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Various typos #18

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8 changes: 4 additions & 4 deletions Script/week_5_content.tex
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Expand Up @@ -215,9 +215,9 @@ \section{Codes}
Given a code for a specific channel, we can study the probability that an error occurs while transmitting a message. For a fixed message $m$, this probability is represented by the following quantity.

\begin{definition}[Probability of error]\label{def:probability-of-error}
Given an $(M,n)$ code for a channel $(\mathcal{X},P_{Y|X},\mathcal{Y})$, the probability of error $\lambda_m$ is the probability that the decoded output is not equal to the input message $m$. More formally,
Given an $(M,n)$ code for a channel $(\mathcal{X},P_{Y|X},\mathcal{Y})$, the probability of error $\lambda_w$ is the probability that the decoded output is not equal to the input message $w$. More formally,
\[
\lambda_w := P[\mathtt{dec}(Y^n) \neq m \mid X^n = \mathtt{enc}(w)].
\lambda_w := P[\mathtt{dec}(Y^n) \neq w \mid X^n = \mathtt{enc}(w)].
\]
Given this quantity, the \term{maximal probability of error} is defined as
\[\lambda^{(n)} := \max_{w \in [M]} \lambda_w.\]
Expand Down Expand Up @@ -404,7 +404,7 @@ \subsection{Generalization: Linear Codes}~\label{sec:linear-codes}
\[
\mathtt{enc}(m) = G^T \cdot m.
\]
The codebook $C$ is the set $\Set{G^T \cdot m}{m \in \mathcal{X}^n}$.
The codebook $C$ is the set $\Set{G^T \cdot m}{m \in \{0,1\}^k}$.
\footnote{The reason for the transposition in $G^T$ is that historically, coding theorists preferred to use row vectors and matrix multiplication from the right instead of column vectors and multiplication from the left, which is more standard in other areas. Notice that for row vectors $c = m \cdot G$, we equivalently have column vectors $c^T = (m \cdot G)^T = G^T \cdot m^T$.}

\begin{example}[{Generator matrix of the $[7,4]$ Hamming code}]
Expand Down Expand Up @@ -479,7 +479,7 @@ \subsection{Generalization: Linear Codes}~\label{sec:linear-codes}
\begin{example}[The 5-bit repetition code]
The generating matrix of $R_5$, the 5-bit repetition code, is
\[G^T = \left(\begin{array}{c}
1\\1\\1\\1
1\\1\\1\\1\\1
\end{array}\right).\]
The parity check matrix is
\[
Expand Down
4 changes: 2 additions & 2 deletions Script/week_5b_content.tex
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Expand Up @@ -26,7 +26,7 @@
\node[anchor=west] at (6,2) {1};
\end{tikzpicture}
\end{center}
We can send two messages, $m_1$ and $m_2$, over the channel by defining $\enc{m_1} = a$ and $\enc{m_2} = b$. The decoding is defined as $\dec{0} = m_1$, and $\dec{1} = \dec{2} = m_2$.
We can send two messages, $m_1$ and $m_2$, over the channel by defining $\enc{m_1} = a$ and $\enc{m_2} = b$. The decoding is defined as $\dec{1} = m_1$, and $\dec{2} = \dec{3} = m_2$.
\end{example}


Expand Down Expand Up @@ -141,7 +141,7 @@ \section{Confusability graphs}
\[V(G) := \mathcal{X},\]
the set of input symbols of the channel, and
\[
E(G) := \{\{x,x'\} \subset \mathcal{X} \mid x \neq x' \mbox{ and } \exists y \in \mathcal{Y} \mbox{ s.t. } P_{Y|X}(y|x) \cdot P_{Y_X}(y|x') > 0\}
E(G) := \{\{x,x'\} \subset \mathcal{X} \mid x \neq x' \mbox{ and } \exists y \in \mathcal{Y} \mbox{ s.t. } P_{Y|X}(y|x) \cdot P_{Y|X}(y|x') > 0\}
\]
is the set of input pairs that are confusable (because they reach a shared output symbol $y \in \mathcal{Y}$).
\end{definition}
Expand Down