Finite tuples admit effective prefix-free binary encodings

First encode a single binary word by $c(w):=1^{\mathrm{\mid }w\mathrm{\mid }}0w\mathrm{.}$ From the initial block of $1$'s and the following $0$, one reads off $\mathrm{\mid }w\mathrm{\mid }$ and then reads exactly that many remaining bits, so $c$ is injective and has a total decoder that either returns the uniquely parsed word or the malformed token of [L1].

For the fixed tuple length $k$, define $E(w_1,\dots ,w_k):=1^{k}0c(w_1)\cdots c(w_k)\mathrm{.}$ The initial block $1^{k}0$ tells the decoder that exactly $k$ self-delimiting blocks must follow. Parsing those $k$ blocks recovers a unique tuple when the input has the required form, and otherwise the decoder returns the malformed token. Hence $E$ is an effective encoding in the sense of [L1].

The encoding $E$ is prefix-free. Indeed, once the parser has read the initial block $1^{k}0$, it knows that the codeword ends exactly after the $k$th self-delimiting block. So a valid codeword cannot be a proper prefix of a different valid codeword with extra unread bits at the end.

Therefore $(\{0,1{\}}^{\ast }{)}^k$ admits an effective prefix-free binary encoding.