Channel Coding/Decoding of Linear Block Codes - Revision history

Guenter at 16:17, 30 November 2022

2022-11-30T16:17:28Z

Guenter at 09:39, 22 November 2022

2022-11-22T09:39:42Z

Hwang at 11:27, 18 November 2022

2022-11-18T11:27:05Z

Hwang at 11:22, 18 November 2022

2022-11-18T11:22:30Z

Hwang at 11:18, 18 November 2022

2022-11-18T11:18:02Z

Hwang at 11:17, 18 November 2022

2022-11-18T11:17:05Z

Hwang at 11:11, 18 November 2022

2022-11-18T11:11:44Z

Hwang at 11:06, 18 November 2022

2022-11-18T11:06:03Z

Hwang at 10:59, 18 November 2022

2022-11-18T10:59:28Z

Hwang at 10:57, 18 November 2022

2022-11-18T10:57:58Z

@@ Line 141: / Line 141: @@
 {{GraueBox|TEXT=
 $\text{Example 3:}$&nbsp; Starting from the systematic&nbsp; $\text{(5,&nbsp;2,&nbsp;3)}$&nbsp; code&nbsp; $\mathcal{C} = \big \{ (0, 0, 0, 0, 0) \hspace{0.05cm},\hspace{0.15cm}(0, 1, 0, 1, 1) \hspace{0.05cm},\hspace{0.15cm}(1, 0, 1, 1, 0)  \hspace{0.05cm},\hspace{0.15cm}(1, 1, 1, 0, 1) \big \}$&nbsp; the syndrome decoding procedure is now described in detail.
-[[File:EN_KC_T_1_5_S3b_neu1.png|right|frame|Example&nbsp; $\text{(5,&nbsp;2,&nbsp;3)}$ syndrome table with cosets '''KORREKTUR: ""'''|class=fit]]
+[[File:EN_KC_T_1_5_S3b_neu2.png|right|frame|Example&nbsp; $\text{(5,&nbsp;2,&nbsp;3)}$ syndrome table with cosets|class=fit]]
 *The generator matrix and the parity-check matrix are:

@@ Line 87: / Line 87: @@
 Comparing the syndrome with the&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Code_definition_by_the_parity-check_matrix|$\text{parity-check equations}$]]&nbsp; of the Hamming code,&nbsp; we see that
-*most likely the fourth symbol&nbsp; $(x_4 = u_4)$&nbsp; of the code word has been corrupted '''KORREKTUR: falsified''',<br>
+*most likely the fourth symbol&nbsp; $(x_4 = u_4)$&nbsp; of the code word has been falsified,<br>
 *the code word estimator will thus yield the result&nbsp; $\underline{z} = (0, 1, 1, 0, 0, 0, 1)$,<br>
-*the decision is correct only if only one bit was corrupted '''KORREKTUR: falsified''' during transmission.<br><br>
+*the decision is correct only if only one bit was falsified during transmission.<br><br>
 Below are the required corrections for the&nbsp; $\text{(7, 4, 3)}$&nbsp; Hamming code resulting from the calculated syndrome&nbsp; $\underline{s}$&nbsp; corresponding to the columns of the parity-check matrix:
@@ Line 200: / Line 200: @@
 <br>
 We now consider the&nbsp; [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_rate|$\text{bit error rate}$]]&nbsp; for the following constellation:
-[[File:EN_KC_T_1_5_S4.png|right|frame|Bit error rate for the&nbsp; $\text{(7, 4, 3)}$ Hamming code '''Korrektur''' Gainn|class=fit]]
+[[File:EN_KC_T_1_5_S4.png|right|frame|Bit error rate for the&nbsp; $\text{(7, 4, 3)}$ Hamming code|class=fit]]
 #Hamming code&nbsp; $\text{HC (7, 4, 3)}$,<br>
 #AWGN&ndash;channel, characterized by the quotient&nbsp; $E_{\rm B}/N_0$&nbsp; (in dB),<br>

@@ Line 250: / Line 250: @@
 The graphic shows the system model and gives exemplary values for the individual vectors.
 *The left part of the picture&nbsp; (yellow background)&nbsp; is valid for&nbsp; "BSC"&nbsp; with one bit error&nbsp; $(0 &#8594; 1)$&nbsp; at the third bit.
-*The right part of the picture&nbsp; (green background)&nbsp; is for&nbsp; "BEC"&nbsp; and shows two&nbsp; Erasures&nbsp; $(\rm 1 &#8594; E)$&nbsp; at the second and the fourth bit.
+*The right part of the picture&nbsp; (green background)&nbsp; is for&nbsp; "BEC"&nbsp; and shows two&nbsp; erasures&nbsp; $(\rm 1 &#8594; E)$&nbsp; at the second and the fourth bit.
 [[File:EN_KC_T_1_5_S5.png|right|frame|Error correction with BSC and BEC]]

@@ Line 194: / Line 194: @@
 $\text{Conclusion:}$&nbsp; From these examples it is already clear that the&nbsp; &raquo;'''syndrome decoding means a considerable effort'''&laquo;,&nbsp; if one cannot use certain properties e.g. cyclic codes.
-*For large block code lengths, this method fails completely. Thus, to decode a&nbsp; [https://en.wikipedia.org/wiki/BCH_code $\text{BCH codes}$]&nbsp; $($the abbreviation stands for their inventors '''B'''ose, '''C'''haudhuri, '''H'''ocquenghem$)$&nbsp; with code parameters&nbsp; $n = 511$,&nbsp; $k = 259$&nbsp; and&nbsp; $d_{\rm min} = 61$,&nbsp;  one has to evaluate and store exactly&nbsp; $2^{511-259} \approx 10^{76}$&nbsp;error patterns of length&nbsp; $511$.
+*For large block code lengths, this method fails completely. Thus, to decode a&nbsp; [https://en.wikipedia.org/wiki/BCH_code $\text{BCH code}$]&nbsp; $($the abbreviation stands for their inventors '''B'''ose, '''C'''haudhuri, '''H'''ocquenghem$)$&nbsp; with code parameters&nbsp; $n = 511$,&nbsp; $k = 259$&nbsp; and&nbsp; $d_{\rm min} = 61$,&nbsp;  one has to evaluate and store exactly&nbsp; $2^{511-259} \approx 10^{76}$&nbsp;error patterns of length&nbsp; $511$.
 *Happily,&nbsp; however,&nbsp; there are special decoding algorithms for these and also for other codes of large block length,&nbsp; which lead to success with less effort}}.

@@ Line 141: / Line 141: @@
 {{GraueBox|TEXT=
 $\text{Example 3:}$&nbsp; Starting from the systematic&nbsp; $\text{(5,&nbsp;2,&nbsp;3)}$&nbsp; code&nbsp; $\mathcal{C} = \big \{ (0, 0, 0, 0, 0) \hspace{0.05cm},\hspace{0.15cm}(0, 1, 0, 1, 1) \hspace{0.05cm},\hspace{0.15cm}(1, 0, 1, 1, 0)  \hspace{0.05cm},\hspace{0.15cm}(1, 1, 1, 0, 1) \big \}$&nbsp; the syndrome decoding procedure is now described in detail.
-[[File:EN_KC_T_1_5_S3b_neu1.png|right|frame|Example&nbsp; $\text{(5,&nbsp;2,&nbsp;3)}$ syndrome table with cosets|class=fit]]
+[[File:EN_KC_T_1_5_S3b_neu1.png|right|frame|Example&nbsp; $\text{(5,&nbsp;2,&nbsp;3)}$ syndrome table with cosets '''KORREKTUR: ""'''|class=fit]]
 *The generator matrix and the parity-check matrix are:

@@ Line 123: / Line 123: @@
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; Note that the error vector&nbsp; $\underline{e}$&nbsp; as well as the received vector&nbsp; $\underline{y}$&nbsp; is an element of&nbsp; ${\rm GF}(2^n)$&nbsp; unlike the syndrome&nbsp; $\underline{s} \in {\rm GF}(2^m)$&nbsp; with number&nbsp; $m = n-k$&nbsp; of parity-check equations. This means,&nbsp; That
+$\text{Conclusion:}$&nbsp; Note that the error vector&nbsp; $\underline{e}$&nbsp; as well as the received vector&nbsp; $\underline{y}$&nbsp; is an element of&nbsp; ${\rm GF}(2^n)$&nbsp; unlike the syndrome&nbsp; $\underline{s} \in {\rm GF}(2^m)$&nbsp; with number&nbsp; $m = n-k$&nbsp; of parity-check equations. This means,&nbsp; that
 *the association between the syndrome&nbsp; $\underline{s}$&nbsp; and the error vector&nbsp; $\underline{e}$&nbsp; is not unique,&nbsp; but

← Older revision		Revision as of 11:11, 18 November 2022
Line 108:		Line 108:


−	Under this assumption,  we briefly summarize the results of the last ~~pages~~:	+	Under this assumption,  we briefly summarize the results of the last sections:
	*Syndrome decoding is a realization possibility of maximum likelihood decoding of block codes.  One decides on the code word  $\underline{x}_i$   with the least Hamming distance to the received word  $\underline{y}$:		*Syndrome decoding is a realization possibility of maximum likelihood decoding of block codes.  One decides on the code word  $\underline{x}_i$   with the least Hamming distance to the received word  $\underline{y}$:

@@ Line 17: / Line 17: @@
 *The vector&nbsp; $\underline{v}$&nbsp; after decoding&nbsp; (at the sink)&nbsp; should match the information word&nbsp; $\underline{u}$&nbsp; as well as possible.
-*That is: &nbsp; The&nbsp; '''block error probability'''&nbsp; should be as small as possible:
+*That is: &nbsp; The&nbsp; &raquo;'''block error probability'''&laquo;&nbsp; should be as small as possible:
 ::<math>{ \rm Pr(block\:error)} = { \rm Pr}( \underline{v} \ne \underline{u}) \stackrel{!}{=} { \rm minimum}\hspace{0.05cm}.</math>
@@ Line 44: / Line 44: @@
 \hspace{0.05cm}.</math>
-A bit error at position&nbsp; $i$ &nbsp; &rArr; &nbsp; $y_i &ne; x_i$&nbsp; is expressed by the&nbsp; '''error coefficient'''&nbsp; $e_i = 1$.<br>
+A bit error at position&nbsp; $i$ &nbsp; &rArr; &nbsp; $y_i &ne; x_i$&nbsp; is expressed by the&nbsp; &raquo;'''error coefficient'''&laquo;&nbsp; $e_i = 1$.<br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The&nbsp; '''syndrome'''&nbsp; $\underline{s} = (s_0, s_1, \hspace{0.05cm}\text{...} \hspace{0.05cm}, s_{m-1})$&nbsp; is calculated&nbsp; (as row resp. column vector)&nbsp; from the received word&nbsp; $\underline{y}$&nbsp; and the parity-check matrix&nbsp; $\boldsymbol{\rm H}$&nbsp; as follows:
+$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''syndrome'''&laquo;&nbsp; $\underline{s} = (s_0, s_1, \hspace{0.05cm}\text{...} \hspace{0.05cm}, s_{m-1})$&nbsp; is calculated&nbsp; (as row resp. column vector)&nbsp; from the received word&nbsp; $\underline{y}$&nbsp; and the parity-check matrix&nbsp; $\boldsymbol{\rm H}$&nbsp; as follows:
 ::<math>\underline{s} = \underline{y} \cdot { \boldsymbol{\rm H} }^{\rm T}\hspace{0.3cm}{\rm bzw.}\hspace{0.3cm}
@@ Line 192: / Line 192: @@
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; From these examples it is already clear that the&nbsp; '''syndrome decoding means a considerable effort''',&nbsp; if one cannot use certain properties e.g. cyclic codes.
+$\text{Conclusion:}$&nbsp; From these examples it is already clear that the&nbsp; &raquo;'''syndrome decoding means a considerable effort'''&laquo;,&nbsp; if one cannot use certain properties e.g. cyclic codes.
 *For large block code lengths, this method fails completely. Thus, to decode a&nbsp; [https://en.wikipedia.org/wiki/BCH_code $\text{BCH codes}$]&nbsp; $($the abbreviation stands for their inventors '''B'''ose, '''C'''haudhuri, '''H'''ocquenghem$)$&nbsp; with code parameters&nbsp; $n = 511$,&nbsp; $k = 259$&nbsp; and&nbsp; $d_{\rm min} = 61$,&nbsp;  one has to evaluate and store exactly&nbsp; $2^{511-259} \approx 10^{76}$&nbsp;error patterns of length&nbsp; $511$.
@@ Line 216: / Line 216: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; We refer as&nbsp; '''coding gain'''&nbsp; of a considered system configuration,&nbsp;
+$\text{Definition:}$&nbsp; We refer as&nbsp; &raquo;'''coding gain'''&laquo;&nbsp; of a considered system configuration,&nbsp;
 *characterized by its code and
 *the way it is decoded,

@@ Line 8: / Line 8: @@
 == Block diagram and requirements ==
 <br>
-We start from the block diagram already shown in the chapter&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures|"Channel Models and Decision Structures"]]&nbsp; where the digital channel model used is mostly the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC|"Binary Symmetric Channel"]]&nbsp;$\rm  (BSC)$.&nbsp;
+We start from the block diagram already shown in the chapter&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures|$\text{Channel Models and Decision Structures}$]]&nbsp; where the digital channel model used is mostly the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC|$\text{Binary Symmetric Channel}$]]&nbsp;$\rm  (BSC)$.&nbsp;
-For code word estimation,&nbsp; we use the&nbsp; "Maximum Likelihood Decision"&nbsp; $\rm (ML)$,&nbsp; which for binary codes &nbsp; &#8658; &nbsp; $\underline{x}  \in {\rm GF}(2^n)$&nbsp; at the block level gives the same result as the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Definitions_of_the_different_optimal_receivers|"MAP Receiver"]].<br>
+For code word estimation,&nbsp; we use the&nbsp; "Maximum Likelihood Decision"&nbsp; $\rm (ML)$,&nbsp; which for binary codes &nbsp; &#8658; &nbsp; $\underline{x}  \in {\rm GF}(2^n)$&nbsp; at the block level gives the same result as the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Definitions_of_the_different_optimal_receivers|$\text{MAP Receiver}$]].<br>
 [[File:EN_KC_T_1_5_S1.png|right|frame|Block diagram for decoding block codes|class=fit]]
@@ Line 27: / Line 27: @@
 ::<math>\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} {\rm Pr}( \underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm}|\hspace{0.05cm} \underline{y} ) \hspace{0.05cm}.</math>
-*For the BSC model,&nbsp; both &nbsp; $\underline{x}_i \in {\rm GF}(2^n)$ &nbsp; and &nbsp; $\underline{y}  \in {\rm GF}(2^n)$,&nbsp; so the maximum likelihood decision rule can also be written using the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding |"Hamming distance"]]&nbsp; $d_{\rm H}( \underline{y}, \, \underline{x}_i)$:
+*For the BSC model,&nbsp; both &nbsp; $\underline{x}_i \in {\rm GF}(2^n)$ &nbsp; and &nbsp; $\underline{y}  \in {\rm GF}(2^n)$,&nbsp; so the maximum likelihood decision rule can also be written using the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding |$\text{Hamming distance}$]]&nbsp; $d_{\rm H}( \underline{y}, \, \underline{x}_i)$:
 ::<math>\underline{z} = {\rm arg} \min_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm}
@@ Line 65: / Line 65: @@
 {{GraueBox|TEXT=
-$\text{Example 1:}$&nbsp; Starting from the systematic&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Some_properties_of_the_.287.2C_4.2C_3.29_Hamming_code|"$\text{(7, 4, 3)}$ Hamming code"]],&nbsp; the following result is obtained for the received vector&nbsp; $\underline{y} = (0, 1, 1, 1, 0, 0, 1)$:
+$\text{Example 1:}$&nbsp; Starting from the systematic&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Some_properties_of_the_.287.2C_4.2C_3.29_Hamming_code|$\text{(7, 4, 3)}$ Hamming code]],&nbsp; the following result is obtained for the received vector&nbsp; $\underline{y} = (0, 1, 1, 1, 0, 0, 1)$:
 ::<math>{ \boldsymbol{\rm H} } \cdot \underline{y}^{\rm T}
@@ Line 86: / Line 86: @@
 	\end{pmatrix} = \underline{s}^{\rm T} \hspace{0.05cm}.</math>
-Comparing the syndrome with the&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Code_definition_by_the_parity-check_matrix| "parity-check equations"]]&nbsp; of the Hamming code,&nbsp; we see that
+Comparing the syndrome with the&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Code_definition_by_the_parity-check_matrix|$\text{parity-check equations}$]]&nbsp; of the Hamming code,&nbsp; we see that
 *most likely the fourth symbol&nbsp; $(x_4 = u_4)$&nbsp; of the code word has been corrupted,<br>
@@ Line 117: / Line 117: @@
 :$$\underline{s} = \underline{y} \cdot { \boldsymbol{\rm H}}^{\rm T} .$$
-*With the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|"Hamming weight"]]&nbsp; $w_{\rm H}(\underline{e})$&nbsp; the second interpretation can also be mathematically formulated as follows:
+*With the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|$\text{Hamming weight}$]]&nbsp; $w_{\rm H}(\underline{e})$&nbsp; the second interpretation can also be mathematically formulated as follows:
 ::<math>\underline{z} = \underline{y} + {\rm arg} \min_{\underline{e}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} {\rm GF}(2^n)} \hspace{0.1cm}
@@ Line 194: / Line 194: @@
 $\text{Conclusion:}$&nbsp; From these examples it is already clear that the&nbsp; '''syndrome decoding means a considerable effort''',&nbsp; if one cannot use certain properties e.g. cyclic codes.
-*For large block code lengths, this method fails completely. Thus, to decode a&nbsp; [https://en.wikipedia.org/wiki/BCH_code "BCH codes"]&nbsp; $($the abbreviation stands for their inventors '''B'''ose, '''C'''haudhuri, '''H'''ocquenghem$)$&nbsp; with code parameters&nbsp; $n = 511$,&nbsp; $k = 259$&nbsp; and&nbsp; $d_{\rm min} = 61$,&nbsp;  one has to evaluate and store exactly&nbsp; $2^{511-259} \approx 10^{76}$&nbsp;error patterns of length&nbsp; $511$.
+*For large block code lengths, this method fails completely. Thus, to decode a&nbsp; [https://en.wikipedia.org/wiki/BCH_code $\text{BCH codes}$]&nbsp; $($the abbreviation stands for their inventors '''B'''ose, '''C'''haudhuri, '''H'''ocquenghem$)$&nbsp; with code parameters&nbsp; $n = 511$,&nbsp; $k = 259$&nbsp; and&nbsp; $d_{\rm min} = 61$,&nbsp;  one has to evaluate and store exactly&nbsp; $2^{511-259} \approx 10^{76}$&nbsp;error patterns of length&nbsp; $511$.
 *Happily,&nbsp; however,&nbsp; there are special decoding algorithms for these and also for other codes of large block length,&nbsp; which lead to success with less effort}}.
 == Coding gain - bit error rate with AWGN ==
 <br>
-We now consider the&nbsp; [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_rate| "bit error rate"]]&nbsp; for the following constellation:
+We now consider the&nbsp; [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_rate|$\text{bit error rate}$]]&nbsp; for the following constellation:
 [[File:EN_KC_T_1_5_S4.png|right|frame|Bit error rate for the&nbsp; $\text{(7, 4, 3)}$ Hamming code '''Korrektur''' Gainn|class=fit]]
 #Hamming code&nbsp; $\text{HC (7, 4, 3)}$,<br>
@@ Line 209: / Line 209: @@
 *The black comparison curve applies e.g. to binary phase modulation&nbsp; $\rm (BPSK)$&nbsp; without coding.&nbsp; For this one needs for the bit error rate &nbsp; $10^{-5}$&nbsp; about&nbsp; $10 \cdot \lg \, E_{\rm B}/N_0 = 9.6 \, \rm dB$.<br>
-*The red circles apply to the&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Some_properties_of_the_.287.2C_4.2C_3.29_Hamming_code|"$\text{HC (7, 4, 3)}$"]]&nbsp; and hard decisions of the maximum likelihood decoder&nbsp; $\text{(ML&ndash;HD)}$.&nbsp; Syndrome decoding is a possible realization form for this.<br>
+*The red circles apply to the&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Some_properties_of_the_.287.2C_4.2C_3.29_Hamming_code|$\text{HC (7, 4, 3)}$]]&nbsp; and hard decisions of the maximum likelihood decoder&nbsp; $\text{(ML&ndash;HD)}$.&nbsp; Syndrome decoding is a possible realization form for this.<br>
 *This configuration brings an improvement over the comparison system only for&nbsp; $10 \cdot \lg \, E_{\rm B}/N_0 >6 \, \rm dB$.&nbsp; For&nbsp; $\rm BER =10^{-5}$&nbsp; one only needs&nbsp; $10 \cdot \lg \, E_{\rm B}/N_0 \approx 9.2 \, \rm dB$.<br>
-*The green crosses for&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Some_properties_of_the_.287.2C_4.2C_3.29_Hamming_code|"$\text{HC (7, 4, 3)}$"]]&nbsp; and&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder| "Soft&ndash;Decision"]]&nbsp; $\text{(ML&ndash;SD)}$&nbsp; are below the red curve throughout the range.&nbsp; For&nbsp; $\rm BER =10^{-5}$&nbsp; this gives&nbsp; $10 \cdot \lg \, E_{\rm B}/N_0 \approx 7.8 \, \rm dB$.<br><br>
+*The green crosses for&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Some_properties_of_the_.287.2C_4.2C_3.29_Hamming_code|$\text{HC (7, 4, 3)}$]]&nbsp; and&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder| $\text{soft decision}$]]&nbsp; $\text{(ML&ndash;SD)}$&nbsp; are below the red curve throughout the range.&nbsp; For&nbsp; $\rm BER =10^{-5}$&nbsp; this gives&nbsp; $10 \cdot \lg \, E_{\rm B}/N_0 \approx 7.8 \, \rm dB$.<br><br>
 {{BlaueBox|TEXT=
@@ Line 238: / Line 238: @@
 <br>
 Finally,&nbsp; it will be shown to what extent the decoder has to be modified
-*if instead of the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC| "BSC model"]]&nbsp; ("Binary Symmetric Channel")&nbsp;
+*if instead of the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC| $\text{BSC model}$]]&nbsp; ("Binary Symmetric Channel")&nbsp;
-*the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Erasure_Channel_.E2.80.93_BEC| "BEC model"]]&nbsp; ("Binary Erasure Channel")&nbsp; is used,&nbsp;
+*the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Erasure_Channel_.E2.80.93_BEC|$\text{BEC model}$]]&nbsp; ("Binary Erasure Channel")&nbsp; is used,&nbsp;
@@ Line 245: / Line 245: @@
 {{GraueBox|TEXT=
-$\text{Example 4:}$&nbsp; We consider again as in&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Generalization_of_syndrome_coding|$\text{"Example 3"}$]]&nbsp; the systematic&nbsp; $\text{(5, 2, 3)}$&nbsp; block code&nbsp;
+$\text{Example 4:}$&nbsp; We consider again as in&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Generalization_of_syndrome_coding|$\text{Example 3}$]]&nbsp; the systematic&nbsp; $\text{(5, 2, 3)}$&nbsp; block code&nbsp;
 :$$\mathcal{C} = \big \{ (0, 0, 0, 0, 0) \hspace{0.05cm},\hspace{0.3cm}(0, 1, 0, 1, 1) \hspace{0.05cm},\hspace{0.3cm}(1, 0, 1, 1, 0)  \hspace{0.05cm},\hspace{0.3cm}(1, 1, 1, 0, 1) \big \}.$$