Channel Coding/General Description of Linear Block Codes - Revision history

Guenter at 09:32, 22 November 2022

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Hwang at 10:46, 18 November 2022

2022-11-18T10:46:37Z

Hwang at 10:39, 18 November 2022

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Hwang at 10:16, 18 November 2022

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Hwang at 10:13, 18 November 2022

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Hwang at 23:14, 12 November 2022

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Hwang at 13:31, 1 November 2022

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Guenter at 12:53, 10 July 2022

2022-07-10T12:53:20Z

Guenter at 09:14, 10 July 2022

2022-07-10T09:14:33Z

Noah at 21:39, 29 June 2022

2022-06-29T21:39:03Z

@@ Line 408: / Line 408: @@
 *This equation holds in general,&nbsp; i.e.,&nbsp; also in the non-binary case.&nbsp; Since we restrict ourselves to binary codes &#8658; &nbsp; $\mathcal{C} \in \text{GF}(2^n)$ &nbsp; &rArr;&nbsp; $ - \boldsymbol{\rm P} = +\boldsymbol{\rm P}$.
-*Thus we obtain the form we will use in the remainder of this paper '''KORREKTUR: page?'''.<br>
+*Thus we obtain the form we will use in the remainder of this chapter.<br>
 ::<math>{ \boldsymbol{\rm H} } =\left(  { \boldsymbol{\rm P} }^{\rm T}  \: ; \: { \boldsymbol{\rm I} }_m \right)\hspace{0.05cm}.

@@ Line 408: / Line 408: @@
 *This equation holds in general,&nbsp; i.e.,&nbsp; also in the non-binary case.&nbsp; Since we restrict ourselves to binary codes &#8658; &nbsp; $\mathcal{C} \in \text{GF}(2^n)$ &nbsp; &rArr;&nbsp; $ - \boldsymbol{\rm P} = +\boldsymbol{\rm P}$.
-*Thus we obtain the form we will use in the remainder of this paper.<br>
+*Thus we obtain the form we will use in the remainder of this paper '''KORREKTUR: page?'''.<br>
 ::<math>{ \boldsymbol{\rm H} } =\left(  { \boldsymbol{\rm P} }^{\rm T}  \: ; \: { \boldsymbol{\rm I} }_m \right)\hspace{0.05cm}.

@@ Line 314: / Line 314: @@
 The analysis shows:
-*The corresponding codes&nbsp; $\mathcal{C}$&nbsp; and&nbsp; $\mathcal{C}_{\rm sys}$&nbsp; contain different code words and are thus&nbsp; &raquo;'''not identical'''&laquo;.:
+*The corresponding codes&nbsp; $\mathcal{C}$&nbsp; and&nbsp; $\mathcal{C}_{\rm sys}$&nbsp; contain different code words and are thus&nbsp; &raquo;'''not identical'''&laquo;:
 ::<math>\mathcal{C} = \big \{ (0, 0, 0, 0) \hspace{0.05cm}, (0, 0, 1, 1) \hspace{0.05cm},(1, 1, 0, 0) \hspace{0.05cm},(1, 1, 1, 1) \big \}\hspace{0.05cm},</math>
 ::<math>\mathcal{C}_{\rm sys}= \big \{ (0, 0, 0, 0) \hspace{0.05cm}, (0, 1, 0, 1) \hspace{0.05cm},(1, 0, 1, 0) \hspace{0.05cm},(1, 1, 1, 1) \big \}\hspace{0.05cm}.</math>

@@ Line 8: / Line 8: @@
 == Linear codes and cyclic codes ==
 <br>
-All codes mentioned so far are&nbsp; '''linear''':
+All codes mentioned so far are&nbsp; &raquo;'''linear'''&laquo;:
 *the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|$\text{single parity-check code}$]]&nbsp; $\rm (SPC)$,
@@ Line 19: / Line 19: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; A&nbsp; '''linear binary block code'''&nbsp; $\mathcal{C}$ &nbsp; is a set of&nbsp; $2^k$&nbsp; code words&nbsp; $\underline{x}= (x_1, x_2, \hspace{0.05cm}\text{...}\hspace{0.05cm}, x_n)$,&nbsp; where the&nbsp; (modulo 2)&nbsp; sum of any two code words&nbsp; $\underline{x}$&nbsp; and&nbsp; $\underline{x}\hspace{0.05cm}'$&nbsp; again gives a valid code word:
+$\text{Definition:}$&nbsp; A&nbsp; &raquo;'''linear binary block code'''&laquo;&nbsp; $\mathcal{C}$ &nbsp; is a set of&nbsp; $2^k$&nbsp; code words&nbsp; $\underline{x}= (x_1, x_2, \hspace{0.05cm}\text{...}\hspace{0.05cm}, x_n)$,&nbsp; where the&nbsp; (modulo 2)&nbsp; sum of any two code words&nbsp; $\underline{x}$&nbsp; and&nbsp; $\underline{x}\hspace{0.05cm}'$&nbsp; again gives a valid code word:
 ::<math>\underline{x}, \underline{x}\hspace{0.05cm}' \in {\rm GF}(2^n),\hspace{0.3cm} \underline{x}, \underline{x}\hspace{0.05cm}' \in  \mathcal{C}
@@ Line 48: / Line 48: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; A linear block code&nbsp; $\mathcal{C}$&nbsp; is called&nbsp; "'''cyclic'''"&nbsp; if every cyclic shift of a code word&nbsp; $\underline{x}$ &nbsp; $($to the left or right$)$ &nbsp; results in a valid code word:
+$\text{Definition:}$&nbsp; A linear block code&nbsp; $\mathcal{C}$&nbsp; is called&nbsp; &raquo;'''cyclic'''&laquo;&nbsp; if every cyclic shift of a code word&nbsp; $\underline{x}$ &nbsp; $($to the left or right$)$ &nbsp; results in a valid code word:
 ::<math>\underline{x}= (x_1, x_2, \hspace{0.05cm}\text{...} \hspace{0.05cm}, x_n) \in   \mathcal{C}
@@ Line 84: / Line 84: @@
 In this equation are used:
-*the&nbsp; '''parity-check matrix'''&nbsp; ${ \boldsymbol{\rm H}}$&nbsp;  with&nbsp; $m = n-k = 3$&nbsp; rows and&nbsp; $n = 7$&nbsp; columns:
+*the&nbsp; &raquo;'''parity-check matrix'''&laquo;&nbsp; ${ \boldsymbol{\rm H}}$&nbsp;  with&nbsp; $m = n-k = 3$&nbsp; rows and&nbsp; $n = 7$&nbsp; columns:
 ::<math>{ \boldsymbol{\rm H}} = \begin{pmatrix}
@@ Line 136: / Line 136: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; A linear binary block code&nbsp; $\mathcal{C}$&nbsp; can be characterized by the&nbsp; '''check matrix'''&nbsp; $\boldsymbol{\rm H}$&nbsp; or with the&nbsp; '''generator matrix'''&nbsp; $\boldsymbol{\rm G}$&nbsp; as follows:
+$\text{Definition:}$&nbsp; A linear binary block code&nbsp; $\mathcal{C}$&nbsp; can be characterized by the&nbsp; &raquo;'''check matrix'''&laquo;&nbsp; $\boldsymbol{\rm H}$&nbsp; or with the&nbsp; &raquo;'''generator matrix'''&laquo;&nbsp; $\boldsymbol{\rm G}$&nbsp; as follows:
 ::<math>\mathcal{C} = \{ \underline{x} \in {\rm GF}(2^n)\text{:} \hspace{0.2cm}{ \boldsymbol{\rm H} } \cdot \underline{x}^{\rm T}= \underline{0}^{\rm T} \}\hspace{0.05cm},</math>
@@ Line 282: / Line 282: @@
 It is gratifying from the point of view of channel coding,&nbsp; that for each code&nbsp; $\mathcal{C}$&nbsp; a systematic&nbsp; $($identical or at least equivalent$)$&nbsp; code&nbsp; $\mathcal{C}_{\rm sys}$&nbsp; can be found.<br>
-In an&nbsp; '''identical systematic code''',&nbsp;&nbsp; $\underline{x}$&nbsp; and&nbsp; $\underline{x}_{\rm sys}$&nbsp; contain the same code words,&nbsp; only the mapping&nbsp; $\underline{u}  &nbsp; &#8594; &nbsp; \underline{x}$&nbsp; is different.&nbsp; One gets from a block code&nbsp; $\mathcal{C}$&nbsp; to the identical systematic code&nbsp; $\mathcal{C}_{\rm sys}$&nbsp; by the following manipulations with respect to the generator matrix&nbsp; $\mathcal{\rm G}$:
+In an&nbsp; &raquo;'''identical systematic code'''&laquo;,&nbsp;&nbsp; $\underline{x}$&nbsp; and&nbsp; $\underline{x}_{\rm sys}$&nbsp; contain the same code words,&nbsp; only the mapping&nbsp; $\underline{u}  &nbsp; &#8594; &nbsp; \underline{x}$&nbsp; is different.&nbsp; One gets from a block code&nbsp; $\mathcal{C}$&nbsp; to the identical systematic code&nbsp; $\mathcal{C}_{\rm sys}$&nbsp; by the following manipulations with respect to the generator matrix&nbsp; $\mathcal{\rm G}$:
 *Swap or permute the rows,<br>
@@ Line 294: / Line 294: @@
-If this is not possible,&nbsp; one can at least find an&nbsp; '''equivalent systematic code'''&nbsp; by swapping or permuting the columns of&nbsp; $\boldsymbol{\rm G}$&nbsp;:
+If this is not possible,&nbsp; one can at least find an&nbsp; &raquo;'''equivalent systematic code'''&laquo;&nbsp; by swapping or permuting the columns of&nbsp; $\boldsymbol{\rm G}$&nbsp;:
 ::<math>\mathcal{C}_{\rm sys} = {\rm \pi} (\mathcal{C})\hspace{0.3cm}{\rm with}\hspace{0.3cm}{\rm \pi}():\hspace{0.15cm}{\rm permutation\:operator}\hspace{0.05cm}.</math>
@@ Line 314: / Line 314: @@
 The analysis shows:
-*The corresponding codes&nbsp; $\mathcal{C}$&nbsp; and&nbsp; $\mathcal{C}_{\rm sys}$&nbsp; contain different code words and are thus&nbsp; '''not identical'''.:
+*The corresponding codes&nbsp; $\mathcal{C}$&nbsp; and&nbsp; $\mathcal{C}_{\rm sys}$&nbsp; contain different code words and are thus&nbsp; &raquo;'''not identical'''&laquo;.:
 ::<math>\mathcal{C} = \big \{ (0, 0, 0, 0) \hspace{0.05cm}, (0, 0, 1, 1) \hspace{0.05cm},(1, 1, 0, 0) \hspace{0.05cm},(1, 1, 1, 1) \big \}\hspace{0.05cm},</math>
 ::<math>\mathcal{C}_{\rm sys}= \big \{ (0, 0, 0, 0) \hspace{0.05cm}, (0, 1, 0, 1) \hspace{0.05cm},(1, 0, 1, 0) \hspace{0.05cm},(1, 1, 1, 1) \big \}\hspace{0.05cm}.</math>
-*But they are&nbsp; '''equivalent''':&nbsp; $\boldsymbol{\rm G}_{\rm sys}$&nbsp; is obtained from&nbsp; $\boldsymbol{\rm G}$&nbsp; by swapping the second and third columns.&nbsp;
+*But they are&nbsp; &raquo;'''equivalent'''&laquo;:&nbsp; $\boldsymbol{\rm G}_{\rm sys}$&nbsp; is obtained from&nbsp; $\boldsymbol{\rm G}$&nbsp; by swapping the second and third columns.&nbsp;
 *It is a&nbsp; $\text{(4, 2, 2)}$ block code in both cases &nbsp;&nbsp;&#8658;&nbsp;&nbsp; $d_{\rm min} = 2$.}}
@@ Line 520: / Line 520: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; Two linear codes&nbsp; $\mathcal{C}$&nbsp; and&nbsp; $\mathcal{C}\hspace{0.05cm}'$,&nbsp; both made of&nbsp; ${\rm GF}(2^n)$,&nbsp; are&nbsp; '''orthogonal'''&nbsp; if all code words&nbsp; $\underline{x} \in \mathcal{C}$&nbsp; to all code words&nbsp; $\underline{x}\hspace{0.05cm}' \in \mathcal{C}\hspace{0.05cm}'$&nbsp; are orthogonal.
+$\text{Definition:}$&nbsp; Two linear codes&nbsp; $\mathcal{C}$&nbsp; and&nbsp; $\mathcal{C}\hspace{0.05cm}'$,&nbsp; both made of&nbsp; ${\rm GF}(2^n)$,&nbsp; are&nbsp; &raquo;'''orthogonal'''&laquo;&nbsp; if all code words&nbsp; $\underline{x} \in \mathcal{C}$&nbsp; to all code words&nbsp; $\underline{x}\hspace{0.05cm}' \in \mathcal{C}\hspace{0.05cm}'$&nbsp; are orthogonal.
-*One then refers to&nbsp; $\mathcal{C}$&nbsp; and&nbsp; $\mathcal{C}\hspace{0.05cm}'$&nbsp; as&nbsp; '''dual codes'''.}}<br>
+*One then refers to&nbsp; $\mathcal{C}$&nbsp; and&nbsp; $\mathcal{C}\hspace{0.05cm}'$&nbsp; as&nbsp; &raquo;'''dual codes'''&laquo;.}}<br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; Two code words&nbsp; $\underline{x} \in{\rm GF}(2^n)$&nbsp; and&nbsp; $\underline{x\hspace{0.05cm}'} \in {\rm GF}(2^n)$&nbsp; are&nbsp; '''orthogonal'''&nbsp; to each other whenever the&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Nomenclature_in_the_fourth_chapter|$\text{inner product}$]]&nbsp; disappears:
+$\text{Definition:}$&nbsp; Two code words&nbsp; $\underline{x} \in{\rm GF}(2^n)$&nbsp; and&nbsp; $\underline{x\hspace{0.05cm}'} \in {\rm GF}(2^n)$&nbsp; are&nbsp; &raquo;'''orthogonal'''&laquo;&nbsp; to each other whenever the&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Nomenclature_in_the_fourth_chapter|$\text{inner product}$]]&nbsp; disappears:
 ::<math>\big \langle \underline{x} \cdot \underline{x}\hspace{0.05cm}' \big \rangle =   \sum_{i=1  }^{n} x_i \cdot x\hspace{0.05cm}'_i = 0 \hspace{0.05cm}, \hspace{0.5cm}

@@ Line 9: / Line 9: @@
 <br>
 All codes mentioned so far are&nbsp; '''linear''':
-*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|"single parity-check code"]]&nbsp; $\rm (SPC)$,
+*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|$\text{single parity-check code}$]]&nbsp; $\rm (SPC)$,
-*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|"repetition code"]]&nbsp; $\rm (RC)$,
+*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|$\text{repetition code}$]]&nbsp; $\rm (RC)$,
-*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes|"Hamming code"]]&nbsp; $\rm (HC)$.
+*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes|$\text{Hamming code}$]]&nbsp; $\rm (HC)$.
@@ Line 177: / Line 177: @@
 *This code is neither suitable for error correction nor for error detection because of&nbsp; $d_{\rm min} = 1$.&nbsp; Nevertheless,&nbsp; this code is also considered exemplarily in the next sections,&nbsp; because the coding results are well interpretable.<br>
-*At this point we would like to draw your attention to the applet&nbsp; [[Applets:Gram-Schmidt-Verfahren|Gram&ndash;Schmidt process]]&nbsp; for the book&nbsp; "Digital Signal Transmission",&nbsp; which teaches the calculation of basis functions,&nbsp; although in a different context than the one used here.<br>
+*At this point we would like to draw your attention to the applet&nbsp; [[Applets:Gram-Schmidt-Verfahren|$\text{Gram&ndash;Schmidt process}$]]&nbsp; for the book&nbsp; "Digital Signal Transmission",&nbsp; which teaches the calculation of basis functions,&nbsp; although in a different context than the one used here.<br>
 == Identical Codes ==
 <br>
-The vectors&nbsp; $\underline{g}_1$,&nbsp; $\underline{g}_2$,&nbsp; ... &nbsp;,&nbsp; $\underline{g}_k$&nbsp; used in&nbsp; $\text{Example 4}$&nbsp; in the last section are the&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Orthonormal_basis_functions| basis vectors]]&nbsp; of the linear block code&nbsp; $\mathcal{C}$.
+The vectors&nbsp; $\underline{g}_1$,&nbsp; $\underline{g}_2$,&nbsp; ... &nbsp;,&nbsp; $\underline{g}_k$&nbsp; used in&nbsp; $\text{Example 4}$&nbsp; in the last section are the&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Orthonormal_basis_functions|$\text{basis vectors}$]]&nbsp; of the linear block code&nbsp; $\mathcal{C}$.
 *The code itself can be viewed as a&nbsp; $k$&ndash;dimensional&nbsp; subspace&nbsp; of&nbsp; $\text{GF}(2^n)$.
@@ Line 478: / Line 478: @@
 == Representation of SPC and RC as dual codes ==
 <br>
-For the codes already dealt with in the chapter&nbsp;  [[Channel_Coding/Examples_of_Binary_Block_Codes|"Examples of binary block codes"]],&nbsp;  the generator matrix&nbsp; $\boldsymbol{\rm G}$&nbsp; and the parity-check matrix&nbsp; $\boldsymbol{\rm H}$&nbsp; are to be given in each case.&nbsp; Let the code length always be&nbsp; $n = 5$ for the following examples,&nbsp; but the results can be interpreted in the same way for other code lengths.&nbsp; It applies to
+For the codes already dealt with in the chapter&nbsp;  [[Channel_Coding/Examples_of_Binary_Block_Codes|$\text{Examples of binary block codes}$]],&nbsp;  the generator matrix&nbsp; $\boldsymbol{\rm G}$&nbsp; and the parity-check matrix&nbsp; $\boldsymbol{\rm H}$&nbsp; are to be given in each case.&nbsp; Let the code length always be&nbsp; $n = 5$ for the following examples,&nbsp; but the results can be interpreted in the same way for other code lengths.&nbsp; It applies to
-*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|"single parity&ndash;check code"]] &nbsp; &#8658; &nbsp; $\text{SPC (5, 4)}$:
+*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|$\text{single parity&ndash;check code}$]] &nbsp; &#8658; &nbsp; $\text{SPC (5, 4)}$:
 ::<math>{ \boldsymbol{\rm H}}
@@ Line 494: / Line 494: @@
 	\hspace{0.05cm};</math>
-*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|"repetition code"]]  &nbsp; &#8658; &nbsp; $\text{RC (5, 1)}$:
+*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|$\text{repetition code}$]]  &nbsp; &#8658; &nbsp; $\text{RC (5, 1)}$:
 ::<math>{ \boldsymbol{\rm G}}
@@ Line 524: / Line 524: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; Two code words&nbsp; $\underline{x} \in{\rm GF}(2^n)$&nbsp; and&nbsp; $\underline{x\hspace{0.05cm}'} \in {\rm GF}(2^n)$&nbsp; are&nbsp; '''orthogonal'''&nbsp; to each other whenever the&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Nomenclature_in_the_fourth_chapter|"inner product"]]&nbsp; disappears:
+$\text{Definition:}$&nbsp; Two code words&nbsp; $\underline{x} \in{\rm GF}(2^n)$&nbsp; and&nbsp; $\underline{x\hspace{0.05cm}'} \in {\rm GF}(2^n)$&nbsp; are&nbsp; '''orthogonal'''&nbsp; to each other whenever the&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Nomenclature_in_the_fourth_chapter|$\text{inner product}$]]&nbsp; disappears:
 ::<math>\big \langle \underline{x} \cdot \underline{x}\hspace{0.05cm}' \big \rangle =   \sum_{i=1  }^{n} x_i \cdot x\hspace{0.05cm}'_i = 0 \hspace{0.05cm}, \hspace{0.5cm}
@@ Line 543: / Line 543: @@
 == Some properties of the (7, 4, 3) Hamming code ==
 <br>
-Let us summarize again the previous results of this chapter using the example of the systematic Hamming code, which has already been described in detail in the chapter&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes|Examples of Binary Block Codes]]&nbsp;. This&nbsp; $\text{(7, 4, 3)}$ code is characterized by
+Let us summarize again the previous results of this chapter using the example of the systematic Hamming code, which has already been described in detail in the chapter&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes|$\text{Examples of Binary Block Codes}$]]&nbsp;. This&nbsp; $\text{(7, 4, 3)}$ code is characterized by
 [[File:P ID2359 KC T 1 4 S7.png|right|frame|Code words of the&nbsp; $\text{HC (7, 4, 3)}$|class=fit]]
 *the number of parity-check equations&nbsp; $m = 3$,<br>

@@ Line 175: / Line 175: @@
 *The basis vectors used here for the calculation&nbsp; $\underline{g}_1$,&nbsp; $\underline{g}_2$&nbsp; and&nbsp; $\underline{g}_3$&nbsp; &ndash; each with length&nbsp; $n = 5$&nbsp; &ndash; correspond to the&nbsp; $k = 3$&nbsp; rows of the generator matrix&nbsp; $\boldsymbol{\rm G}$.
-*This code is neither suitable for error correction nor for error detection because of&nbsp; $d_{\rm min} = 1$.&nbsp; Nevertheless,&nbsp; this code is also considered exemplarily on the next pages,&nbsp; because the coding results are well interpretable.<br>
+*This code is neither suitable for error correction nor for error detection because of&nbsp; $d_{\rm min} = 1$.&nbsp; Nevertheless,&nbsp; this code is also considered exemplarily in the next sections,&nbsp; because the coding results are well interpretable.<br>
 *At this point we would like to draw your attention to the applet&nbsp; [[Applets:Gram-Schmidt-Verfahren|Gram&ndash;Schmidt process]]&nbsp; for the book&nbsp; "Digital Signal Transmission",&nbsp; which teaches the calculation of basis functions,&nbsp; although in a different context than the one used here.<br>
@@ Line 182: / Line 182: @@
 == Identical Codes ==
 <br>
-The vectors&nbsp; $\underline{g}_1$,&nbsp; $\underline{g}_2$,&nbsp; ... &nbsp;,&nbsp; $\underline{g}_k$&nbsp; used in&nbsp; $\text{Example 4}$&nbsp; on the last page are the&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Orthonormal_basis_functions| basis vectors]]&nbsp; of the linear block code&nbsp; $\mathcal{C}$.
+The vectors&nbsp; $\underline{g}_1$,&nbsp; $\underline{g}_2$,&nbsp; ... &nbsp;,&nbsp; $\underline{g}_k$&nbsp; used in&nbsp; $\text{Example 4}$&nbsp; in the last section are the&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Orthonormal_basis_functions| basis vectors]]&nbsp; of the linear block code&nbsp; $\mathcal{C}$.
 *The code itself can be viewed as a&nbsp; $k$&ndash;dimensional&nbsp; subspace&nbsp; of&nbsp; $\text{GF}(2^n)$.
@@ Line 247: / Line 247: @@
 (1, 1, 1, 1, 1) \hspace{0.1cm}= \hspace{0.1cm} \underline{x}_7\hspace{0.05cm}.</math>
-*The corresponding code words&nbsp; $\underline{x}_i = \underline{u}_i \cdot  \boldsymbol{\rm G}$&nbsp; of the code &nbsp; $\mathcal{C}$ &nbsp; &rArr; &nbsp; generator matrix&nbsp;  $\boldsymbol{\rm G}$&nbsp; are give in&nbsp;  $\text{Example 4}$&nbsp; (previous page).}}<br>
+*The corresponding code words&nbsp; $\underline{x}_i = \underline{u}_i \cdot  \boldsymbol{\rm G}$&nbsp; of the code &nbsp; $\mathcal{C}$ &nbsp; &rArr; &nbsp; generator matrix&nbsp;  $\boldsymbol{\rm G}$&nbsp; are give in&nbsp;  $\text{Example 4}$&nbsp; (previous section).}}<br>
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp; The code tables of&nbsp; $\text{Example 4}$&nbsp; and&nbsp; $\text{Example 5}$&nbsp; make clear:
-*$\mathcal{C}$&nbsp; and&nbsp; $\mathcal{C}\hspace{0.05cm}'$&nbsp; contain the same code words.&nbsp; They are thus&nbsp; "identical codes"&nbsp; and both have the same capability of correction&nbsp; (see next page).<br>
+*$\mathcal{C}$&nbsp; and&nbsp; $\mathcal{C}\hspace{0.05cm}'$&nbsp; contain the same code words.&nbsp; They are thus&nbsp; "identical codes"&nbsp; and both have the same capability of correction&nbsp; (see next section).<br>
 *$\mathcal{C}\hspace{0.05cm}'$&nbsp;  is now a&nbsp; "systematic code"&nbsp; since the first&nbsp; $k$&nbsp; binary digits of each code word&nbsp; $\underline{x}\hspace{0.05cm}'_i$&nbsp; match the binary digits of the information word.}}<br>
@@ Line 266: / Line 266: @@
 *The generator matrix in this case has the form &nbsp; $\boldsymbol{\rm G_{\rm sys} } =\left({ \boldsymbol{\rm I}_{\rm k} \ ; \ \boldsymbol{\rm P} }\right)$ &nbsp; with the&nbsp; $k&times;k$&nbsp; "identity matrix" &nbsp; $\boldsymbol{\rm I}_{\rm k}$ &nbsp; and a suitably chosen&nbsp; $(n-1)&times;k$&nbsp; matrix &nbsp; $\boldsymbol{\rm P}$.}}<br>
-So,&nbsp; for &nbsp; $\text{Example 5}$&nbsp; on the last page can also be written:
+So,&nbsp; for &nbsp; $\text{Example 5}$&nbsp; in the last section can also be written:
 ::<math> \boldsymbol{\rm G_{sys}} =\left({ \boldsymbol{\rm I}}_3 \: ; \: { \boldsymbol{\rm P}}\right)\hspace{0.3cm}{\rm with}\hspace{0.3cm}
@@ Line 474: / Line 474: @@
 #This code offers no protection against a transmission error with respect to the third bit&nbsp; $(x_3 = u_3)$&nbsp;.
 #This,&nbsp; of course,&nbsp; does not allow for error detection,&nbsp; and much less error correction.
-#But the same is true for the non-systematic code according to&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Relationship_between_generator_and_parity-check_matrix|$\text{Example 7}$]]&nbsp; on the last page.}}<br>
+#But the same is true for the non-systematic code according to&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Relationship_between_generator_and_parity-check_matrix|$\text{Example 7}$]]&nbsp; in the last section.}}<br>
 == Representation of SPC and RC as dual codes ==

@@ Line 562: / Line 562: @@
 *There are seven code words resulting from&nbsp; "$0011101$"&nbsp; each by cyclic shift&nbsp; (all highlighted in green).
-*For each codeword also exists the "negated" codeword, for example, besides&nbsp; "$0001011$"&nbsp; there is also the code word&nbsp; "$1110100$".
+*For each code word also exists the "negated" code word, for example, besides&nbsp; "$0001011$"&nbsp; there is also the code word&nbsp; "$1110100$".
 *The parity-check matrix can be written as follows:

@@ Line 264: / Line 264: @@
 (u_1,\ u_2,\ ... \hspace{0.05cm},\ u_k,\ x_{k+1}, \hspace{0.05cm}\text{...}\hspace{0.05cm},\ x_n)\hspace{0.05cm}.$
-*The generator matrix in this case has the form &nbsp; $\boldsymbol{\rm G_{\rm sys} } =\left({ \boldsymbol{\rm I}_{\rm k} \ ; \ \boldsymbol{\rm P} }\right)$ &nbsp; with the&nbsp; $k&times;k$&nbsp; unit matrix &nbsp; $\boldsymbol{\rm I}_{\rm k}$ &nbsp; and a suitably chosen&nbsp; $(n-1)&times;k$&nbsp; matrix &nbsp; $\boldsymbol{\rm P}$.}}<br>
+*The generator matrix in this case has the form &nbsp; $\boldsymbol{\rm G_{\rm sys} } =\left({ \boldsymbol{\rm I}_{\rm k} \ ; \ \boldsymbol{\rm P} }\right)$ &nbsp; with the&nbsp; $k&times;k$&nbsp; "identity matrix" &nbsp; $\boldsymbol{\rm I}_{\rm k}$ &nbsp; and a suitably chosen&nbsp; $(n-1)&times;k$&nbsp; matrix &nbsp; $\boldsymbol{\rm P}$.}}<br>
 So,&nbsp; for &nbsp; $\text{Example 5}$&nbsp; on the last page can also be written:

@@ Line 556: / Line 556: @@
 One can see from this:
-*The code includes both the all zero word&nbsp; $(0000000)$&nbsp; and the all one word&nbsp;  $(1111111)$.
+*The code includes both the all zero word&nbsp; "$0000000$"&nbsp; and the all one word&nbsp;  "$1111111$".
-*There are seven code words resulting from&nbsp; $(0001011)$&nbsp; each by cyclic shift&nbsp; (all highlighted in yellow).
-*There are seven code words resulting from&nbsp; $(0011101)$&nbsp; each by cyclic shift&nbsp; (all highlighted in green).
+*There are seven code words resulting from&nbsp; "$0001011$"&nbsp; each by cyclic shift&nbsp; (all highlighted in yellow).
-*For each codeword also exists the "negated" codeword, for example, besides&nbsp; $(0001011)$&nbsp; there is also the codeword&nbsp; $(1110100)$.
 *The parity-check matrix can be written as follows:
@@ Line 578: / Line 582: @@
 	\end{pmatrix}\hspace{0.05cm}.</math>
-Accordingly, the following applies to the generator matrix:
+*Accordingly, the following applies to the generator matrix:
 ::<math>{ \boldsymbol{\rm G}} =\left({ \boldsymbol{\rm I}}_4 \: ; \: { \boldsymbol{\rm P}}\right)

@@ Line 544: / Line 544: @@
 <br>
 Let us summarize again the previous results of this chapter using the example of the systematic Hamming code, which has already been described in detail in the chapter&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes|Examples of Binary Block Codes]]&nbsp;. This&nbsp; $\text{(7, 4, 3)}$ code is characterized by
-[[File:P ID2359 KC T 1 4 S7.png|right|frame|Code words of thes&nbsp; $\text{HC (7, 4, 3)}$|class=fit]]
+[[File:P ID2359 KC T 1 4 S7.png|right|frame|Code words of the&nbsp; $\text{HC (7, 4, 3)}$|class=fit]]
 *the number of parity-check equations&nbsp; $m = 3$,<br>
 *the code length&nbsp; $n = 2^m-1 = 7$,<br>
@@ Line 553: / Line 553: @@
-In obiger Tabelle sind die&nbsp; $2^4 = 16$&nbsp; Codeworte angegeben <br>(schwarz:&nbsp; vier Informationsbits, rot:&nbsp; drei Prüfbits).
+In the above table the&nbsp; $2^4 = 16$&nbsp; code words are given <br>(black:&nbsp; four information bits, red:&nbsp; three parity bits).
-Man erkennt daraus:
+One can see from this:
-*Der Code  beinhaltet sowohl das Null&ndash;Wort&nbsp; $(0000000)$&nbsp; als auch das Eins&ndash;Wort&nbsp;  $(1111111)$.
+*The code includes both the all zero word&nbsp; $(0000000)$&nbsp; and the all one word&nbsp;  $(1111111)$.
-*Es gibt sieben Codeworte, die sich aus&nbsp; $(0001011)$&nbsp; jeweils durch zyklische Verschiebung ergeben&nbsp; (alle gelb hinterlegt).
+*There are seven code words resulting from&nbsp; $(0001011)$&nbsp; each by cyclic shift&nbsp; (all highlighted in yellow).
-*Es gibt sieben Codeworte, die sich aus&nbsp; $(0011101)$&nbsp; jeweils durch zyklische Verschiebung ergeben&nbsp; (alle grün hinterlegt).
+*There are seven code words resulting from&nbsp; $(0011101)$&nbsp; each by cyclic shift&nbsp; (all highlighted in green).
-*Zu jedem Codewort  existiert auch das "negierte" Codewort, zum Beispiel gibt es neben&nbsp; $(0001011)$&nbsp; auch das Codewort&nbsp; $(1110100)$.
+*For each codeword also exists the "negated" codeword, for example, besides&nbsp; $(0001011)$&nbsp; there is also the codeword&nbsp; $(1110100)$.
-*Die Prüfmatrix kann wie folgt geschrieben werden:
+*The parity-check matrix can be written as follows:
 ::<math>{ \boldsymbol{\rm H}} =  \begin{pmatrix}
@@ Line 578: / Line 578: @@
 	\end{pmatrix}\hspace{0.05cm}.</math>
-Dementsprechend gilt für die Generatormatrix:
+Accordingly, the following applies to the generator matrix:
 ::<math>{ \boldsymbol{\rm G}} =\left({ \boldsymbol{\rm I}}_4 \: ; \: { \boldsymbol{\rm P}}\right)