two elsewhere. For equally likely messages, giving each the same number of bits is optimal; uneven reallocation always penalizes someone. That accounting is the geometric heart of why entropy rates are tight limits .