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Improved ECC handout

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2023-06-19 15:42:33 -07:00
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52
Advanced/Error-Correcting Codes/parts/00 detection.tex
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@ -4,7 +4,7 @@ An ISBN\footnote{International Standard Book Number} is a unique numeric book id
\vspace{3mm}
Say we have a sequence of nine digits, forming a partial ISBN-10: $n_1 n_2 ... n_9$. \\
Say we have a sequence of nine digits, forming a partial ISBN-10: $n_1 n_2 ... n_9$. \par
The final digit, $n_{10}$, is calculated as follows:
$$
@ -41,8 +41,8 @@ $$
\vspace{2mm}
$10n_1 + 9n_2 + ... + 2n_9 + n_{10} \equiv$ \\
$(-n_1) + (-2n_2) + ... + (-9n_9) + n_{10} =$ \\
$10n_1 + 9n_2 + ... + 2n_9 + n_{10} \equiv$ \par
$(-n_1) + (-2n_2) + ... + (-9n_9) + n_{10} =$ \par
$-n_{10} + n_{10} \equiv 0$
\vspace{2mm}
@ -54,7 +54,7 @@ $$
\vfill
\problem{}
Take a valid ISBN-10 and change one digit. Is it possible that you get another valid ISBN-10? \\
Take a valid ISBN-10 and change one digit. Is it possible that you get another valid ISBN-10? \par
Provide an example or a proof.
\begin{solution}
@ -62,29 +62,29 @@ Provide an example or a proof.
\vspace{3mm}
If you change one digit of the ISBN, $S$ changes by $km$, where $k \in \{1,2,...,10\}$ and $|m| \leq 10$. \\
If you change one digit of the ISBN, $S$ changes by $km$, where $k \in \{1,2,...,10\}$ and $|m| \leq 10$. \par
$k$ and $m$ cannot be divisible by 11, thus $km$ cannot be divisible by 11.
\vspace{3mm}
We know that $S \equiv 0 \text{ (mod 11)}$. \\
We know that $S \equiv 0 \text{ (mod 11)}$. \par
After the change, the checksum is $S + km \equiv km \not\equiv 0 \text{ (mod 11)}$.
\end{solution}
\vfill
\problem{}
Take a valid ISBN-10 and swap two adjacent digits. When will the result be a valid ISBN-10? \\
Take a valid ISBN-10 and swap two adjacent digits. When will the result be a valid ISBN-10? \par
This is called a \textit{transposition error}.
\begin{solution}
Let $n_1n_2...n_{10}$ be a valid ISBN-10. \\
Let $n_1n_2...n_{10}$ be a valid ISBN-10. \par
When we swap $n_i$ and $n_{i+1}$, we subtract $n_i$ and add $n_{i+1}$ to the checksum.
\vspace{3mm}
If the new ISBN is to be valid, we must have that $n_{i+1} - n_i \equiv 0 \text{ (mod 11)}$. \\
If the new ISBN is to be valid, we must have that $n_{i+1} - n_i \equiv 0 \text{ (mod 11)}$. \par
This is impossible unless $n_i = n_{i+1}$. Figure out why yourself.
\end{solution}
@ -98,7 +98,7 @@ $$
n_{13} = \Biggr[ \sum_{i=1}^{12} n_i \times (2 + (-1)^i) \Biggl] \text{ mod } 10
$$
What is the last digit of the following ISBN-13? \\
What is the last digit of the following ISBN-13? \par
\texttt{978-0-380-97726-?}
\begin{solution}
@ -108,54 +108,54 @@ What is the last digit of the following ISBN-13? \\
\vfill
\problem{}
Take a valid ISBN-13 and change one digit. Is it possible that you get another valid ISBN-13? \\
Provide an example or a proof.
Take a valid ISBN-13 and change one digit. Is it possible that you get another valid ISBN-13? \par
If you can, provide an example; if you can't, provide a proof.
\begin{solution}
Let $n_1n_2...n_{13}$ be a valid ISBN-13. Choose some $n_i$ and change it to $m_i$. \\
Let $n_1n_2...n_{13}$ be a valid ISBN-13. Choose some $n_i$ and change it to $m_i$. \par
\vspace{3mm}
Since $n_i$, $m_i$ $\in \{0, 1, 2, ..., 9\}$, $-9 \leq n_i - m_i \leq 9$. \\
Since $n_i$, $m_i$ $\in \{0, 1, 2, ..., 9\}$, $-9 \leq n_i - m_i \leq 9$. \par
\vspace{2mm}
Case 0: $i$ is 13 \\
Case 0: $i$ is 13 \par
This is trivial.
\vspace{2mm}
Case 1: $i$ is odd \\
For the new ISBN to be valid, we need $n_i - m_i \equiv 0 \text{ (mod 10)}$. \\
Case 1: $i$ is odd \par
For the new ISBN to be valid, we need $n_i - m_i \equiv 0 \text{ (mod 10)}$. \par
This cannot happen if $n_i \neq m_i$.
\vspace{2mm}
Case 2: $i$ is even \\
For the new ISBN to be valid, we need $3(n_i - m_i) \equiv 0 \text{ (mod 10)}$ \\
Case 2: $i$ is even \par
For the new ISBN to be valid, we need $3(n_i - m_i) \equiv 0 \text{ (mod 10)}$ \par
This cannot happen, 10 and 3 are coprime.
\end{solution}
\vfill
\problem{}
Take a valid ISBN-13 and swap two adjacent digits. When will the result be a valid ISBN-13? \\
Take a valid ISBN-13 and swap two adjacent digits. When will the result be a valid ISBN-13? \par
\hint{The answer here is more interesting than it was last time.}
\begin{solution}
Say we swap $n_i$ and $n_{i+1}$, where $i \in \{1, 2, ..., 11\}$. \\
The checksum changes by $2(n_{i+1} - n_i)$, and will \\
Say we swap $n_i$ and $n_{i+1}$, where $i \in \{1, 2, ..., 11\}$. \par
The checksum changes by $2(n_{i+1} - n_i)$, and will \par
remain the same if this value is $\equiv 0 \text{ (mod 10)}$.
\end{solution}
\vfill
\problem{}
\texttt{978-0-08-2066-46-6} was a valid ISBN until I changed a single digit. \\
Can you tell me which digit I changed?
\problem{}<isbn-nocorrect>
\texttt{978-0-08-2066-46-6} was a valid ISBN until I changed a single digit. \par
Can you find the digit I changed? Can you recover the original ISBN?
\begin{solution}
Nope, unless you look at the meaning of each digit in the spec. \\
Nope, unless you look at the meaning of each digit in the spec. \par
If you're unlucky, maybe not even then.
\end{solution}