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

Guenter at 14:11, 10 October 2022

2022-10-10T14:11:31Z

Guenter at 15:47, 9 October 2022

2022-10-09T15:47:13Z

Guenter at 15:46, 9 October 2022

2022-10-09T15:46:13Z

Guenter at 13:01, 8 October 2022

2022-10-08T13:01:39Z

Guenter at 15:08, 5 October 2022

2022-10-05T15:08:27Z

Noah at 15:57, 2 September 2022

2022-09-02T15:57:39Z

Noah at 15:56, 2 September 2022

2022-09-02T15:56:47Z

Guenter at 12:29, 2 September 2022

2022-09-02T12:29:53Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.08: Generatorpolynome für Reed-Solomon to Aufgaben:Exercise 2.08: Generator Polynomials for Reed-Solomon

2022-09-02T12:20:49Z

Guenter moved page Aufgabe 2.08: Generatorpolynome für Reed-Solomon to Exercise 2.08: Generator Polynomials for Reed-Solomon

Noah at 11:29, 2 September 2022

2022-09-02T11:29:41Z

@@ Line 47: / Line 47: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are the <u>solutions 2 and 3</u> &nbsp;&#8658;&nbsp; matrices $\mathbf{G}_{\rm B}$ and $\mathbf{G}_{\rm C}$.
+'''(1)'''&nbsp; Correct are the&nbsp; <u>solutions 2 and 3</u> &nbsp; &#8658; &nbsp; matrices&nbsp; $\mathbf{G}_{\rm B}$&nbsp; and&nbsp; $\mathbf{G}_{\rm C}$.
-*In the matrix $\mathbf{G}_{\rm C}$ the allowed transformations&nbsp; $\alpha^8 = \alpha, \ \alpha^{10} = \alpha^3$&nbsp; and&nbsp; $\alpha^{12} = \alpha^5$&nbsp; have already been considered.
+*In the matrix&nbsp; $\mathbf{G}_{\rm C}$&nbsp; the allowed transformations&nbsp; $\alpha^8 = \alpha, \ \alpha^{10} = \alpha^3$&nbsp; and&nbsp; $\alpha^{12} = \alpha^5$&nbsp; 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.
-'''(2)'''&nbsp; 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)'''&nbsp; In the&nbsp; $\rm RSC \, (7, \, 3, \, 5)_8$,&nbsp;  in each coding step are processed&nbsp;  $k = 3$&nbsp; information symbols, &nbsp; <br>in coding step 1 according to the specification the symbols&nbsp; $\alpha^4, \ 1$ and $\alpha^3$.
 *With the generator matrix $\mathbf{G}_{\rm C}$ thus holds:
 :$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm C} =
@@ Line 67: / Line 66: @@
 & \alpha^2 & \alpha^4 & \alpha^6 & \alpha^1 & \alpha^{3} & \alpha^{5}
 \end{pmatrix}\hspace{0.05cm}. $$
+[[File:EN_KC_Z_2_5_neu.png|right|frame|$\rm GF(2^3)$&nbsp; representation as powers, polynomials, vectors]]
 *This results according to the adjacent auxiliary table:
 :$$c_0 \hspace{-0.15cm} \ = \ \hspace{-0.15cm}   \alpha^{4}\cdot 1  + 1 \cdot 1 +  \alpha^{3}\cdot 1 =
@@ Line 84: / Line 84: @@
   (\alpha^{2} + \alpha) + (\alpha^2 +1) + \alpha = 1 \hspace{0.05cm}.$$
-*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&nbsp; '''(4)'''&nbsp; of&nbsp; [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8|"Exercise 2.7"]].&nbsp; Correct are the&nbsp; <u>solutions 1 and 2</u>.
-*So it is not $c_6 = 0$, but $c_6 = 1$.
+*So it is not&nbsp; $c_6 = 0$,&nbsp; but $c_6 = 1$.
-'''(3)'''&nbsp; 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)'''&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 $\mathbf{G}_{\rm D}$ one obtains:
+*With the generator matrix&nbsp; $\mathbf{G}_{\rm D}$&nbsp; one obtains:
 :$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm D} =
 \begin{pmatrix}
@@ Line 107: / Line 107: @@
 :$$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}.$$
-*This takes into account that the bracket expression $[ \ \text{...} \ ]$ corresponds exactly to the result $c_1$ of subtask (2).
+*This takes into account that the bracket expression&nbsp; $[ \ \text{...} \ ]$&nbsp; corresponds exactly to the result&nbsp; $c_1$&nbsp; of subtask&nbsp; '''(2)'''.
 *Corresponding is also considered in the following calculations:
@@ Line 125: / Line 125: @@
   =  (001) + (100) = (101) = \alpha^{6} \hspace{0.05cm}.$$
-*This means: <u>All proposed solutions</u> are correct.
+*This means:&nbsp; <u>All proposed solutions</u>&nbsp; are correct.
 {{ML-Fuß}}

@@ Line 32: / Line 32: @@
 - the matrix&nbsp; $\mathbf{G}_{\rm D}$.
-{The information sequence starts with&nbsp; $\alpha^4, \, 1, \, \alpha^3, \, 0, \, \alpha^6$. Determine the first codeword for the&nbsp; $\rm RSC \, (7, \, 3, \, 5)_8$.
+{The information sequence starts with&nbsp; $\alpha^4, \, 1, \, \alpha^3, \, 0, \, \alpha^6$.&nbsp; Determine the first code word for the&nbsp; $\rm RSC \, (7, \, 3, \, 5)_8$.
 |type="[]"}
 + It holds&nbsp; $c_0 = \alpha^2$,
@@ Line 38: / Line 38: @@
 - It holds&nbsp; $c_6 = 0$.
-{What is the codeword for the&nbsp; $\rm RSC \, (7, \, 5, \, 3)_8$ given the same sequence of information?
+{What is the code word for the&nbsp; $\rm RSC \, (7, \, 5, \, 3)_8$ given the same sequence of information?
 |type="[]"}
 + It holds&nbsp; $c_0 = 1$,

@@ Line 2: / Line 2: @@
 [[File:P_ID2525__KC_A_2_8.png|right|frame|Four generator matrices, three of which describe Reed-Solomon codes]]
-In the&nbsp; [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8|"Exercise 2.7"]]&nbsp; you should determine the codewords of the&nbsp; $\rm RSC \, (7, \, 3, \, 5)_8$&nbsp; via a polynomial. However, you can also determine the codeword&nbsp; $\underline{c}$&nbsp; from the information word&nbsp; $\underline{u}$&nbsp; and the generator matrix&nbsp; $\mathbf{G}$&nbsp; according to the following equation:
+In the&nbsp; [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8|"Exercise 2.7"]]&nbsp; you should determine the code words of the&nbsp; $\rm RSC \, (7, \, 3, \, 5)_8$&nbsp; via a polynomial.&nbsp; However,&nbsp; you can also determine the code word&nbsp; $\underline{c}$ &nbsp; from the information word&nbsp; $\underline{u}$&nbsp; and the generator matrix&nbsp; $\mathbf{G}$&nbsp; according to the following equation:
 :$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}
 \hspace{0.05cm}.$$
-*Two of these generator matrices describe the&nbsp; $\rm RSC \, (7, \, 3, \, 5)_8$. In the subtask '''(1)''' is explicitly asked which.
+*Two of these generator matrices describe the&nbsp; $\rm RSC \, (7, \, 3, \, 5)_8$.&nbsp; In the subtask&nbsp; '''(1)'''&nbsp; is explicitly asked which.
 *Another generator matrix belongs to&nbsp; $\rm RSC \, (7, \, 5, \, 3)_8$, which is considered in subtask '''(3)'''.
@@ Line 13: / Line 14: @@
 * Important information about Reed&ndash;Solomon codes can also be found in&nbsp; [[Aufgaben:Exercise_2.07:_Reed-Solomon_Code_(7,_3,_5)_to_Base_8| "Exercise 2.7"]].

@@ Line 55: / Line 55: @@
 '''(2)'''&nbsp; 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|$\rm GF(2^3)$&nbsp; representation as powers, polynomials, vectors]]
+[[File:EN_KC_Z_2_5_neu.png|right|frame|$\rm GF(2^3)$&nbsp; representation as powers, polynomials, vectors]]
 *With the generator matrix $\mathbf{G}_{\rm C}$ thus holds:

@@ Line 55: / Line 55: @@
 '''(2)'''&nbsp; 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|$\rm GF(2^3)$&nbsp; representation as powers, polynomials, vectors]]
 *With the generator matrix $\mathbf{G}_{\rm C}$ thus holds:
 :$$\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}_{\rm C} =

← Older revision		Revision as of 12:29, 2 September 2022
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−	[[Category:Channel Coding: Exercises\|^2.3 ~~Zu den Reed–Solomon–Codes~~^]]	+	[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]

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 Hints:
-* This task belongs to the chapter&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes| "Definition und Eigenschaften von Reed&ndash;Solomon Codes"]].
+* This task belongs to the chapter&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes| "Definition and Properties of Reed-Solomon Codes"]].
 * Important information about Reed&ndash;Solomon codes can also be found in the&nbsp; [[Aufgaben:Aufgabe_2.07:_Reed–Solomon–Code_(7,_3,_5)_zur_Basis_8| "Exercise 2.7"]].