Difference between revisions of "Aufgaben:Exercise 1.1: For Labeling Books"

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{{quiz-Header|Buchseite=Kanalcodierung/Zielsetzung_der_Kanalcodierung
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{{quiz-Header|Buchseite=Channel_Coding/Objective_of_Channel_Coding
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}}
 
}}
  
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[[File:P_ID2380__KC_A_1_1.png|right|frame|ISBN–10? Oder ISBN–13?]]
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[[File:EN_KC_A_1_1.png|right|frame|'''ISBN–10'''? Or '''ISBN–13'''?]]
  
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Seit den 1960er Jahren werden alle Bücher mit einer 10–stelligen ''International Standard Book Number'' versehen. Die letzte Ziffer dieser sog. '''ISBN–10–Angabe''' berechnet sich dabei entsprechend folgender Regel:
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Since the 1960s,  all books are provided with a 10-digit  "International Standard Book Number"  $\rm (ISBN)$.  The last digit of this so-called  '''ISBN-10 specification'''  is calculated according to the following rule:
  
 
:$$ z_{10}= \left ( \sum_{i=1}^{9} \hspace{0.2cm} i \cdot z_i \right ) \hspace{-0.2cm} \mod 11 \hspace{0.05cm}.$$
 
:$$ z_{10}= \left ( \sum_{i=1}^{9} \hspace{0.2cm} i \cdot z_i \right ) \hspace{-0.2cm} \mod 11 \hspace{0.05cm}.$$
 
   
 
   
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Seit 2007 ist zusätzlich die Angabe entsprechend des Standards '''ISBN–13''' verpflichtend, wobei die Prüfziffer $z_{\rm 13}$ sich dann wie folgt ergibt:
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Since 2007,  the specification according to the standard  '''ISBN-13'''  is additionally mandatory,  whereby the check digit  $z_{\rm 13}$  then results as follows:
  
 
:$$z_{13} = 10 - \left ( \sum_{i=1}^{12} \hspace{0.2cm} z_i \cdot 3^{(i+1)\mod 2} \right ) \hspace{-0.2cm} \mod 10 \hspace{0.05cm}.$$
 
:$$z_{13} = 10 - \left ( \sum_{i=1}^{12} \hspace{0.2cm} z_i \cdot 3^{(i+1)\mod 2} \right ) \hspace{-0.2cm} \mod 10 \hspace{0.05cm}.$$
 
   
 
   
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Nebenstehend sind einige beispielhafte ISBN angegeben. Hierauf beziehen sich die folgenden Fragen.
 
  
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''Hinweis'': Die Aufgabe gehört zum Themengebiet von Kapitel [[Kanalcodierung/Zielsetzung_der_Kanalcodierung|Zielsetzung_der_Kanalcodierung]]
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Some exemplary  "ISBNs"  are given opposite.  The following questions refer to these.
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 +
 
 +
 
 +
 
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Hints:  This exercise belongs to the chapter  [[Channel_Coding/Objective_of_Channel_Coding|"Objective of Channel Coding"]]
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===Fragebogen===
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===Questions===
  
 
<quiz display=simple>
 
<quiz display=simple>
  
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Um welchen Standard handelt es sich bei Beispiel 1?   
 
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|type="[]"}
 
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- ISBN–10,
 
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+ISBN–13.
 
  
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{Entsprechend Beispiel 2 sind zwei Ziffern einer ISBN–13 ausgelöscht. Kann man die ISBN rekonstruieren? Wenn Ja: Geben Sie die ISBN–13 an.
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What is the standard for &nbsp;$\text{Example 1}$?  
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|type="[]"}
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|type="()"}
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- ja,
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- ISBN-10,
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+Nein.
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+ISBN-13.
  
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{Entsprechend Beispiel 3 ist eine Ziffer einer ISBN–13 ausgelöscht. Kann die ISBN rekonstruiert werden? Wenn Ja: Geben Sie die ISBN–13 an.
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{Accordingly &nbsp;$\text{Example 2}$&nbsp; two digits of an ISBN-13 are deleted.&nbsp; Is it possible to reconstruct the ISBN?&nbsp; If yes: &nbsp; Specify the ISBN-13.
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|type="[]"}
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|type="()"}
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+Ja,
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- Yes,
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-Nein.
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+No.
  
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{Wieviele verschiedene Werte kann die Prüfziffer  $z_{\rm 10}$ bei ISBN–10 annehmen?
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{According to &nbsp;$\text{Example 3}$&nbsp; one digit of an ISBN-13 is erased.&nbsp; Can the ISBN be reconstructed?&nbsp; If Yes: &nbsp; Specify the ISBN-13.
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|type="()"}
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+Yes,
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-No.
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{How many different values &nbsp;$(M)$&nbsp; can the check digit&nbsp; $z_{\rm 10}$&nbsp; take for ISBN-10?
 
|type="{}"}
 
|type="{}"}
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$M \ = \ $ { 11 3% } $\ \rm$  
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$M \ = \ $ { 11 3% } $\ \rm$  
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{Mitgeteilt als ISBN–10 wird 3–8273–7064–7. Welche Aussage trifft zu?
 
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|type="[]"}
 
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- Dies ist keine zulässige ISBN.
 
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+ Die ISBN könnte richtig sein.
 
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- Die ISBN ist mit Sicherheit richtig.
 
  
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{Assumed as ISBN-10 is&nbsp; "3-8273-7064-7".&nbsp; Which statement is true?
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|type="()"}
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- This is not a valid ISBN.
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+ The ISBN could be correct.
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- The ISBN is certainly correct.
 
</quiz>
 
</quiz>
  
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===Musterlösung===
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===Solution===
 
{{ML-Kopf}}
 
{{ML-Kopf}}
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'''(1)'''&nbsp; Allein durch Abzählen der ISBN–Ziffern erkennt man, dass Antwort 2 richtig ist. Die gewichtete Summe über alle Ziffern ergibt ein Vielfaches von 10:
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'''(1)'''&nbsp; Just by counting the ISBN digits,&nbsp; you can tell that&nbsp; <u>answer 2</u>&nbsp; is correct.&nbsp; The weighted sum over all digits is a multiple of 10:
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S \hspace{-0.1cm}& = & \hspace{-0.1cm} \sum_{i=1}^{13} \hspace{0.2cm} z_i \cdot 3^{(i+1) \hspace{-0.2cm} \mod 2}   =\\
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:$$S \ = \ \hspace{-0.1cm} \sum_{i=1}^{13} \hspace{0.2cm} z_i \cdot 3^{(i+1) \hspace{-0.2cm} \cdot 2} = (9+8+8+7+6+8) \cdot 1 + (7+3+2+3+0+4) \cdot 3 = 110\hspace{0.3cm}
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\hspace{-0.1cm}& = & \hspace{-0.1cm}  (9+8+8+7+7+6+8) \cdot 1 +
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\Rightarrow \hspace{0.3cm} S \hspace{-0.2cm} \mod 10 \hspace{0.15cm}\underline {= 0} \hspace{0.05cm}.$$
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  (7+3+2+3+0+4) \cdot 3 = 110
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'''(2)'''&nbsp; Die Antwort ist Nein. Mit einer einzigen Prüfziffer lässt sich nur eine Auslöschung rekonstruieren.
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'''(2)'''&nbsp; The answer is&nbsp; <u>No</u>.&nbsp; Only one cancellation can be reconstructed with a single check digit.
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'''3.'''
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'''4.'''
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'''5.'''
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'''(3)'''&nbsp; One digit can be reconstructed &nbsp;⇒ &nbsp; <u>Yes</u>.&nbsp; For the digit&nbsp; $z_{\rm 8}$,&nbsp; it must hold:
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'''6.'''
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:$$[(9+8+4+3+0+1+2) \cdot 1 + (7+3+5+z_8+7+5) \cdot 3] \hspace{-0.2cm} \mod 10 = 0\hspace{0.3cm}
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'''7.'''
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\Rightarrow \hspace{0.3cm} [108 + 3z_8] \hspace{-0.2cm} \mod 10 = 0 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} z_8 \hspace{0.15cm}\underline {= 4} \hspace{0.05cm}.$$
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'''(4)'''&nbsp; By the modulo 11 operation,&nbsp; $z_{10}$&nbsp; can take the values&nbsp; $0,\ 1,\ \text{...} ,\ 10$ &nbsp; ⇒ &nbsp; $\underline{M =11}$.
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*Since&nbsp; "10"&nbsp; is not a digit,&nbsp; one makes do with&nbsp; $z_{10} = \rm X$.
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*This corresponds to the Roman representation of the number&nbsp; "10".
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 +
 
 +
 
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'''(5)'''&nbsp; The test condition is:
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:$$\ \ S= \left ( \sum_{i=1}^{10} \hspace{0.2cm} i \cdot z_i \right ) \hspace{-0.2cm} \mod 11 = 0 \hspace{0.05cm}.$$
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*The given ISBN satisfies this condition:
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:$$3 \cdot 1 + 8 \cdot 2 + 2 \cdot 3 + 7 \cdot 4 + 3 \cdot 5 + 7 \cdot 6 + 0 \cdot 7 + 6 \cdot 8 + 4 \cdot 9 + 7 \cdot 10 = 264\hspace{0.3cm}
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⇒\hspace{0.3cm} S= 264 \hspace{-0.3cm} \mod 11 = 0 \hspace{0.05cm}.$$ 
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*Correct is&nbsp; <u>statement 2</u>,&nbsp; since the check sum&nbsp; $S = 0$&nbsp; could result even with more than one error.
 
{{ML-Fuß}}
 
{{ML-Fuß}}
  
  
  
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[[Category:Aufgaben zu  Kanalcodierung|^1.1 Zielsetzung der Kanalcodierung
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[[Category:Channel Coding: Exercises|^1.1 Objective of Channel Coding^]]
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^]]
 

Latest revision as of 13:53, 6 June 2022

ISBN–10? Or ISBN–13?

Since the 1960s,  all books are provided with a 10-digit  "International Standard Book Number"  $\rm (ISBN)$.  The last digit of this so-called  ISBN-10 specification  is calculated according to the following rule:

$$ z_{10}= \left ( \sum_{i=1}^{9} \hspace{0.2cm} i \cdot z_i \right ) \hspace{-0.2cm} \mod 11 \hspace{0.05cm}.$$

Since 2007,  the specification according to the standard  ISBN-13  is additionally mandatory,  whereby the check digit  $z_{\rm 13}$  then results as follows:

$$z_{13} = 10 - \left ( \sum_{i=1}^{12} \hspace{0.2cm} z_i \cdot 3^{(i+1)\mod 2} \right ) \hspace{-0.2cm} \mod 10 \hspace{0.05cm}.$$


Some exemplary  "ISBNs"  are given opposite.  The following questions refer to these.



Hints:  This exercise belongs to the chapter  "Objective of Channel Coding"


Questions

1

What is the standard for  $\text{Example 1}$?

ISBN-10,
ISBN-13.

2

Accordingly  $\text{Example 2}$  two digits of an ISBN-13 are deleted.  Is it possible to reconstruct the ISBN?  If yes:   Specify the ISBN-13.

Yes,
No.

3

According to  $\text{Example 3}$  one digit of an ISBN-13 is erased.  Can the ISBN be reconstructed?  If Yes:   Specify the ISBN-13.

Yes,
No.

4

How many different values  $(M)$  can the check digit  $z_{\rm 10}$  take for ISBN-10?

$M \ = \ $

$\ \rm$

5

Assumed as ISBN-10 is  "3-8273-7064-7".  Which statement is true?

This is not a valid ISBN.
The ISBN could be correct.
The ISBN is certainly correct.


Solution

Solution

(1)  Just by counting the ISBN digits,  you can tell that  answer 2  is correct.  The weighted sum over all digits is a multiple of 10:

$$S \ = \ \hspace{-0.1cm} \sum_{i=1}^{13} \hspace{0.2cm} z_i \cdot 3^{(i+1) \hspace{-0.2cm} \cdot 2} = (9+8+8+7+6+8) \cdot 1 + (7+3+2+3+0+4) \cdot 3 = 110\hspace{0.3cm} \Rightarrow \hspace{0.3cm} S \hspace{-0.2cm} \mod 10 \hspace{0.15cm}\underline {= 0} \hspace{0.05cm}.$$


(2)  The answer is  No.  Only one cancellation can be reconstructed with a single check digit.


(3)  One digit can be reconstructed  ⇒   Yes.  For the digit  $z_{\rm 8}$,  it must hold:

$$[(9+8+4+3+0+1+2) \cdot 1 + (7+3+5+z_8+7+5) \cdot 3] \hspace{-0.2cm} \mod 10 = 0\hspace{0.3cm} \Rightarrow \hspace{0.3cm} [108 + 3z_8] \hspace{-0.2cm} \mod 10 = 0 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} z_8 \hspace{0.15cm}\underline {= 4} \hspace{0.05cm}.$$


(4)  By the modulo 11 operation,  $z_{10}$  can take the values  $0,\ 1,\ \text{...} ,\ 10$   ⇒   $\underline{M =11}$.

  • Since  "10"  is not a digit,  one makes do with  $z_{10} = \rm X$.
  • This corresponds to the Roman representation of the number  "10".


(5)  The test condition is:

$$\ \ S= \left ( \sum_{i=1}^{10} \hspace{0.2cm} i \cdot z_i \right ) \hspace{-0.2cm} \mod 11 = 0 \hspace{0.05cm}.$$
  • The given ISBN satisfies this condition:
$$3 \cdot 1 + 8 \cdot 2 + 2 \cdot 3 + 7 \cdot 4 + 3 \cdot 5 + 7 \cdot 6 + 0 \cdot 7 + 6 \cdot 8 + 4 \cdot 9 + 7 \cdot 10 = 264\hspace{0.3cm} ⇒\hspace{0.3cm} S= 264 \hspace{-0.3cm} \mod 11 = 0 \hspace{0.05cm}.$$
  • Correct is  statement 2,  since the check sum  $S = 0$  could result even with more than one error.

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