Channel Coding/Some Basics of Algebra - Revision history

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← Older revision		Revision as of 14:09, 19 November 2022
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	*For understanding the Reed–Solomon codes, however, this knowledge is not absolutely necessary.		*For understanding the Reed–Solomon codes, however, this knowledge is not absolutely necessary.

−	*So you could jump directly to the chapter  [[Channel_Coding/Extension_Field#Verallgemeinerte_Definition_eines_Erweiterungsk.C3.B6rpers\| ~~$\text{~~Extension Field}$]] .<br>	+	*So you could jump directly to the chapter  [[Channel_Coding/Extension_Field#Verallgemeinerte_Definition_eines_Erweiterungsk.C3.B6rpers\| "Extension Field"]] .<br>

@@ Line 165: / Line 165: @@
 *A Galois field&nbsp; ${\rm GF}(q)$&nbsp; can be formed in the manner described here as&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_ring| $\text{ring}$]]&nbsp; of integer sizes modulo&nbsp; $q$&nbsp; only if&nbsp; $q$&nbsp; is a prime number: &nbsp; <br> &nbsp; &nbsp; $q = 2$,&nbsp; $q = 3$,&nbsp; $q = 5$,&nbsp; $q = 7$,&nbsp; $q = 11$, ...<br>
-*But if the order&nbsp; $q$&nbsp; can be expressed in the form&nbsp; $q = P^m$&nbsp; with a prime&nbsp; $P$&nbsp; and integer&nbsp; $m$,&nbsp; the Galois field&nbsp; ${\rm GF}(q)$&nbsp; can be found via an&nbsp; [[Channel_Coding/Extension_Field#Generalized_definition_of_an_extension_field|$\text{extension field}$]]&nbsp; find.}}<br>
+*But if the order&nbsp; $q$&nbsp; can be expressed in the form&nbsp; $q = P^m$&nbsp; with a prime&nbsp; $P$&nbsp; and integer&nbsp; $m$,&nbsp; the Galois field&nbsp; ${\rm GF}(q)$&nbsp; can be found via an&nbsp; [[Channel_Coding/Extension_Field#Generalized_definition_of_an_extension_field|$\text{extension field}$]].&nbsp; }}<br>
 == Group, ring, field - basic algebraic concepts ==

@@ Line 165: / Line 165: @@
 *A Galois field&nbsp; ${\rm GF}(q)$&nbsp; can be formed in the manner described here as&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_ring| $\text{ring}$]]&nbsp; of integer sizes modulo&nbsp; $q$&nbsp; only if&nbsp; $q$&nbsp; is a prime number: &nbsp; <br> &nbsp; &nbsp; $q = 2$,&nbsp; $q = 3$,&nbsp; $q = 5$,&nbsp; $q = 7$,&nbsp; $q = 11$, ...<br>
-*But if the order&nbsp; $q$&nbsp; can be expressed in the form&nbsp; $q = P^m$&nbsp; with a prime&nbsp; $P$&nbsp; and integer&nbsp; $m$,&nbsp; the Galois field&nbsp; ${\rm GF}(q)$&nbsp; can be found via an&nbsp; [[Channel_Coding/Extension_Field#Generalized_Definition_of_an_Extension. C3.B6rpers|$\text{extension field}$]]&nbsp; find.}}<br>
+*But if the order&nbsp; $q$&nbsp; can be expressed in the form&nbsp; $q = P^m$&nbsp; with a prime&nbsp; $P$&nbsp; and integer&nbsp; $m$,&nbsp; the Galois field&nbsp; ${\rm GF}(q)$&nbsp; can be found via an&nbsp; [[Channel_Coding/Extension_Field#Generalized_definition_of_an_extension_field|$\text{extension field}$]]&nbsp; find.}}<br>
 == Group, ring, field - basic algebraic concepts ==

@@ Line 157: / Line 157: @@
 & 0 & 2 & 0 & 2\\
 & 0 & 3 & 2 & 1
-\end{array} \right]\hspace{0.25cm} \Rightarrow\hspace{0.25cm}\text{kein }{\rm GF}(4) .
+\end{array} \right]\hspace{0.25cm} \Rightarrow\hspace{0.25cm}\text{no }{\rm GF}(4) .
 $$}}

@@ Line 64: / Line 64: @@
 ::<math>\forall \hspace{0.15cm}  z_i \in {\rm GF}(q),\hspace{0.15cm} \exists \hspace{0.15cm} {\rm Inv_A}(z_i) \in {\rm GF}(q)\text{:}
-\hspace{0.25cm}z_i + {\rm Inv_A}(z_i) = N_{\rm A} = {0}  \hspace{0.3cm}\Rightarrow \hspace{0.3cm}\text{kurz:}\hspace{0.3cm}
+\hspace{0.25cm}z_i + {\rm Inv_A}(z_i) = N_{\rm A} = {0}  \hspace{0.3cm}\Rightarrow \hspace{0.3cm}\text{short:}\hspace{0.3cm}
 {\rm Inv_A}(z_i) = - z_i \hspace{0.05cm}. </math>
@@ Line 70: / Line 70: @@
 ::<math>\forall \hspace{0.15cm}  z_i \in {\rm GF}(q),\hspace{0.15cm} z_i \ne N_{\rm A}, \hspace{0.15cm} \exists \hspace{0.15cm} {\rm Inv_M}(z_i) \in {\rm GF}(q)\text{:}
-\hspace{0.25cm}z_i \cdot {\rm Inv_M}(z_i) = N_{\rm M} = {1}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\text{kurz:}\hspace{0.3cm}
+\hspace{0.25cm}z_i \cdot {\rm Inv_M}(z_i) = N_{\rm M} = {1}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\text{short:}\hspace{0.3cm}
 {\rm Inv_M}(z_i) = z_i^{-1}
 \hspace{0.05cm}. </math>

@@ Line 8: / Line 8: @@
 == # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 <br>
-This chapter discusses the&nbsp; &raquo;Reed-Solomon codes&laquo;,&nbsp; invented in the early 1960s by&nbsp; [https://en.wikipedia.org/wiki/Irving_S._Reed $\text{Irving Stoy Reed}$]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Gustave_Solomon $\text{Gustave Solomon}$].&nbsp; Unlike binary block codes,&nbsp; these codes are based on a Galois field &nbsp; ${\rm GF}(2^m)$ &nbsp; with&nbsp; $m > 1$.&nbsp; So they work with multilevel symbols instead of binary characters&nbsp; ("bits").
+This chapter discusses the&nbsp; &raquo;Reed-Solomon codes&laquo;,&nbsp; invented in the early 1960s by&nbsp; [https://en.wikipedia.org/wiki/Irving_S._Reed $\text{Irving Stoy Reed}$]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Gustave_Solomon $\text{Gustave Solomon}$].&nbsp; Unlike binary block codes,&nbsp; these codes are based on a Galois field &nbsp; ${\rm GF}(2^m)$ &nbsp; with&nbsp; $m > 1$.&nbsp; So they work with multi-level symbols instead of binary characters&nbsp; ("bits").
 Specifically,&nbsp; this chapter covers:

@@ Line 8: / Line 8: @@
 == # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 <br>
-This chapter discusses the&nbsp; &raquo;Reed-Solomon codes&laquo;,&nbsp; invented in the early 1960s by&nbsp; [https://en.wikipedia.org/wiki/Irving_S._Reed Irving Stoy Reed]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Gustave_Solomon Gustave Solomon].&nbsp; Unlike binary block codes,&nbsp; these codes are based on a Galois field &nbsp; ${\rm GF}(2^m)$ &nbsp; with&nbsp; $m > 1$.&nbsp; So they work with multilevel symbols instead of binary characters&nbsp; ("bits").
+This chapter discusses the&nbsp; &raquo;Reed-Solomon codes&laquo;,&nbsp; invented in the early 1960s by&nbsp; [https://en.wikipedia.org/wiki/Irving_S._Reed $\text{Irving Stoy Reed}$]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Gustave_Solomon $\text{Gustave Solomon}$].&nbsp; Unlike binary block codes,&nbsp; these codes are based on a Galois field &nbsp; ${\rm GF}(2^m)$ &nbsp; with&nbsp; $m > 1$.&nbsp; So they work with multilevel symbols instead of binary characters&nbsp; ("bits").
 Specifically,&nbsp; this chapter covers:
@@ Line 32: / Line 32: @@
 == Definition of a Galois field ==
 <br>
-Before we can turn to the description of&nbsp; Reed&ndash;Solomon codes,&nbsp; we need some basic algebraic notions.&nbsp; We begin with the properties of the Galois field &nbsp; ${\rm GF}(q)$,&nbsp; named after the Frenchman&nbsp; [https://en.wikipedia.org/wiki/%C3%89variste_Galois Évariste Galois], whose biography is rather unusual for a mathematician.<br>
+Before we can turn to the description of&nbsp; Reed&ndash;Solomon codes,&nbsp; we need some basic algebraic notions.&nbsp; We begin with the properties of the Galois field &nbsp; ${\rm GF}(q)$,&nbsp; named after the Frenchman&nbsp; [https://en.wikipedia.org/wiki/%C3%89variste_Galois $\text{Évariste Galois}$], whose biography is rather unusual for a mathematician.<br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp;  A &nbsp; $\rm Galois\:field$&nbsp; ${\rm GF}(q)$&nbsp; is a&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Group.2C_ring.2C_field_-_basic_algebraic_concepts| "finite field"]]&nbsp; with&nbsp; $q$&nbsp; elements&nbsp; $z_0$,&nbsp; $z_1$,&nbsp; ... ,&nbsp; $z_{q-1}$, if the eight statements listed below &nbsp;$\rm (A)$&nbsp; ... &nbsp;$\rm (H)$&nbsp; with respect to&nbsp; "addition" &nbsp; &rArr; &nbsp; "$+$" &nbsp; and&nbsp; "multiplication"&nbsp; &rArr; &nbsp;"$\hspace{0.05cm}\cdot \hspace{0.05cm}$" &nbsp; are true.
+$\text{Definition:}$&nbsp;  A &nbsp; $\rm Galois\:field$&nbsp; ${\rm GF}(q)$&nbsp; is a&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Group.2C_ring.2C_field_-_basic_algebraic_concepts|$\text{finite field}$]]&nbsp; with&nbsp; $q$&nbsp; elements&nbsp; $z_0$,&nbsp; $z_1$,&nbsp; ... ,&nbsp; $z_{q-1}$, if the eight statements listed below &nbsp;$\rm (A)$&nbsp; ... &nbsp;$\rm (H)$&nbsp; with respect to&nbsp; "addition" &nbsp; &rArr; &nbsp; "$+$" &nbsp; and&nbsp; "multiplication"&nbsp; &rArr; &nbsp;"$\hspace{0.05cm}\cdot \hspace{0.05cm}$" &nbsp; are true.
 *Addition and multiplication are to be understood here modulo&nbsp; $q$&nbsp;.
@@ Line 140: / Line 140: @@
 {{GraueBox|TEXT=
-$\text{Example 2:}$&nbsp;  In contrast,&nbsp; the number set&nbsp; $Z_4 = \{0, 1, 2, 3\}$ &nbsp; &#8658; &nbsp; $q = 4$&nbsp; is&nbsp; '''not'''&nbsp; a Galois field.
+$\text{Example 2:}$&nbsp;  In contrast,&nbsp; the number set&nbsp; $Z_4 = \{0, 1, 2, 3\}$ &nbsp; &#8658; &nbsp; $q = 4$&nbsp; is&nbsp; &raquo;'''not'''&laquo;&nbsp; a Galois field.
 *The reason for this is that here is no multiplicative inverse to the element&nbsp; $z_2 = 2$.&nbsp; For a Galois field it would have to be true: &nbsp; $2 \cdot {\rm Inv_M}(2) = 1$.
@@ Line 163: / Line 163: @@
 {{BlaueBox|TEXT=
 $\text{Generalization (without proof for now):}$
-*A Galois field&nbsp; ${\rm GF}(q)$&nbsp; can be formed in the manner described here as&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_ring| "ring"]]&nbsp; of integer sizes modulo&nbsp; $q$&nbsp; only if&nbsp; $q$&nbsp; is a prime number: &nbsp; <br> &nbsp; &nbsp; $q = 2$,&nbsp; $q = 3$,&nbsp; $q = 5$,&nbsp; $q = 7$,&nbsp; $q = 11$, ...<br>
+*A Galois field&nbsp; ${\rm GF}(q)$&nbsp; can be formed in the manner described here as&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_ring| $\text{ring}$]]&nbsp; of integer sizes modulo&nbsp; $q$&nbsp; only if&nbsp; $q$&nbsp; is a prime number: &nbsp; <br> &nbsp; &nbsp; $q = 2$,&nbsp; $q = 3$,&nbsp; $q = 5$,&nbsp; $q = 7$,&nbsp; $q = 11$, ...<br>
-*But if the order&nbsp; $q$&nbsp; can be expressed in the form&nbsp; $q = P^m$&nbsp; with a prime&nbsp; $P$&nbsp; and integer&nbsp; $m$,&nbsp; the Galois field&nbsp; ${\rm GF}(q)$&nbsp; can be found via an&nbsp; [[Channel_Coding/Extension_Field#Generalized_Definition_of_an_Extension. C3.B6rpers|"extension field"]]&nbsp; find.}}<br>
+*But if the order&nbsp; $q$&nbsp; can be expressed in the form&nbsp; $q = P^m$&nbsp; with a prime&nbsp; $P$&nbsp; and integer&nbsp; $m$,&nbsp; the Galois field&nbsp; ${\rm GF}(q)$&nbsp; can be found via an&nbsp; [[Channel_Coding/Extension_Field#Generalized_Definition_of_an_Extension. C3.B6rpers|$\text{extension field}$]]&nbsp; find.}}<br>
 == Group, ring, field - basic algebraic concepts ==
@@ Line 173: / Line 173: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-*A&nbsp; $\rm Galois\, field$&nbsp; ${\rm GF}(q)$ is a&nbsp; "field"&nbsp; with a finite number&nbsp; $(q)$&nbsp; of elements &nbsp; &#8658; &nbsp; '''finite field'''.&nbsp;
+*A&nbsp; $\rm Galois\, field$&nbsp; ${\rm GF}(q)$ is a&nbsp; "field"&nbsp; with a finite number&nbsp; $(q)$&nbsp; of elements &nbsp; &#8658; &nbsp; &raquo;'''finite field'''&laquo;.&nbsp;
 *Each field is again a special case of a&nbsp; "ring",&nbsp; which itself can be represented again as a special case of an&nbsp; "Abelian group".}}<br>
@@ Line 188: / Line 188: @@
 *For understanding the Reed&ndash;Solomon codes, however, this knowledge is not absolutely necessary.
-*So you could jump directly to the chapter&nbsp; [[Channel_Coding/Extension_Field#Verallgemeinerte_Definition_eines_Erweiterungsk.C3.B6rpers| "Extension Field"]]&nbsp;.<br>
+*So you could jump directly to the chapter&nbsp; [[Channel_Coding/Extension_Field#Verallgemeinerte_Definition_eines_Erweiterungsk.C3.B6rpers| $\text{Extension Field}$]]&nbsp;.<br>
@@ Line 203: / Line 203: @@
 #For all&nbsp; $z_i &#8712; \mathcal{G}$&nbsp; there is an inverse element&nbsp; ${\rm Inv_A}(z_i) &#8712; \mathcal{G}$&nbsp; with respect to addition:&nbsp; $z_i + {\rm Inv_A}(z_i)= N_{\rm A} $.&nbsp; For a number group:&nbsp; ${\rm Inv_A}(z_i) =- z_i$.<br>
 #For all&nbsp; $z_i, z_j, z_k &#8712; \mathcal{G}$&nbsp; holds:&nbsp; $z_i + (z_j + z_k)= (z_i + z_j) + z_k$ &nbsp; &#8658; &nbsp; "Associative law&nbsp; for&nbsp; $+$".<br>
-#If in addition for all&nbsp; $z_i, z_j &#8712; \mathcal{G}$&nbsp; the&nbsp; "commutative law"&nbsp; $z_i + z_j= z_j + z_i$&nbsp; is satisfied,&nbsp; one speaks of a&nbsp;  "commutative group"&nbsp; or an&nbsp;  "Abelian group",&nbsp; named after the Norwegian mathematician&nbsp; [https://en.wikipedia.org/wiki/Niels_Henrik_Abel Niels Hendrik Abel].}}<br>
+#If in addition for all&nbsp; $z_i, z_j &#8712; \mathcal{G}$&nbsp; the&nbsp; "commutative law"&nbsp; $z_i + z_j= z_j + z_i$&nbsp; is satisfied,&nbsp; one speaks of a&nbsp;  "commutative group"&nbsp; or an&nbsp;  "Abelian group",&nbsp; named after the Norwegian mathematician&nbsp; [https://en.wikipedia.org/wiki/Niels_Henrik_Abel $\text{Niels Hendrik Abel}$].}}<br>
 {{GraueBox|TEXT=
@@ Line 224: / Line 224: @@
 == Definition and examples of an algebraic ring  ==
 <br>
-According to the&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Group.2C_ring.2C_field_-_basic_algebraic_concepts| "overview graphic"]]&nbsp;
+According to the&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Group.2C_ring.2C_field_-_basic_algebraic_concepts|$\text{overview graphic}$]]&nbsp;
 *one gets from the group&nbsp; $(\mathcal{G}, +)$&nbsp;
@@ Line 239: / Line 239: @@
 The following properties must be satisfied:
-#In terms of addition,&nbsp; the ring&nbsp; $(\mathcal{R}, +, \cdot)$&nbsp; is an&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_group|"Abelian group"]]&nbsp; $(\mathcal{G}, +)$.<br>
+#In terms of addition,&nbsp; the ring&nbsp; $(\mathcal{R}, +, \cdot)$&nbsp; is an&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_group|$\text{Abelian group}$]]&nbsp; $(\mathcal{G}, +)$.<br>
 #In addition,&nbsp; for all&nbsp; $z_i, z_j &#8712; \mathcal{R}$&nbsp; also&nbsp; $(z_i \cdot z_j) &#8712; \mathcal{R}$ &nbsp; &#8658; &nbsp; "Closure criterion for&nbsp; $\cdot$".<br>
 #There is always also a neutral element&nbsp; $N_{\rm M} &#8712; \mathcal{R}$&nbsp; with respect to multiplication,&nbsp; so that for all&nbsp; $z_i &#8712; \mathcal{R}$&nbsp; holds: &nbsp; $z_i \cdot N_{\rm M} = z_i$.&nbsp; For a number group:&nbsp;  $N_{\rm M} \equiv 1$.<br>
@@ Line 246: / Line 246: @@
 Further the following agreements shall hold:
-*A ring&nbsp; $(\mathcal{R}, +, \cdot)$&nbsp; is not necessarily commutative.&nbsp; If in fact the&nbsp; "commutative law"&nbsp; also holds for all elements&nbsp; $z_i, z_j &#8712; \mathcal{R}$&nbsp; with respect to multiplication&nbsp; $(z_i \cdot z_j= z_j \cdot z_i)$&nbsp; then it is called  in the technical literature a&nbsp; "'''commutative ring'''".
+*A ring&nbsp; $(\mathcal{R}, +, \cdot)$&nbsp; is not necessarily commutative.&nbsp; If in fact the&nbsp; "commutative law"&nbsp; also holds for all elements&nbsp; $z_i, z_j &#8712; \mathcal{R}$&nbsp; with respect to multiplication&nbsp; $(z_i \cdot z_j= z_j \cdot z_i)$&nbsp; then it is called  in the technical literature a&nbsp; &raquo;'''commutative ring'''&laquo;.
-*Exists for each element&nbsp; $z_i &#8712; \mathcal{R}$&nbsp; except&nbsp; $N_{\rm A}$&nbsp; $($neutral element of addition,&nbsp; zero element$)$&nbsp; an element&nbsp; ${\rm Inv_M}(z_i)$&nbsp;  inverse with respect to multiplication such that&nbsp; $z_i \cdot {\rm Inv_M}(z_i) = 1$&nbsp; holds, then there is a&nbsp; "<b>division ring</b>".<br>
+*Exists for each element&nbsp; $z_i &#8712; \mathcal{R}$&nbsp; except&nbsp; $N_{\rm A}$&nbsp; $($neutral element of addition,&nbsp; zero element$)$&nbsp; an element&nbsp; ${\rm Inv_M}(z_i)$&nbsp;  inverse with respect to multiplication such that&nbsp; $z_i \cdot {\rm Inv_M}(z_i) = 1$&nbsp; holds, then there is a&nbsp; &raquo;<b>division ring</b>&laquo;.<br>
-*The ring is&nbsp; "<b>free of zero divisors</b>"&nbsp; if from&nbsp; $z_i \cdot z_j = 0$&nbsp; follows necessarily&nbsp; $z_i = 0$&nbsp; or&nbsp; $z_j = 0$.&nbsp; In abstract algebra,&nbsp; a zero divisor of a ring is an element&nbsp; $z_i$ different from the zero element if there exists an element&nbsp; $z_j \ne 0$&nbsp; such that the product&nbsp; $z_i \cdot z_j = 0$&nbsp;.<br>
+*The ring is&nbsp; &raquo;<b>free of zero divisors</b>&laquo;&nbsp; if from&nbsp; $z_i \cdot z_j = 0$&nbsp; follows necessarily&nbsp; $z_i = 0$&nbsp; or&nbsp; $z_j = 0$.&nbsp; In abstract algebra,&nbsp; a zero divisor of a ring is an element&nbsp; $z_i$ different from the zero element if there exists an element&nbsp; $z_j \ne 0$&nbsp; such that the product&nbsp; $z_i \cdot z_j = 0$&nbsp;.<br>
-*A commutative ring free of zero divisors is called&nbsp; "<b>integral domain</b>".<br><br>
+*A commutative ring free of zero divisors is called&nbsp; &raquo;<b>integral domain</b>&laquo;.<br><br>
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp;  Comparing the definitions of&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_group| "group"]],&nbsp; "ring"&nbsp; (see above), [[Aufgaben:Aufgabe_2.1:_Gruppe,_Ring,_Körper|"field"]]&nbsp; and&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_of_a_Galois_field| "Galois field"]],&nbsp; we recognize that a&nbsp; '''Galois field'''&nbsp; ${\rm GF}(q)$
+$\text{Conclusion:}$&nbsp;  Comparing the definitions of&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_group| $\text{group}$]],&nbsp; "ring"&nbsp; (see above), [[Aufgaben:Aufgabe_2.1:_Gruppe,_Ring,_Körper|$\text{field}$]]&nbsp; and&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_of_a_Galois_field|$\text{Galois field}$]],&nbsp; we recognize that a&nbsp; &raquo;'''Galois field'''&laquo;&nbsp; ${\rm GF}(q)$
 #is a finite field with&nbsp; $q$&nbsp; elements,<br>
 #is at the same time a&nbsp; "commutative division ring",&nbsp; and <br>

@@ Line 95: / Line 95: @@
 <br>
 We first check that for the binary number set&nbsp; $Z_2 = \{0, 1\}$ &nbsp; &#8658; &nbsp; $q=2$ &nbsp; $($valid for the simple binary code$)$
-*the eight criteria mentioned on the last page are met,
+*the eight criteria mentioned in the last section are met,
 *so that we can indeed speak of&nbsp; "${\rm GF}(2)$".
@@ Line 169: / Line 169: @@
 == Group, ring, field - basic algebraic concepts ==
 <br>
-On the first pages,&nbsp; some basic algebraic terms have already been mentioned,&nbsp; without their meanings having been explained in more detail.&nbsp; This is to be made up now in all shortness from view of a communication engineer, whereby we mainly refer to the representation in&nbsp; [Fri96]<ref name='Fri96'>Friedrichs, B.:&nbsp; Kanalcodierung – Grundlagen und Anwendungen in modernen Kommunikationssystemen.&nbsp; Berlin – Heidelberg: Springer, 1996.</ref>&nbsp; and&nbsp;  [KM+08]<ref name='KM+08'>Kötter, R.; Mayer, T.; Tüchler, M.; Schreckenbach, F.; Brauchle, J.:&nbsp; Channel Coding.&nbsp; Lecture manuscript, Chair of Communications Engineering, TU Munich, 2008.</ref>.&nbsp; To summarize:<br>
+In the first sections,&nbsp; some basic algebraic terms have already been mentioned,&nbsp; without their meanings having been explained in more detail.&nbsp; This is to be made up now in all shortness from view of a communication engineer, whereby we mainly refer to the representation in&nbsp; [Fri96]<ref name='Fri96'>Friedrichs, B.:&nbsp; Kanalcodierung – Grundlagen und Anwendungen in modernen Kommunikationssystemen.&nbsp; Berlin – Heidelberg: Springer, 1996.</ref>&nbsp; and&nbsp;  [KM+08]<ref name='KM+08'>Kötter, R.; Mayer, T.; Tüchler, M.; Schreckenbach, F.; Brauchle, J.:&nbsp; Channel Coding.&nbsp; Lecture manuscript, Chair of Communications Engineering, TU Munich, 2008.</ref>.&nbsp; To summarize:<br>
 {{BlaueBox|TEXT=
@@ Line 185: / Line 185: @@
-On the next two pages, the algebraic terms mentioned here will be discussed in more detail.
+In the next two sections, the algebraic terms mentioned here will be discussed in more detail.
 *For understanding the Reed&ndash;Solomon codes, however, this knowledge is not absolutely necessary.

@@ Line 270: / Line 270: @@
 [[Aufgaben:Exercise_2.2Z:_Galois_Field_GF(5)|Exercise 2.2Z: Galois Field GF(5)]]
-==Sources==
+==References==
 <references/>
 {{Display}}

← Older revision		Revision as of 12:09, 27 August 2022
Line 188:		Line 188:
	*For understanding the Reed–Solomon codes, however, this knowledge is not absolutely necessary.		*For understanding the Reed–Solomon codes, however, this knowledge is not absolutely necessary.

−	*So you could jump directly to the chapter  [[Channel_Coding/Extension_Field#Verallgemeinerte_Definition_eines_Erweiterungsk.C3.B6rpers\| "Extension ~~Fields~~"]] .<br>	+	*So you could jump directly to the chapter  [[Channel_Coding/Extension_Field#Verallgemeinerte_Definition_eines_Erweiterungsk.C3.B6rpers\| "Extension Field"]] .<br>