Binary prefix codes on finite supports

Let $M$ be a finite nonempty message set. A binary prefix code on $M$ is an injective assignment $c:M\to \{0,1{\}}^{\ast }$ such that no assigned string is a proper initial segment of another assigned string. Write $\mathrm{\ell }(m):=\mathrm{\mid }c(m)\mathrm{\mid }$ for its codeword lengths. For a probability vector $p$ on $M$, its expected length is $L_p(c):={\sum }_{m\in M}p(m)\mathrm{\ell }(m)\mathrm{.}$

The empty string is permitted only when $M$ is a singleton. In that case it describes one known source block, or a known number of copies of the sole message; the message count is part of the framing convention. For arbitrary concatenations whose message count is not supplied, require every codeword to be nonempty. Thus a singleton source can retain a zero-bit code for one known block without claiming unknown-count decoding.