I know this is borderline computer science, but since it is the math I have questions about i decided to ask on here.
On slide 10 in the following they state:
Let $X_1, X_2,...,X_n$ be i.i.d r.v. with generic $X$. A coding scheme is
$$\mathcal{C} : \mathcal{X}^n \to \mathcal{D}^{n'}$$
Where $\mathcal{X}$ is some sample space, and the same for $\mathcal{D}$. My question is then - what does the superscript $n$ denote?
Based on the context they use it in I imaging it is something like the following: If we denote the $n$ r.v.'s as a vector $\mathbf{X}$, then $\mathcal{X}^n$ denote the set of all possible realizations of $\mathbf{X}$. However this is just a guess.
$\mathcal{X}^n$ is here the $n$-fold Cartesian product of the set $\mathcal{X}$; that is, $$ \mathcal{X}^n = \underbrace{\mathcal{X}\times\mathcal{X}\times\cdots\times \mathcal{X}}_{n\text{ times}} $$ What this means here is that $\mathcal{C}$ is a function which takes as input (a tuple of) $n$ values in $\mathcal{X}$, and outputs (a tuple of) $n'$ values in $\mathcal{D}^{n'}$: $$ \mathcal{C}(x_1,\dots,x_n) = (d_1,\dots,d_{n'}) $$ for $x_1,\dots,x_n\in \mathcal{X}$ and for some $d_1,\dots,d_{n'}\in \mathcal{D}$.
Note that we can write either $x_1,\dots,x_n\in \mathcal{X}$ (all $n$ elements are in $\mathcal{X}$) or $(x_1,\dots,x_n)\in \mathcal{X}^n$ (the tuple of $n$ elements is in $\mathcal{X}^n$). These have equivalent meanings.