Aufgaben:Exercise 2.08: Generator Polynomials for Reed-Solomon - Revision history

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	[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]		[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]
	[[de:Aufgaben:2.~~08_RS–Generatorpolynome~~]]		[[de:Aufgaben:Aufgabe 2.08: Generatorpolynome für Reed-Solomon]]

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 *With the generator matrix $\mathbf{G}_{\rm C}$ thus holds:
-:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm C} =
+:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm C} =\begin{pmatrix}\alpha^4 & 1 & \alpha^3\end{pmatrix} \cdot\begin{pmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1\\1 & \alpha^1 & \alpha^2 & \alpha^3 & \alpha^4 & \alpha^5 & \alpha^6\\1 & \alpha^2 & \alpha^4 & \alpha^6 & \alpha^1 & \alpha^{3} & \alpha^{5}\end{pmatrix}\hspace{0.05cm}. $$
 [[File:EN_KC_Z_2_5_neu.png|right|frame|$\rm GF(2^3)$&nbsp; representation as powers, polynomials, vectors]]
@@ Line 83: / Line 75: @@
 '''(3)'''&nbsp; At the&nbsp; $\rm RSC \, (7, \, 5, \, 3)_8$,&nbsp; the information word&nbsp; $\underline{u} = (u_0, \, u_1, \, u_2, \, u_3, \, u_4)$&nbsp; must be considered.
 *With the generator matrix&nbsp; $\mathbf{G}_{\rm D}$&nbsp; one obtains:
-:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm D} =
+:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm D} =\begin{pmatrix}\alpha^4 & 1 & \alpha^3 & 0 & \alpha^6\end{pmatrix} \cdot\begin{pmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1\\1 & \alpha^1 & \alpha^2 & \alpha^3 & \alpha^4 & \alpha^5 & \alpha^6\\1 & \alpha^2 & \alpha^4 & \alpha^6 & \alpha^1 & \alpha^{3} & \alpha^{5}\\1 & \alpha^3 & \alpha^6 & \alpha^2 & \alpha^5 & \alpha^{1} & \alpha^{4}\\1 & \alpha^4 & \alpha^1 & \alpha^5 & \alpha^2 & \alpha^{6} & \alpha^{3}\end{pmatrix}\hspace{0.05cm}. $$
 *From this it follows:

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	[[File:P_ID2525__KC_A_2_8.png\|right\|frame\|Four generator matrices, three of which describe Reed-Solomon codes]]		[[File:P_ID2525__KC_A_2_8.png\|right\|frame\|Four generator matrices, three of which describe Reed-Solomon codes]]
	In the  [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8\|"Exercise 2.7"]]  you should determine the code words of the  $\rm RSC \, (7, \, 3, \, 5)_8$  via a polynomial.  However,  you can also determine the code word  $\underline{c}$   from the information word  $\underline{u}$  and the generator matrix  $\mathbf{G}$  according to the following equation:		In the  [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8\|"Exercise 2.7"]]  you should determine the code words of the  $\rm RSC \, (7, \, 3, \, 5)_8$  via a polynomial.  However,  you can also determine the code word  $\underline{c}$   from the information word  $\underline{u}$  and the generator matrix  $\mathbf{G}$  according to the following equation:
	:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}		:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	*Two of these generator matrices describe the  $\rm RSC \, (7, \, 3, \, 5)_8$.  In the subtask  '''(1)'''  is explicitly asked which.		*Two of these generator matrices describe the  $\rm RSC \, (7, \, 3, \, 5)_8$.  In the subtask  '''(1)'''  is explicitly asked which.
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	*This results according to the adjacent auxiliary table:		*This results according to the adjacent auxiliary table:
	:$$c_0 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot 1 + \alpha^{3}\cdot 1 =		:$$c_0 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot 1 + \alpha^{3}\cdot 1 =(110) + (001) + (011)= (100) = \alpha^{2} \hspace{0.05cm},$$
	(110) + (001) + (011)= (100) = \alpha^{2} \hspace{0.05cm},$$		:$$c_1 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha + \alpha^{3}\cdot \alpha^{2}=(110) + (010) + (110) = (011) = \alpha^{3} \hspace{0.05cm},$$
	:$$c_1 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha + \alpha^{3}\cdot \alpha^{2}=		:$$c_2 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha^{2} + \alpha^{3}\cdot \alpha^{4}=(110) + (100) + (001) = (011) = \alpha^{3} \hspace{0.05cm},$$
	(110) + (010) + (110) = (011) = \alpha^{3} \hspace{0.05cm},$$		:$$c_3 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha^{3} + \alpha^{3}\cdot \alpha^{6}=$(110) + (011) + (100) = (001) = 1 \hspace{0.05cm},$$
	:$$c_2 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha^{2} + \alpha^{3}\cdot \alpha^{4}=		:$$c_4 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha^{4} + \alpha^{3}\cdot \alpha^{1}= \alpha^{4} \hspace{0.05cm},$$
	(110) + (100) + (001) = (011) = \alpha^{3} \hspace{0.05cm},$$		:$$c_5 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha^{5} + \alpha^{3}\cdot \alpha^{3}=(110) + (111) + (101) = (100) = \alpha^{2} \hspace{0.05cm},$$
	:$$c_3 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha^{3} + \alpha^{3}\cdot \alpha^{6}=$		:$$c_6 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha^{6} + \alpha^{3}\cdot \alpha^{5}=(\alpha^{2} + \alpha) + (\alpha^2 +1) + \alpha = 1 \hspace{0.05cm}.$$
	(110) + (011) + (100) = (001) = 1 \hspace{0.05cm},$$
	:$$c_4 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha^{4} + \alpha^{3}\cdot \alpha^{1}
	= \alpha^{4} \hspace{0.05cm},$$
	:$$c_5 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha^{5} + \alpha^{3}\cdot \alpha^{3}=
	(110) + (111) + (101) = (100) = \alpha^{2} \hspace{0.05cm},$$
	:$$c_6 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot \alpha^{6} + \alpha^{3}\cdot \alpha^{5}=
	(\alpha^{2} + \alpha) + (\alpha^2 +1) + \alpha = 1 \hspace{0.05cm}.$$

	*You get exactly the same result as in subtask  '''(4)'''  of  [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8\|"Exercise 2.7"]].  Correct are the  <u>solutions 1 and 2</u>.		*You get exactly the same result as in subtask  '''(4)'''  of  [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8\|"Exercise 2.7"]].  Correct are the  <u>solutions 1 and 2</u>.
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	*Corresponding is also considered in the following calculations:		*Corresponding is also considered in the following calculations:
	:$$c_2 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ \alpha^{3} \right ] + \alpha^{6}\cdot \alpha^{1}=		:$$c_2 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ \alpha^{3} \right ] + \alpha^{6}\cdot \alpha^{1}=\left [ \alpha^{3} \right ] + \alpha^{7} =(011) + (001) = (010) = \alpha^{1} \hspace{0.05cm},$$
	\left [ \alpha^{3} \right ] + \alpha^{7} =		:$$c_3 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ 1 \right ] + \alpha^{6}\cdot \alpha^{5}=\left [ 1 \right ] + \alpha^{4}=(001) + (110) = (111) = \alpha^{5} \hspace{0.05cm},$$
	(011) + (001) = (010) = \alpha^{1} \hspace{0.05cm},$$		:$$c_4 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ \alpha^{4} \right ] + \alpha^{6}\cdot \alpha^{2}=\left [ \alpha^{4} \right ] + \alpha^{1} = (110) + (010) = (100)= \alpha^{2} \hspace{0.05cm},$$
	:$$c_3 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ 1 \right ] + \alpha^{6}\cdot \alpha^{5}=		:$$c_5 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ \alpha^{2} \right ] + \alpha^{6}\cdot \alpha^{6}=\left [ \alpha^{2} \right ] + \alpha^{5} =(100) + (111) = (011) = \alpha^{3} \hspace{0.05cm},$$
	\left [ 1 \right ] + \alpha^{4}=		:$$c_6 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ 1 \right ] + \alpha^{6}\cdot \alpha^{3}= \left [ 1 \right ] + \alpha^{2}= (001) + (100) = (101) = \alpha^{6} \hspace{0.05cm}.$$
	(001) + (110) = (111) = \alpha^{5} \hspace{0.05cm},$$
	:$$c_4 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ \alpha^{4} \right ] + \alpha^{6}\cdot \alpha^{2}=
	\left [ \alpha^{4} \right ] + \alpha^{1} = (110) + (010) = (100)
	= \alpha^{2} \hspace{0.05cm},$$
	:$$c_5 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ \alpha^{2} \right ] + \alpha^{6}\cdot \alpha^{6}=
	\left [ \alpha^{2} \right ] + \alpha^{5} =
	(100) + (111) = (011) = \alpha^{3} \hspace{0.05cm},$$
	:$$c_6 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ 1 \right ] + \alpha^{6}\cdot \alpha^{3}= \left [ 1 \right ] + \alpha^{2}
	= (001) + (100) = (101) = \alpha^{6} \hspace{0.05cm}.$$

	*This means:  <u>All proposed solutions</u>  are correct.		*This means:  <u>All proposed solutions</u>  are correct.

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	[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]		[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]
			[[de:Aufgaben:2.08_RS–Generatorpolynome]]

Guenter at 14:11, 10 October 2022

2022-10-10T14:11:31Z

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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  Correct are the <u>solutions 2 and 3</u>  ⇒  matrices $\mathbf{G}_{\rm B}$ and $\mathbf{G}_{\rm C}$.		'''(1)'''  Correct are the  <u>solutions 2 and 3</u>   ⇒   matrices  $\mathbf{G}_{\rm B}$  and  $\mathbf{G}_{\rm C}$.
	*In the matrix $\mathbf{G}_{\rm C}$ the allowed transformations  $\alpha^8 = \alpha, \ \alpha^{10} = \alpha^3$  and  $\alpha^{12} = \alpha^5$  have already been considered.		*In the matrix  $\mathbf{G}_{\rm C}$  the allowed transformations  $\alpha^8 = \alpha, \ \alpha^{10} = \alpha^3$  and  $\alpha^{12} = \alpha^5$  have already been considered.
	*The matrix $\mathbf{G}_{\rm A}$ holds for the $(7, \, 5, \, 3)$&~~ndash~~;Hamming code and $\mathbf{G}_{\rm D}$ belongs to the $\rm RSC \, (7, \, 5, \, 3)_8$. See subtask (3) for more details.
			*The matrix  $\mathbf{G}_{\rm A}$  holds for the  $(7, \, 5, \, 3)$  Hamming code and  $\mathbf{G}_{\rm D}$  belongs to the  $\rm RSC \, (7, \, 5, \, 3)_8$.  See subtask  '''(3)'''  for more details.



	'''(2)'''  In the $\rm RSC \, (7, \, 3, \, 5)_8$ ~~information symbols are processed~~ in each coding step $k = 3$, in coding step 1 according to the specification the symbols $\alpha^4, \ 1$ and $\alpha^3$.		'''(2)'''  In the  $\rm RSC \, (7, \, 3, \, 5)_8$,  in each coding step are processed  $k = 3$  information symbols,   <br>in coding step 1 according to the specification the symbols  $\alpha^4, \ 1$ and $\alpha^3$.

	~~[[File:EN_KC_Z_2_5_neu.png\|right\|frame\|$\rm GF(2^3)$  representation as powers, polynomials, vectors]]~~

	*With the generator matrix $\mathbf{G}_{\rm C}$ thus holds:		*With the generator matrix $\mathbf{G}_{\rm C}$ thus holds:
	:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm C} =		:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm C} =
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	1 & \alpha^2 & \alpha^4 & \alpha^6 & \alpha^1 & \alpha^{3} & \alpha^{5}		1 & \alpha^2 & \alpha^4 & \alpha^6 & \alpha^1 & \alpha^{3} & \alpha^{5}
	\end{pmatrix}\hspace{0.05cm}. $$		\end{pmatrix}\hspace{0.05cm}. $$
			[[File:EN_KC_Z_2_5_neu.png\|right\|frame\|$\rm GF(2^3)$  representation as powers, polynomials, vectors]]

	*This results according to the adjacent auxiliary table:		*This results according to the adjacent auxiliary table:
	:$$c_0 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot 1 + \alpha^{3}\cdot 1 =		:$$c_0 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \alpha^{4}\cdot 1 + 1 \cdot 1 + \alpha^{3}\cdot 1 =
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	(\alpha^{2} + \alpha) + (\alpha^2 +1) + \alpha = 1 \hspace{0.05cm}.$$		(\alpha^{2} + \alpha) + (\alpha^2 +1) + \alpha = 1 \hspace{0.05cm}.$$

	*You get exactly the same result as in subtask (4) of [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8\|"Exercise 2.7"]]. Correct are <u>solutions 1 and 2</u>.		*You get exactly the same result as in subtask  '''(4)'''  of  [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8\|"Exercise 2.7"]].  Correct are the  <u>solutions 1 and 2</u>.
	*So it is not $c_6 = 0$, but $c_6 = 1$.		*So it is not  $c_6 = 0$,  but $c_6 = 1$.



	'''(3)'''  At $\rm RSC \, (7, \, 5, \, 3)_8$, the information word $\underline{u} = (u_0, \, u_1, \, u_2, \, u_3, \, u_4)$ must be considered.		'''(3)'''  At the  $\rm RSC \, (7, \, 5, \, 3)_8$,  the information word  $\underline{u} = (u_0, \, u_1, \, u_2, \, u_3, \, u_4)$  must be considered.
	*With the generator matrix $\mathbf{G}_{\rm D}$ one obtains:		*With the generator matrix  $\mathbf{G}_{\rm D}$  one obtains:
	:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm D} =		:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm D} =
	\begin{pmatrix}		\begin{pmatrix}
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	:$$c_1 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ \alpha^{4}\cdot 1 + 1 \cdot \alpha + \alpha^{3}\cdot \alpha^{2} \right ] + 0 \cdot \alpha^{3} + \alpha^{6}\cdot \alpha^{4}= \left [ \alpha^{3} \right ] + \alpha^{3} = 0 \hspace{0.05cm}.$$		:$$c_1 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} \left [ \alpha^{4}\cdot 1 + 1 \cdot \alpha + \alpha^{3}\cdot \alpha^{2} \right ] + 0 \cdot \alpha^{3} + \alpha^{6}\cdot \alpha^{4}= \left [ \alpha^{3} \right ] + \alpha^{3} = 0 \hspace{0.05cm}.$$

	*This takes into account that the bracket expression $[ \ \text{...} \ ]$ corresponds exactly to the result $c_1$ of subtask (2).		*This takes into account that the bracket expression  $[ \ \text{...} \ ]$  corresponds exactly to the result  $c_1$  of subtask  '''(2)'''.

	*Corresponding is also considered in the following calculations:		*Corresponding is also considered in the following calculations:
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	= (001) + (100) = (101) = \alpha^{6} \hspace{0.05cm}.$$		= (001) + (100) = (101) = \alpha^{6} \hspace{0.05cm}.$$

	*This means: <u>All proposed solutions</u> are correct.		*This means:  <u>All proposed solutions</u>  are correct.
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 15:47, 9 October 2022

2022-10-09T15:47:13Z

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	- the matrix  $\mathbf{G}_{\rm D}$.		- the matrix  $\mathbf{G}_{\rm D}$.

	{The information sequence starts with  $\alpha^4, \, 1, \, \alpha^3, \, 0, \, \alpha^6$. Determine the first ~~codeword~~ for the  $\rm RSC \, (7, \, 3, \, 5)_8$.		{The information sequence starts with  $\alpha^4, \, 1, \, \alpha^3, \, 0, \, \alpha^6$.  Determine the first code word for the  $\rm RSC \, (7, \, 3, \, 5)_8$.
	\|type="[]"}		\|type="[]"}
	+ It holds  $c_0 = \alpha^2$,		+ It holds  $c_0 = \alpha^2$,
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	- It holds  $c_6 = 0$.		- It holds  $c_6 = 0$.

	{What is the ~~codeword~~ for the  $\rm RSC \, (7, \, 5, \, 3)_8$ given the same sequence of information?		{What is the code word for the  $\rm RSC \, (7, \, 5, \, 3)_8$ given the same sequence of information?
	\|type="[]"}		\|type="[]"}
	+ It holds  $c_0 = 1$,		+ It holds  $c_0 = 1$,

Guenter at 15:46, 9 October 2022

2022-10-09T15:46:13Z

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	[[File:P_ID2525__KC_A_2_8.png\|right\|frame\|Four generator matrices, three of which describe Reed-Solomon codes]]		[[File:P_ID2525__KC_A_2_8.png\|right\|frame\|Four generator matrices, three of which describe Reed-Solomon codes]]
	In the  [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8\|"Exercise 2.7"]]  you should determine the ~~codewords~~ of the  $\rm RSC \, (7, \, 3, \, 5)_8$  via a polynomial. However, you can also determine the ~~codeword~~  $\underline{c}$  from the information word  $\underline{u}$  and the generator matrix  $\mathbf{G}$  according to the following equation:		In the  [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8\|"Exercise 2.7"]]  you should determine the code words of the  $\rm RSC \, (7, \, 3, \, 5)_8$  via a polynomial.  However,  you can also determine the code word  $\underline{c}$   from the information word  $\underline{u}$  and the generator matrix  $\mathbf{G}$  according to the following equation:
	:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}		:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	*Two of these generator matrices describe the  $\rm RSC \, (7, \, 3, \, 5)_8$. In the subtask '''(1)''' is explicitly asked which.		*Two of these generator matrices describe the  $\rm RSC \, (7, \, 3, \, 5)_8$.  In the subtask  '''(1)'''  is explicitly asked which.
	*Another generator matrix belongs to  $\rm RSC \, (7, \, 5, \, 3)_8$, which is considered in subtask '''(3)'''.
			*Another generator matrix belongs to  $\rm RSC \, (7, \, 5, \, 3)_8$,  which is considered in subtask  '''(3)'''.


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			Hints:
			* This exercise belongs to the chapter  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes\| "Definition and Properties of Reed-Solomon Codes"]].

			* Important information about Reed–Solomon codes can also be found in  [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8\| "Exercise 2.7"]].
	~~Hints:~~
	* This task belongs to the chapter  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes\| "Definition and Properties of Reed-Solomon Codes"]].
	* Important information about Reed–Solomon codes can also be found in ~~the~~  [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8\| "Exercise 2.7"]].

Guenter at 13:01, 8 October 2022

2022-10-08T13:01:39Z

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	'''(2)'''  In the $\rm RSC \, (7, \, 3, \, 5)_8$ information symbols are processed in each coding step $k = 3$, in coding step 1 according to the specification the symbols $\alpha^4, \ 1$ and $\alpha^3$.		'''(2)'''  In the $\rm RSC \, (7, \, 3, \, 5)_8$ information symbols are processed in each coding step $k = 3$, in coding step 1 according to the specification the symbols $\alpha^4, \ 1$ and $\alpha^3$.

	[[File:~~EN_KC_A_2_7~~.png\|right\|frame\|$\rm GF(2^3)$  representation as powers, polynomials, vectors]]		[[File:EN_KC_Z_2_5_neu.png\|right\|frame\|$\rm GF(2^3)$  representation as powers, polynomials, vectors]]

	*With the generator matrix $\mathbf{G}_{\rm C}$ thus holds:		*With the generator matrix $\mathbf{G}_{\rm C}$ thus holds:

Guenter at 15:08, 5 October 2022

2022-10-05T15:08:27Z

← Older revision		Revision as of 17:08, 5 October 2022
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	'''(2)'''  In the $\rm RSC \, (7, \, 3, \, 5)_8$ information symbols are processed in each coding step $k = 3$, in coding step 1 according to the specification the symbols $\alpha^4, \ 1$ and $\alpha^3$.		'''(2)'''  In the $\rm RSC \, (7, \, 3, \, 5)_8$ information symbols are processed in each coding step $k = 3$, in coding step 1 according to the specification the symbols $\alpha^4, \ 1$ and $\alpha^3$.

	[[File:~~P_ID2584__KC_T_2_5_Darstellung~~.png\|right\|frame\|$\rm GF(2^3)$ as powers, polynomials ~~and~~ vectors ]]		[[File:EN_KC_A_2_7.png\|right\|frame\|$\rm GF(2^3)$  representation as powers, polynomials, vectors]]

	*With the generator matrix $\mathbf{G}_{\rm C}$ thus holds:		*With the generator matrix $\mathbf{G}_{\rm C}$ thus holds:
	:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm C} =		:$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm C} =

Noah at 15:57, 2 September 2022

2022-09-02T15:57:39Z

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	(\alpha^{2} + \alpha) + (\alpha^2 +1) + \alpha = 1 \hspace{0.05cm}.$$		(\alpha^{2} + \alpha) + (\alpha^2 +1) + \alpha = 1 \hspace{0.05cm}.$$

	*You get exactly the same result as in subtask (4) of [[Aufgaben:~~Aufgabe_2~~.07:~~_Reed–Solomon–Code_~~(7,_3,_5)~~_zur_Basis_8~~\|"Exercise 2.7"]]. Correct are <u>solutions 1 and 2</u>.		*You get exactly the same result as in subtask (4) of [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8\|"Exercise 2.7"]]. Correct are <u>solutions 1 and 2</u>.
	*So it is not $c_6 = 0$, but $c_6 = 1$.		*So it is not $c_6 = 0$, but $c_6 = 1$.