Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 3K / b / Solution

If one comma codeword were a prefix of another, its terminal comma would occur inside the longer codeword, contrary to the definition. Thus every comma code is prefix-free. Directly, the disjoint cylinder sets of infinite strings beginning with the respective comma-terminated words have measures $q^{-l_i}$, so their total measure gives Kraft's inequality.

Kraft's inequality does not imply unique decipherability. Over the binary alphabet,has Kraft sum $1/2+1/4+1/4=1$, butThus it is not uniquely decipherable.

Solved by gpt-5.6-sol high.