Today's lectures

CodingAndCryptography/01_noiseless_coding.tex

@@ -341,13 +341,14 @@ \subsection{Huffman coding}
     which yields a prefix-free code with strictly smaller expected word length.
 \end{proof}
 
-\begin{theorem}
+\begin{theorem}[Huffman 1952]
     Huffman codes are optimal.
 \end{theorem}
 
 \begin{proof}
     The proof is by induction on $m$.
     If $m = 2$, then the codewords are 0 and 1, which is clearly optimal.
+
     Assume $m > 2$, and let $c_m$ be the Huffman code for $X_m$ which takes values $\mu_1, \dots, \mu_m$ with probabilities $p_1 \geq \dots \geq p_m$.
     $c_m$ is constructed from a Huffman code $c_{m-1}$ with random variable $X_{m-1}$ taking values $\mu_1, \dots, \mu_{n-2}, \nu$ with probabilities $p_1, \dots, p_{m-2}, p_{m-1} + p_m$.
     The code $c_{m-1}$ is optimal by the inductive hypothesis.

CodingAndCryptography/02_noisy_channels.tex

@@ -1,22 +1,26 @@
 \section{Noisy channels}
 
 \subsection{Decoding rules}
-\begin{definition}
+\begin{definition}[Binary {$[n,m]$}-code]
     A \vocab{binary $[n,m]$-code} is a subset $C$ of $\qty{0,1}^n$ of size $m = \abs{C}$.
     We say $n$ is the \vocab{length} of the code, and elements of $C$ are called \vocab{codewords}.
 \end{definition}
+
 We use an $[n,m]$-code to send one of $m$ messages through a channel using $n$ bits.
 For instance, if the channel is a binary symmetric channel, we use the channel $n$ times.
 Note that $1 \leq m \leq 2^n$, so the information rate $\rho(C) = \frac{1}{n} \log m$ satisfies $0 \leq \rho(C) \leq 1$.
 If $m = 1$, $\rho(C) = 0$, and if $C = \qty{0,1}^n$, $\rho(C) = 1$.
+
 \begin{definition}
     Let $x, y \in \qty{0,1}^n$.
     The \vocab{Hamming distance} between $x$ and $y$ is
     \begin{align*}
         d(x,y) = \abs{\qty{i \mid x_i \neq y_i}}
     \end{align*}
 \end{definition}
+
 In this section, we consider only the binary symmetric channel with probability $p$.
+
 \begin{definition}
     Let $C$ be a binary $[n,m]$-code.
     \begin{itemize}

CodingAndCryptography/cc.tex

@@ -15,10 +15,10 @@
     \maketitle
     \tableofcontents
 
-    \input{00_modelling_communication.tex}
-    \input{01_noiseless_coding.tex}
-    \input{02_noisy_channels.tex}
-    \input{03_information_theory.tex}
-    \input{04_algebraic_coding_theory.tex}
-    \input{05_cryptography.tex}
+    \include{00_modelling_communication.tex}
+    \include{01_noiseless_coding.tex}
+    \include{02_noisy_channels.tex}
+    \include{03_information_theory.tex}
+    \include{04_algebraic_coding_theory.tex}
+    \include{05_cryptography.tex}
 \end{document}