Aufgaben:Exercise 2.2Z: Galois Field GF(5) - Revision history

Guenter at 15:04, 28 August 2022

2022-08-28T15:04:39Z

Guenter at 14:41, 28 August 2022

2022-08-28T14:41:20Z

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Guenter at 13:52, 28 August 2022

2022-08-28T13:52:31Z

Noah at 20:31, 26 August 2022

2022-08-26T20:31:35Z

Noah at 20:30, 26 August 2022

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Guenter at 14:44, 24 August 2022

2022-08-24T14:44:01Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.2Z: Galoisfeld GF(5) to Aufgaben:Exercise 2.2Z: Galois Field GF(5)

2022-08-24T14:40:32Z

Guenter moved page Aufgabe 2.2Z: Galoisfeld GF(5) to Exercise 2.2Z: Galois Field GF(5)

Javier: Text replacement - "„" to """

2021-05-28T14:24:40Z

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Javier: Text replacement - "”" to """

2021-05-28T14:24:07Z

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Javier: Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

2021-03-23T12:52:19Z

Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

@@ Line 10: / Line 10: @@
 * an addition table modulo 5,
-* a multiplication table modulo 5,
+* a multiplication table modulo 5.
@@ Line 27: / Line 27: @@
-Hints:&nbsp; The exercise is related to the topic of the chapter&nbsp; [[Channel_Coding/Some_Basics_of_Algebra| "Some Basics of Algebra"]].
+Hint:&nbsp; The exercise is related to the topic of the chapter&nbsp; [[Channel_Coding/Some_Basics_of_Algebra| "Some Basics of Algebra"]].

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Some_Basics_of_Algebra}}
-[[File:P_ID2494__KC_Z_2_2.png|right|frame|Addition/multiplication for&nbsp; $\{a, \, b, \, c, \, d, \, e\}$]]
+[[File:P_ID2494__KC_Z_2_2.png|right|frame|Addition/multiplication for&nbsp; $\rm \{a, \, b, \, c, \, d, \, e\}$]]
-As in the&nbsp; [[Aufgaben:Exercise_2.2:_Properties_of_Galois_Fields|Exercise 2.2]]&nbsp; we consider a finite field of order&nbsp; $q = 5$&nbsp; and thus the Galois field
+As in the&nbsp; [[Aufgaben:Exercise_2.2:_Properties_of_Galois_Fields|"Exercise 2.2"]]&nbsp; we consider a finite field of order&nbsp; $q = 5$&nbsp; and thus the Galois field
-:$${\rm GF}(5) = \{{a}, { b},{c},{d},{e}\}\hspace{0.05cm}.$$
+:$$\rm GF(5) = \{{a}, { b},{c},{d},{e}\}\hspace{0.05cm}.$$
 No further statements are made about the elements. They can be integers or any mathematical expressions.
@@ Line 9: / Line 9: @@
 The Galois field is determined exclusively by
 * an addition table modulo 5,
 * a multiplication table modulo 5,
-The main properties of a Galois field are compiled on the&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_of_a_Galois_field|"first theory page"]]&nbsp;. Here reference is made to
+The main properties of a Galois field are compiled on the&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_of_a_Galois_field|"first theory page"]].&nbsp; There,&nbsp; reference is made to
 * the commutative and distributive laws,
 * the neutral elements of addition and multiplication,
 * the inverse elements of addition and multiplication, and
 * the determination of primitive elements.
-In the present example&nbsp; $\beta$&nbsp; would be a primitive element if&nbsp; $\beta^2, \ \beta^3$&nbsp; and&nbsp; $\beta^4$&nbsp; (in general: &nbsp;$\beta^{q-1})$&nbsp; yield the remaining elements of the Galois field&nbsp; $\rm GF(5)$&nbsp; except for the zero element.
+In the present example&nbsp; $\beta$&nbsp; would be a primitive element if&nbsp; $\beta^2, \ \beta^3$&nbsp; and&nbsp; $\beta^4$&nbsp; $($in general: &nbsp;$\beta^{q-1})$&nbsp; yield the remaining elements of the Galois field&nbsp; $\rm GF(5)$&nbsp; except for the zero element.
@@ Line 36: / Line 34: @@
 {Determine the neutral element of addition.
 |type="()"}
-- $N_{\rm A} = a$,
+- $N_{\rm A} = \rm a$,
-- $N_{\rm A} = b$,
+- $N_{\rm A} = \rm b$,
-- $N_{\rm A} = c$,
+- $N_{\rm A} = \rm c$,
-+ $N_{\rm A} = d$,
++ $N_{\rm A} = \rm d$,
-- $N_{\rm A} = e$.
+- $N_{\rm A} = \rm e$.
 {Determine the neutral element of multiplication.
 |type="()"}
-- $N_{\rm M} = a$,
+- $N_{\rm M} = \rm a$,
-- $N_{\rm M} = b$,
+- $N_{\rm M} = \rm b$,
-+ $N_{\rm M} = c$,
++ $N_{\rm M} = \rm c$,
-- $N_{\rm M} = d$,
+- $N_{\rm M} = \rm d$,
-- $N_{\rm M} = e$.
+- $N_{\rm M} = \rm e$.
 {If the commutative law is satisfied,
 |type="[]"}
-+ with respect to addition, for example&nbsp; $a + b = b + a, \hspace{0.05cm}\text{ ...} \hspace{0.1cm}, \ d + e = e + d$,
++ with respect to addition,&nbsp; e.g.&nbsp; $\rm a + b = b + a, \hspace{0.05cm}\text{ ...} \hspace{0.1cm}, \ d + e = e + d$,
-+ with respect to multiplication, for example&nbsp; $a \cdot b = b \cdot a, \hspace{0.05cm}\text{ ...} \hspace{0.1cm}, \ d \cdot e = e \cdot d$.
++ with respect to multiplication,&nbsp; e.g.&nbsp; $\rm a \cdot b = b \cdot a, \hspace{0.05cm}\text{ ...} \hspace{0.1cm}, \ d \cdot e = e \cdot d$.
 {For which expressions is the distributive law satisfied?
 |type="[]"}
-+ $a \cdot (b + c) = a \cdot b + a \cdot c$,
++ $\rm a \cdot (b + c) = a \cdot b + a \cdot c$,
-+ $d \cdot (b + c) = d \cdot b + d \cdot c$,
++ $\rm d \cdot (b + c) = d \cdot b + d \cdot c$,
-+ $e \cdot (a + b) = e \cdot a + e \cdot b$.
++ $\rm e \cdot (a + b) = e \cdot a + e \cdot b$.
-{Replace&nbsp; $a, \ b, \ c, \ d, \ e$&nbsp; with elements of the number set&nbsp; $\{0, \, 1, \, 2, \, 3, \, 4\}$ so that equal operation tables result.
+{Replace&nbsp; $\rm a, \ b, \ c, \ d, \ e$&nbsp; with elements of the number set&nbsp; $\{0, \, 1, \, 2, \, 3, \, 4\}$&nbsp; so that equal operation tables result.
 |type="{}"}
-$a \hspace{0.2cm} = \ ${ 3 }
+$\rm a \hspace{0.2cm} = \ ${ 3 }
-$b \hspace{0.25cm}  = \ ${ 2 }
+$\rm b \hspace{0.25cm}  = \ ${ 2 }
-$c \hspace{0.25cm}  = \ ${ 1 }
+$\rm c \hspace{0.25cm}  = \ ${ 1 }
-$d \hspace{0.2cm}  = \ ${ 0. }
+$\rm d \hspace{0.2cm}  = \ ${ 0. }
-$e \hspace{0.2cm}  = \ ${ 4 }
+$\rm e \hspace{0.2cm}  = \ ${ 4 }
 {What statements are true regarding inverse elements?
@@ Line 78: / Line 76: @@
 {Which of the elements are primitive?
 |type="[]"}
-+ $a = 3$.
++ $\rm a = 3$.
-+ $b = 2$,
++ $\rm b = 2$,
-- $e = 4$.
+- $\rm e = 4$.
 </quiz>

@@ Line 28: / Line 28: @@
 Hints:
 * The exercise is related to the topic of the chapter [[Channel_Coding/Some_Basics_of_Algebra| "Some Basics of Algebra"]].
-* Please don't be confused that in the text the set&nbsp; $ \{a}, { b},{c},{d},{e}\}$&nbsp; is used and in the tables the set&nbsp; $ \rm \{a}, { b},{c},{d},{e}\}$. Apologies.
+* Please don't be confused that in the text the set&nbsp; $ \{{a}, { b},{c},{d},{e}\}$&nbsp; is used and in the tables the set&nbsp; $ \rm \{{a}, { b},{c},{d},{e}\}$. Apologies.

← Older revision		Revision as of 14:44, 24 August 2022
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−	[[Category:Channel Coding: Exercises\|^2.1 ~~Einige Grundlagen der~~ Algebra^]]	+	[[Category:Channel Coding: Exercises\|^2.1 Some Basics of Algebra^]]

@@ Line 163: / Line 163: @@
 \hspace{0.05cm},\hspace{0.2cm} e^3 = \hspace{0.05cm} ...\hspace{0.05cm}= 4\hspace{0.05cm},\hspace{0.2cm} e^4 =\hspace{0.05cm} ...\hspace{0.05cm} = 1\hspace{0.13cm} \Rightarrow \hspace{0.13cm}{\rm nicht\hspace{0.15cm} primitiv}\hspace{0.05cm}.$$
-Von der Menge $Z_5 = \{0, \, 1, \, 2, \, 3, \, 4\}$ sind &bdquo;$2$" und &bdquo;$3$" primitive Elemente &nbsp;&#8658;&nbsp; <u>Lösungsvorschläge 1 und 2</u>.
+Von der Menge $Z_5 = \{0, \, 1, \, 2, \, 3, \, 4\}$ sind "$2$" und "$3$" primitive Elemente &nbsp;&#8658;&nbsp; <u>Lösungsvorschläge 1 und 2</u>.
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← Older revision	Revision as of 14:40, 24 August 2022
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← Older revision		Revision as of 12:52, 23 March 2021
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−	[[Category:~~Aufgaben zu Kanalcodierung~~\|^2.1 Einige Grundlagen der Algebra^]]	+	[[Category:Channel Coding: Exercises\|^2.1 Einige Grundlagen der Algebra^]]