Aufgaben:Exercise 1.6: (7, 4) Hamming Code - Revision history

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	[[Category:Channel Coding: Exercises\|^1.3 Examples of Binary Block Codes^]]		[[Category:Channel Coding: Exercises\|^1.3 Examples of Binary Block Codes^]]
	[[de:Aufgaben:1.~~6_Zum_~~(7,_4)–Hamming–Code]]		[[de:Aufgaben:Aufgabe 1.6: Zum (7, 4)–Hamming–Code]]

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	[[Category:Channel Coding: Exercises\|^1.3 Examples of Binary Block Codes^]]		[[Category:Channel Coding: Exercises\|^1.3 Examples of Binary Block Codes^]]
			[[de:Aufgaben:1.6_Zum_(7,_4)–Hamming–Code]]

Hwang at 16:56, 1 November 2022

2022-11-01T16:56:17Z

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	[[File:P_ID2388__KC_A_1_6_neu.png\|right\|frame\|Code table of the  $\text{HC (7, 4)}$]]		[[File:P_ID2388__KC_A_1_6_neu.png\|right\|frame\|Code table of the  $\text{HC (7, 4)}$]]

	In 1962,  [https://en.wikipedia.org/wiki/Richard_Hamming Richard Wesley Hamming]  specified a class of binary block codes that differ in the number  $m$  of check bits supplied. The ~~codeword~~ length is always  $n = 2^m - 1$. The information word consists of  $k = n - m$  bits:		In 1962,  [https://en.wikipedia.org/wiki/Richard_Hamming Richard Wesley Hamming]  specified a class of binary block codes that differ in the number  $m$  of check bits supplied. The code word length is always  $n = 2^m - 1$. The information word consists of  $k = n - m$  bits:

	*$m = 2$:   $\text{(3, 1)}$ Hamming code   ⇒   identical to  [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes\|$\text{RC (3, 1)}$]],		*$m = 2$:   $\text{(3, 1)}$ Hamming code   ⇒   identical to  [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes\|$\text{RC (3, 1)}$]],

Guenter at 09:22, 15 June 2022

2022-06-15T09:22:10Z

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	Where:		Where:
	*$m = 2$:  $\rm (3, 1)$  Hamming code, ⇒ identical to$  [[Channel_Coding/Examples_of_Binary_Block_Codes#RepetitionCodes\|$\text{RC (3, 1)}$,]]		*$m = 2$:  $\rm (3, 1)$  Hamming code, ⇒ identical to  [[Channel_Coding/Examples_of_Binary_Block_Codes#RepetitionCodes\|$\text{RC (3, 1)}$]],

	*$m = 3$:  $\rm (7, 4)$  Hamming code,		*$m = 3$:  $\rm (7, 4)$  Hamming code,
	*$m = 4$:  $\rm (15, 11)$  Hamming code,		*$m = 4$:  $\rm (15, 11)$  Hamming code,

Guenter at 09:19, 15 June 2022

2022-06-15T09:19:28Z

Show changes

Guenter at 08:06, 15 June 2022

2022-06-15T08:06:15Z

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	*Reference is made to the section  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|"Hamming Codes"]].		*Reference is made to the section  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|"Hamming Codes"]].

	*For this Hamming code,  different ~~test~~ equations were used than in the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|theory section]].  Therefore,  the code tables also differ.		*For this Hamming code,  different parity-check equations were used than in the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|theory section]].  Therefore,  the code tables also differ.

	*In the  [[Aufgaben:Exercise_1.07:_Check_and_Generator_Matrix_of_the_HC_(7,_4,_3)\|Exercise 1.7]],  where the same code is used,  the chart of the check equations is given.		*In the  [[Aufgaben:Exercise_1.07:_Check_and_Generator_Matrix_of_the_HC_(7,_4,_3)\|Exercise 1.7]],  where the same code is used,  the chart of the parity-check equations is given.

Guenter at 08:03, 15 June 2022

2022-06-15T08:03:04Z

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	In the course of this ~~task~~ there are questions		In the course of this exercise there are questions
	*to the code size  $ \|\hspace{0.05cm}\mathcal{C}\hspace{0.05cm}\|$,		*to the code size  $ \|\hspace{0.05cm}\mathcal{C}\hspace{0.05cm}\|$,
	*to the code rate  $R$, and		*to the code rate  $R$,  and
	*to the minimum distance  $d_{\rm min}$.		*to the minimum distance  $d_{\rm min}$.


	Furthermore, we want to clarify whether the  $\underline{u}_{i} ⇒ \underline{x}_{i}$  ~~given for this task by its code table~~  ~~$\text{(7, 4)}$ Hamming code is systematic,~~ and whether it is a so called "perfect code". The ~~running~~ index can take the values  $i = 1, \text{...}\hspace{0.05cm} , 2^k =16$ .		Furthermore,  we want to clarify whether the  $\text{(7, 4)}$  Hamming code given for this task by its code table  $\underline{u}_{i} ⇒ \underline{x}_{i}$  is systematic,  and whether it is a so called  "perfect code".  The control index can take the values  $i = 1, \text{...}\hspace{0.05cm} , 2^k =16$ .

	One speaks of a  ''perfect code'' if the following condition is satisfied:		One speaks of a  "perfect code"  if the following condition is satisfied:
	:$$2^k = \frac{2^n} {\sum_{f=0}^t {n \choose f}}\hspace{0.3cm} \Rightarrow \hspace{0.3cm} 2^m = {\sum_{f=0}^t {n \choose f}} \hspace{0.05cm}.$$		:$$2^k = \frac{2^n} {\sum_{f=0}^t {n \choose f}}\hspace{0.3cm} \Rightarrow \hspace{0.3cm} 2^m = {\sum_{f=0}^t {n \choose f}} \hspace{0.05cm}.$$

	Here  $t$  denotes the number of correctable errors. For odd minimum distance  $d_{\rm min}$  holds:		Here  $t$  denotes the number of correctable errors.  For odd minimum distance  $d_{\rm min}$  holds:
	:$$ t = \frac{d_{\rm min}-1 } {2} \hspace{0.05cm}.$$		:$$ t = \frac{d_{\rm min}-1 } {2} \hspace{0.05cm}.$$

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			Hints:
			*This exercise belongs to the chapter  [[Channel_Coding/Examples_of_Binary_Block_Codes\|"Examples of Binary Block Codes"]].

			*Reference is made to the section  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|"Hamming Codes"]].
	~~Hints:~~
	*~~This exercise belongs~~ to the ~~chapter~~  [[Channel_Coding/Examples_of_Binary_Block_Codes\|~~Examples of Binary Block~~ Codes]].		*For this Hamming code,  different test equations were used than in the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|theory section]].  Therefore,  the code tables also differ.
	*~~Reference is made to~~ the ~~chapter~~  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|~~Hamming Codes~~]].

	*For this Hamming code, different test equations were used than in the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|theory section]]. Therefore, the code tables also differ.		*In the  [[Aufgaben:Exercise_1.07:_Check_and_Generator_Matrix_of_the_HC_(7,_4,_3)\|Exercise 1.7]],  where the same code is used,  the chart of the check equations is given.
	*In the  [[Aufgaben:Exercise_1.07:_Check_and_Generator_Matrix_of_the_HC_(7,_4,_3)\|Exercise 1.7]], where the same code is used, the chart of the check equations is given.


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	<quiz display=simple>		<quiz display=simple>

	{Specify the characteristics of the given code  $\mathcal{C}$~~ ~~:		{Specify the characteristics of the given code  $\mathcal{C}$:
	\|type="{}"}		\|type="{}"}
	$ \hspace{0.05cm}\mathcal{C}\hspace{0.05cm}\| \ = \ $ { 16 }		$ \|\hspace{0.05cm}\mathcal{C}\hspace{0.05cm}\| \ = \ $ { 16 }
	$ k \hspace{0.46cm} = \ $ { 4 }		$ k \hspace{0.46cm} = \ $ { 4 }
	$ n \hspace{0.44cm} = \ $ { 7 }		$ n \hspace{0.44cm} = \ $ { 7 }
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	{is this a systematic code?		{Is this a systematic code?
	\|type="()"}		\|type="()"}
	+ Yes,		+ Yes,
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	$t\hspace{0.28cm} = \ $ { 1 }		$t\hspace{0.28cm} = \ $ { 1 }

	{Is the ~~hamming~~ code considered here perfect?		{Is the Hamming code considered here  "perfect"?
	\|type="()"}		\|type="()"}
	+ Yes,		+ Yes,
	- No.		- No.

	{Which statements are true regarding a perfect code?		{Which statements are true regarding a  "perfect code"?
	\|type="[]"}		\|type="[]"}
	- A perfect code always results in zero block error probability.		- A perfect code always results in zero block error probability.
	+ All receive words  $\underline{y}$  are assignable to a valid ~~codeword~~.		+ All receive words  $\underline{y}$  are assignable to a valid code word.
	+ For perfect codes, the minimum Hamming distance is odd.		+ For perfect codes,  the minimum Hamming distance is odd.

	{Which of the codes listed below are perfect?		{Which of the codes listed below are perfect?

Noah at 03:46, 11 June 2022

2022-06-11T03:46:47Z

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	Furthermore, we want to clarify whether the  $\underline{u}_{i} ⇒ \underline{x}_{i}$  given for this task by its code table  $\text{(7, 4)}$ Hamming code is systematic, and whether it is a so-called "perfect code". The running index can take the values  $i = 1, \text{...}\hspace{0.05cm} , 2^k =16$ .		Furthermore, we want to clarify whether the  $\underline{u}_{i} ⇒ \underline{x}_{i}$  given for this task by its code table  $\text{(7, 4)}$ Hamming code is systematic, and whether it is a so called "perfect code". The running index can take the values  $i = 1, \text{...}\hspace{0.05cm} , 2^k =16$ .

	One speaks of a  ''perfect code'' if the following condition is satisfied:		One speaks of a  ''perfect code'' if the following condition is satisfied:

Noah at 03:45, 11 June 2022

2022-06-11T03:45:12Z

Noah at 03:24, 11 June 2022

2022-06-11T03:24:44Z

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	*Reference is made to the chapter  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|Hamming Codes]].		*Reference is made to the chapter  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|Hamming Codes]].

	*For this Hamming code, different test equations were used than in the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|~~Theory part~~]]. Therefore, the code tables also differ.		*For this Hamming code, different test equations were used than in the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|theory section]]. Therefore, the code tables also differ.
	*In the  [[Aufgaben:Exercise_1.07:_Check_and_Generator_Matrix_of_the_HC_(7,_4,_3)\|Exercise 1.7]], where the same code is used, the chart of the check equations is given.		*In the  [[Aufgaben:Exercise_1.07:_Check_and_Generator_Matrix_of_the_HC_(7,_4,_3)\|Exercise 1.7]], where the same code is used, the chart of the check equations is given.

← Older revision		Revision as of 10:06, 15 June 2022
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	*Reference is made to the section  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|"Hamming Codes"]].		*Reference is made to the section  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|"Hamming Codes"]].

	*For this Hamming code,  different ~~test~~ equations were used than in the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|theory section]].  Therefore,  the code tables also differ.		*For this Hamming code,  different parity-check equations were used than in the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|theory section]].  Therefore,  the code tables also differ.

	*In the  [[Aufgaben:Exercise_1.07:_Check_and_Generator_Matrix_of_the_HC_(7,_4,_3)\|Exercise 1.7]],  where the same code is used,  the chart of the check equations is given.		*In the  [[Aufgaben:Exercise_1.07:_Check_and_Generator_Matrix_of_the_HC_(7,_4,_3)\|Exercise 1.7]],  where the same code is used,  the chart of the parity-check equations is given.

← Older revision		Revision as of 05:24, 11 June 2022
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	*Reference is made to the chapter  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|Hamming Codes]].		*Reference is made to the chapter  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|Hamming Codes]].

	*For this Hamming code, different test equations were used than in the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|~~Theory part~~]]. Therefore, the code tables also differ.		*For this Hamming code, different test equations were used than in the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|theory section]]. Therefore, the code tables also differ.
	*In the  [[Aufgaben:Exercise_1.07:_Check_and_Generator_Matrix_of_the_HC_(7,_4,_3)\|Exercise 1.7]], where the same code is used, the chart of the check equations is given.		*In the  [[Aufgaben:Exercise_1.07:_Check_and_Generator_Matrix_of_the_HC_(7,_4,_3)\|Exercise 1.7]], where the same code is used, the chart of the check equations is given.

← Older revision		Revision as of 05:45, 11 June 2022
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	Furthermore, we want to clarify whether the  $\underline{u}_{i} ⇒ \underline{x}_{i}$  given for this task by its code table  $\text{(7, 4)}$-Hamming code is systematic, and whether it is a so-called "perfect code". The running index can take the values  $i = 1, \text{...}\hspace{0.05cm} , 2^k =16$ .		Furthermore, we want to clarify whether the  $\underline{u}_{i} ⇒ \underline{x}_{i}$  given for this task by its code table  $\text{(7, 4)}$ Hamming code is systematic, and whether it is a so-called "perfect code". The running index can take the values  $i = 1, \text{...}\hspace{0.05cm} , 2^k =16$ .

	One speaks of a  ''perfect code'' if the following condition is satisfied:		One speaks of a  ''perfect code'' if the following condition is satisfied:
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	- A perfect code always results in zero block error probability.		- A perfect code always results in zero block error probability.
	+ All receive words  $\underline{y}$  are assignable to a valid codeword.		+ All receive words  $\underline{y}$  are assignable to a valid codeword.
	+ For perfect codes, the minimum Hamming~~–~~distance is odd.		+ For perfect codes, the minimum Hamming distance is odd.

	{Which of the codes listed below are perfect?		{Which of the codes listed below are perfect?
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	*Each codeword $\underline{x}$ first contains the $k = 4$ bits of the information word $\underline{u}$. This is followed by $m = 3$ check bits:		*Each codeword $\underline{x}$ first contains the $k = 4$ bits of the information word $\underline{u}$. This is followed by $m = 3$ check bits:
	:$$\underline{x} = ( x_1, x_2, x_3, x_4, x_5, x_5,x_7) = ( u_1, u_2, u_3, u_4, p_1, p_2, p_3) \hspace{0.05cm}.$$		:$$\underline{x} = ( x_1, x_2, x_3, x_4, x_5, x_5,x_7) = ( u_1, u_2, u_3, u_4, p_1, p_2, p_3) \hspace{0.05cm}.$$
	*This corresponds exactly to the definition of a [[Channel_Coding/~~General_Description_linear_Block_codes~~\|systematic code]] .		*This corresponds exactly to the definition of a [[Channel_Coding/General_Description_of_Linear_Block_Codes\|systematic code]] .


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	:$$ 2^m = {\sum_{f=0}^t {n \choose f}} \hspace{0.05cm}.$$		:$$ 2^m = {\sum_{f=0}^t {n \choose f}} \hspace{0.05cm}.$$

	*In the (7, 4)-Hamming code considered here, $n = 7$, $m = 3$, and $t = 1$, giving a value of 8 on both sides of the equation:		*In the (7, 4) Hamming code considered here, $n = 7$, $m = 3$, and $t = 1$, giving a value of 8 on both sides of the equation:

	:$$ 2^3 = 8\hspace{0.05cm}, \hspace{0.35cm} {\sum_{f=0}^1 {7 \choose f}} = {7 \choose 0} + {7 \choose 1} = 1 + 7 = 8 \hspace{0.05cm}.$$		:$$ 2^3 = 8\hspace{0.05cm}, \hspace{0.35cm} {\sum_{f=0}^1 {7 \choose f}} = {7 \choose 0} + {7 \choose 1} = 1 + 7 = 8 \hspace{0.05cm}.$$
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	[[File:P_ID2389__KC_A_1_6_ML.png\|right\|frame\|illustration of a perfect code]]		[[File:P_ID2389__KC_A_1_6_ML.png\|right\|frame\|illustration of a perfect code]]

	Let's illustrate statement 2 with a graph. Here, the high-dimensional space is shown in a highly simplified way (in 2D). We assume the numerical values $k = 4$, $n = 7$, $m = 3$ and $t = 1$ of the (7, 4, 3)-Hamming code:		Let's illustrate statement 2 with a graph. Here, the high-dimensional space is shown in a highly simplified way (in 2D). We assume the numerical values $k = 4$, $n = 7$, $m = 3$ and $t = 1$ of the (7, 4, 3) Hamming code:

	*For the receive word, $2^7 = 128$ points are possible in 7-dimensional space. The red dots mark the $2^4 = 16$ valid codewords.		*For the receive word, $2^7 = 128$ points are possible in 7 dimensional space. The red dots mark the $2^4 = 16$ valid codewords.

	*The circles include 8 points each, namely a valid codeword and $n = 7$ receive words after only one error, which are assigned to exactly this codeword during decoding.		*The circles include 8 points each, namely a valid codeword and $n = 7$ receive words after only one error, which are assigned to exactly this codeword during decoding.
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	Where:		Where:
	*$m = 2$: (3, 1) Hamming code, ⇒ identical to [[Channel_Coding/~~Examples_binary_BlockCodes~~#~~RepeatCodes~~\|'''RC (3, 1)''',]]		*$m = 2$: (3, 1) Hamming code, ⇒ identical to [[Channel_Coding/Examples_of_Binary_Block_Codes#RepetitionCodes\|'''RC (3, 1)''',]]
	*$m = 3$: (7, 4) Hamming code,		*$m = 3$: (7, 4) Hamming code,
	*$m = 4$: (15, 11) Hamming code,		*$m = 4$: (15, 11) Hamming code,