Channel Coding/Definition and Properties of Reed-Solomon Codes - Revision history

Hwang at 19:46, 23 November 2022

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Hwang at 19:32, 23 November 2022

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Hwang at 16:22, 23 November 2022

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Hwang at 23:28, 12 November 2022

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Guenter at 14:32, 8 October 2022

2022-10-08T14:32:20Z

Guenter at 16:48, 5 October 2022

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Guenter at 14:52, 5 October 2022

2022-10-05T14:52:45Z

@@ Line 127: / Line 127: @@
 ::<math>c_2 =u (x = \alpha^2) =  1 + \alpha^2 \cdot \alpha^2 = 1 + \alpha =  \alpha^2  \hspace{0.05cm}.</math>
-<u>Hints:</u>
+<u>Notes:</u>
 *From the element set&nbsp; $\{0, \ \alpha^0 = 1, \ \alpha^1 = \alpha, \ \alpha^2\}$&nbsp; it should not be concluded that for this code the three-dimensional representation with axes &nbsp; $\alpha^0 = 1$, &nbsp; $\alpha^1 = \alpha$, &nbsp; $\alpha^2$ &nbsp; applies.&nbsp; See section &nbsp; [[Channel_Coding/Extension_Field#Binary_extension_fields_.E2.80.93_Primitive_polynomials|"Primitive Polynomials"]]&nbsp;&ndash; Example 5.

@@ Line 274: / Line 274: @@
 <br>
 An important parameter of any block code is the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|$\text{minimum distance}$]]&nbsp; between any two code words&nbsp; $\underline {c}_i$&nbsp; and&nbsp; $\underline {c}_j$.
-*The Reed&ndash;Solomon&ndash;codes belong to the class of&nbsp; '''linear&nbsp; and&nbsp; cyclic codes'''.&nbsp; For these,&nbsp; one can start from the zero word&nbsp; $\underline {c}_0 = (0, 0, \hspace{0.05cm} \text{...} \hspace{0.05cm}, 0)$&nbsp; as the reference.
+*The Reed&ndash;Solomon&ndash;codes belong to the class of&nbsp; &raquo;'''linear&nbsp; and&nbsp; cyclic codes'''&laquo;.&nbsp; For these,&nbsp; one can start from the zero word&nbsp; $\underline {c}_0 = (0, 0, \hspace{0.05cm} \text{...} \hspace{0.05cm}, 0)$&nbsp; as the reference.
 *From the number of zeros in the other code words&nbsp; $\underline {c}_j &ne; \underline {c}_0$&nbsp; the&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code|$\text{distance spectrum}$]]&nbsp; $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$&nbsp; can be specified.<br>

@@ Line 8: / Line 8: @@
 == Coding principle and code parameters ==
 <br>
-A&nbsp; "Reed&ndash;Solomon code"&nbsp; &ndash; hereafter sometimes shortened to&nbsp; "RS code" &nbsp; &ndash; is a&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Linear_codes_and_cyclic_codes| "linear block code"]] which assigns
+A&nbsp; "Reed&ndash;Solomon code"&nbsp; &ndash; hereafter sometimes shortened to&nbsp; "RS code" &nbsp; &ndash; is a&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Linear_codes_and_cyclic_codes|$\text{linear block code}$]] which assigns
 *an information block&nbsp; $\underline{u}$&nbsp; with&nbsp; $k$&nbsp; symbols
@@ Line 16: / Line 16: @@
 [[File:P ID2515 KC T 2 3 S1 v2.png|right|frame|Linear&nbsp; $(n, \, k)$&nbsp; block code with new nomenclature|class=fit]]
 These codes,&nbsp; still widely used today,&nbsp; were invented as early as the early 1960s
-*by&nbsp; [https://en.wikipedia.org/wiki/Irving_S._Reed Irving Stoy Reed]&nbsp;
+*by&nbsp; [https://en.wikipedia.org/wiki/Irving_S._Reed $\text{Irving Stoy Reed}$]&nbsp;
-*and&nbsp; [https://en.wikipedia.org/wiki/Gustave_Solomon Gustave Solomon].<br>
+*and&nbsp; [https://en.wikipedia.org/wiki/Gustave_Solomon $\text{Gustave Solomon}$].<br>
@@ Line 58: / Line 58: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp; An &nbsp; $(n,\ k)\text{ Reed&ndash;Solomon code}$&nbsp; for the Galois field&nbsp; ${\rm GF}(2^m)$&nbsp; is defined by
-*the&nbsp; $n= 2^{m-1}$&nbsp; elements of&nbsp; ${\rm GF}(2^m) \hspace{-0.05cm}\setminus \hspace{-0.05cm} \{0\} =\{ \alpha^0, \alpha^1, \, \text{...} \, , \alpha^{n-1}\}$,&nbsp; where&nbsp; $\alpha$&nbsp; denotes a&nbsp; [[Channel_Coding/Extension_Field#Binary_extension_fields_.E2.80.93_Primitive_polynomials|"primitive element"]]&nbsp; of&nbsp; ${\rm GF}(2^m)$,&nbsp;
+*the&nbsp; $n= 2^{m-1}$&nbsp; elements of&nbsp; ${\rm GF}(2^m) \hspace{-0.05cm}\setminus \hspace{-0.05cm} \{0\} =\{ \alpha^0, \alpha^1, \, \text{...} \, , \alpha^{n-1}\}$,&nbsp; where&nbsp; $\alpha$&nbsp; denotes a&nbsp; [[Channel_Coding/Extension_Field#Binary_extension_fields_.E2.80.93_Primitive_polynomials|$\text{primitive element}$]]&nbsp; of&nbsp; ${\rm GF}(2^m)$,&nbsp;
-* a&nbsp; [[Channel_Coding/Extension_Field#Generalized_definition_of_an_extension_field|"polynomial"]]&nbsp; of degree&nbsp; $k-1$&nbsp; matched to an information block&nbsp; $\underline{u}$:&nbsp;
+* a&nbsp; [[Channel_Coding/Extension_Field#Generalized_definition_of_an_extension_field|$\text{polynomial}$]]&nbsp; of degree&nbsp; $k-1$&nbsp; matched to an information block&nbsp; $\underline{u}$:&nbsp;
 :::<math>u(x) = \sum_{i = 0}^{k-1} u_i \cdot x^{i}  = u_0 + u_1 \cdot x + u_2 \cdot x^2 + \hspace{0.1cm}... \hspace{0.1cm}+ u_{k-1} \cdot x^{k-1}   \hspace{0.05cm},\hspace{0.4cm} u_i \in {\rm GF}(2^m)
 \hspace{0.05cm}.</math>
@@ Line 135: / Line 135: @@
 == Generator matrix of Reed-Solomon codes ==
 <br>
-Since the Reed&ndash;Solomon code is a linear block code,&nbsp; the relationship between the information word&nbsp; $\underline {u}$&nbsp; and the  code word&nbsp; $\underline {c}$&nbsp; is given by the&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Code_definition_by_the_generator_matrix| "generator matrix"]]&nbsp; $\boldsymbol{\rm G}$:
+Since the Reed&ndash;Solomon code is a linear block code,&nbsp; the relationship between the information word&nbsp; $\underline {u}$&nbsp; and the  code word&nbsp; $\underline {c}$&nbsp; is given by the&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Code_definition_by_the_generator_matrix|$\text{generator matrix}$]]&nbsp; $\boldsymbol{\rm G}$:
 ::<math>\underline {c} = \underline {u} \cdot { \boldsymbol{\rm G}}
@@ Line 201: / Line 201: @@
 {{BlaueBox|TEXT=
-&rArr; &nbsp; The&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Code_definition_by_the_generator_matrix| "generator matrix"]]&nbsp; $\boldsymbol{\rm G}$&nbsp; $(k$ rows,&nbsp; $n$ columns$)$&nbsp; and the&nbsp;  [[Channel_Coding/General_Description_of_Linear_Block_Codes#Code_definition_by_the_parity-check_matrix| "parity-check matrix"]]&nbsp; $\boldsymbol{\rm H}$&nbsp; $(n-k$ rows,&nbsp; $n$&nbsp; columns$)$&nbsp; must jointly satisfy the following equation:
+&rArr; &nbsp; The&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Code_definition_by_the_generator_matrix|$\text{generator matrix}$]]&nbsp; $\boldsymbol{\rm G}$&nbsp; $(k$ rows,&nbsp; $n$ columns$)$&nbsp; and the&nbsp;  [[Channel_Coding/General_Description_of_Linear_Block_Codes#Code_definition_by_the_parity-check_matrix|$\text{parity-check matrix}$]]&nbsp; $\boldsymbol{\rm H}$&nbsp; $(n-k$ rows,&nbsp; $n$&nbsp; columns$)$&nbsp; must jointly satisfy the following equation:
 :$${\boldsymbol{\rm G} } \cdot { \boldsymbol{\rm H } }^{\rm T}= { \boldsymbol{\rm 0} }\hspace{0.05cm}.$$
@@ Line 273: / Line 273: @@
 == Singleton bound and minimum distance ==
 <br>
-An important parameter of any block code is the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding| "minimum distance"]]&nbsp; between any two code words&nbsp; $\underline {c}_i$&nbsp; and&nbsp; $\underline {c}_j$.
+An important parameter of any block code is the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|$\text{minimum distance}$]]&nbsp; between any two code words&nbsp; $\underline {c}_i$&nbsp; and&nbsp; $\underline {c}_j$.
 *The Reed&ndash;Solomon&ndash;codes belong to the class of&nbsp; '''linear&nbsp; and&nbsp; cyclic codes'''.&nbsp; For these,&nbsp; one can start from the zero word&nbsp; $\underline {c}_0 = (0, 0, \hspace{0.05cm} \text{...} \hspace{0.05cm}, 0)$&nbsp; as the reference.
-*From the number of zeros in the other code words&nbsp; $\underline {c}_j &ne; \underline {c}_0$&nbsp; the&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code| "distance spectrum"]]&nbsp; $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$&nbsp; can be specified.<br>
+*From the number of zeros in the other code words&nbsp; $\underline {c}_j &ne; \underline {c}_0$&nbsp; the&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code|$\text{distance spectrum}$]]&nbsp; $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$&nbsp; can be specified.<br>
@@ Line 283: / Line 283: @@
 *Compared to the previous graphs,&nbsp; the symbol set is simplified by &nbsp; $\{0,\ 1,\ 2,\ 3\}$ &nbsp; instead of &nbsp; $\{0,\ \alpha^0,\ \alpha^1,\ \alpha^2\}$.
-*The distance&nbsp; $d$&nbsp; between&nbsp; $\underline {c}_j$&nbsp; and the null word&nbsp; $\underline {c}_0$&nbsp; is identical to the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|"Hamming weight"]]&nbsp; $w_{\rm H}(\underline {c}_j)$.<br>
+*The distance&nbsp; $d$&nbsp; between&nbsp; $\underline {c}_j$&nbsp; and the null word&nbsp; $\underline {c}_0$&nbsp; is identical to the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|$\text{Hamming weight}$]]&nbsp; $w_{\rm H}(\underline {c}_j)$.<br>
 [[File:EN_KC_T_2_3_S3.png|right|frame|To derive the distance spectrum for the&nbsp; $\text{RSC (3, 2, 2)}_4$|class=fit]]
@@ Line 301: / Line 301: @@
 #For&nbsp; "error correcting codes"&nbsp; one usually chooses&nbsp; $d_{\rm min} $&nbsp; odd &nbsp; &#8658; &nbsp; $n-k$&nbsp; even.&nbsp;
 #For Reed&ndash;Solomon codes then up to &nbsp; $t =(n-k)/2$ &nbsp; symbol errors can be corrected.<br>
-#The&nbsp; [https://en.wikipedia.org/wiki/Singleton_bound "Singleton bound"]&nbsp; says that&nbsp; $d_{\rm min} \le n-k+1$&nbsp; holds for all linear codes.
+#The&nbsp; [https://en.wikipedia.org/wiki/Singleton_bound $\text{Singleton bound}$]&nbsp; says that&nbsp; $d_{\rm min} \le n-k+1$&nbsp; holds for all linear codes.
-#Reed&ndash;Solomon codes reach this bound with equality;&nbsp; they are so-called&nbsp; [https://en.wikipedia.org/wiki/Singleton_bound#MDS_codes "maximum distance separable codes"]&nbsp; (short: "MDS codes").
+#Reed&ndash;Solomon codes reach this bound with equality;&nbsp; they are so-called&nbsp; [https://en.wikipedia.org/wiki/Singleton_bound#MDS_codes $\text{maximum distance separable codes}$]&nbsp; (short: "MDS codes").
-#The&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code| "distance spectrum"]]&nbsp; is composed of&nbsp; $W_0 = 1$&nbsp; and other weight factors&nbsp; $W_i$&nbsp; with&nbsp; $d &#8804; i &#8804; n$,&nbsp; where in the following equation&nbsp; $d_{\rm min}$&nbsp; is abbreviated as&nbsp; $d$:
+#The&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code|$\text{distance spectrum}$]]&nbsp; is composed of&nbsp; $W_0 = 1$&nbsp; and other weight factors&nbsp; $W_i$&nbsp; with&nbsp; $d &#8804; i &#8804; n$,&nbsp; where in the following equation&nbsp; $d_{\rm min}$&nbsp; is abbreviated as&nbsp; $d$:
 :::<math>W_i =   {n \choose i} \cdot \sum_{j = 0}^{i-d}\hspace{0.15cm}(-1)^j \cdot {i \choose j} \cdot \bigg  [\hspace{0.03cm}q^{i\hspace{0.03cm}-\hspace{0.03cm}j\hspace{0.03cm}-\hspace{0.03cm}d\hspace{0.03cm}+\hspace{0.03cm}1}-1 \hspace{0.03cm} \bigg ]\hspace{0.05cm}.</math>}}
@@ Line 343: / Line 343: @@
 *Both codes have the same  minimum distance &nbsp;$d_{\rm min} = 2$.
-*But the two codes differ in terms of the distance spectrum&nbsp; $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$&nbsp; and the&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code|"weight enumerator function"]]&nbsp; $W(X)$:
+*But the two codes differ in terms of the distance spectrum&nbsp; $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$&nbsp; and the&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code|$\text{weight enumerator function}$]]&nbsp; $W(X)$:
 :$$\text{RSC (3, 2, 2)}_4\text{:} \hspace{0.3cm} W_0 =  1\hspace{0.05cm}, \hspace{0.2cm}W_2 =  9\hspace{0.05cm}, \hspace{0.2cm}W_3 =  6\hspace{0.3cm}$$
@@ Line 371: / Line 371: @@
 # Another design parameter is the code rate&nbsp; $R=k/n$,&nbsp; where the code word length&nbsp; $n = 2^m -1$&nbsp; is not completely arbitrary.&nbsp; By extension,&nbsp; shortening and puncturing &ndash; see&nbsp; [[Aufgaben:Exercise_1.09Z:_Extension_and/or_Puncturing|"Exercise 1.9Z"]]&nbsp; &ndash; the variety of possible codes can be further increased.<br>
 # For Reed&ndash;Solomon codes,&nbsp; the weight distribution is known exactly and adaptation to the error structure of the channel is possible.&nbsp; These codes are particularly well suited for burst error channels,&nbsp; which are often present in mobile radio systems due to temporary shadowing.<br>
-# In the case of statistically independent errors,&nbsp; [https://en.wikipedia.org/wiki/BCH_code "BCH codes"]&nbsp; $($from the inventors&nbsp; $\rm B$ose&ndash;$\rm C$haudhuri&ndash;$\rm H$ocquenghem$)$&nbsp; are more suitable.&nbsp; These codes are closely related to Reed&ndash;Solomon codes,&nbsp; but they do not always satisfy the Singleton criterion.&nbsp; A detailed description can be found in [Fri96]<ref name='Fri96'>Friedrichs, B.:&nbsp; Kanalcodierung - Grundlagen und Anwendungen in modernen Kommunikations- systemen.&nbsp; Berlin - Heidelberg: Springer, 1996.</ref>.<br>
+# In the case of statistically independent errors,&nbsp; [https://en.wikipedia.org/wiki/BCH_code $\text{BCH codes}$]&nbsp; $($from the inventors&nbsp; $\rm B$ose&ndash;$\rm C$haudhuri&ndash;$\rm H$ocquenghem$)$&nbsp; are more suitable.&nbsp; These codes are closely related to Reed&ndash;Solomon codes,&nbsp; but they do not always satisfy the Singleton criterion.&nbsp; A detailed description can be found in [Fri96]<ref name='Fri96'>Friedrichs, B.:&nbsp; Kanalcodierung - Grundlagen und Anwendungen in modernen Kommunikations- systemen.&nbsp; Berlin - Heidelberg: Springer, 1996.</ref>.<br>
-# Decoding according to the&nbsp; [https://en.wikipedia.org/wiki/Lattice_problem "BDD Principle"]&nbsp; ("Bounded Distance Decoding")&nbsp; can be done computationally very easily,&nbsp; for example with the&nbsp; [https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm "Berlekamp&ndash;Massey Algorithm"].&nbsp; Moreover,&nbsp; the decoder can also process&nbsp; [https://en.wikipedia.org/wiki/Soft-decision_decoder "Soft Decision Information"]&nbsp;  without much additional effort.<br>
+# Decoding according to the&nbsp; [https://en.wikipedia.org/wiki/Lattice_problem $\text{BDD Principle}$]&nbsp; ("Bounded Distance Decoding")&nbsp; can be done computationally very easily,&nbsp; for example with the&nbsp; [https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm $\text{Berlekamp&ndash;Massey Algorithm}$].&nbsp; Moreover,&nbsp; the decoder can also process&nbsp; [https://en.wikipedia.org/wiki/Soft-decision_decoder $\text{Soft Decision Information}$]&nbsp;  without much additional effort.<br>
 == Exercises for the chapter==

@@ Line 152: / Line 152: @@
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; We consider as in&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Coding_principle_and_code_parameters|$\text{Example 2}$]]&nbsp; (see previous page)&nbsp; the&nbsp; $\text{RSC (3, 2, 2)}_4$&nbsp; whose generator matrix has the following form:
+$\text{Example 3:}$&nbsp; We consider as in&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Coding_principle_and_code_parameters|$\text{Example 2}$]]&nbsp; (see previous section)&nbsp; the&nbsp; $\text{RSC (3, 2, 2)}_4$&nbsp; whose generator matrix has the following form:
 ::<math> { \boldsymbol{\rm G} } =
@@ Line 187: / Line 187: @@
 #Of course,&nbsp; with the same generator matrix you get exactly the same code table&nbsp; $\underline {u} &nbsp; &#8596; &nbsp; \underline {c}$&nbsp; as after the calculation via the function&nbsp; $u(x)$.
-#The corresponding code table on bit level&nbsp; $($see&nbsp; $\text{Example 2}$&nbsp; on the previous page$)$&nbsp; results again,&nbsp; if the elements are not given in exponent representation,&nbsp; but in coefficient form:
+#The corresponding code table on bit level&nbsp; $($see&nbsp; $\text{Example 2}$&nbsp; in the previous section$)$&nbsp; results again,&nbsp; if the elements are not given in exponent representation,&nbsp; but in coefficient form:
 :::<math>(0, \hspace{0.1cm}\alpha^{0}, \hspace{0.1cm}\alpha^{1}, \hspace{0.1cm}\alpha^{2}) \hspace{0.3cm}\Leftrightarrow\hspace{0.3cm}(00, \hspace{0.1cm}01, \hspace{0.1cm}10, \hspace{0.1cm}11)
@@ Line 194: / Line 194: @@
 == Generator matrix and parity-check matrix ==
 <br>
-We now generalize the result of the last page for a Reed&ndash;Solomon code with
+We now generalize the result of the last section for a Reed&ndash;Solomon code with
 *dimension&nbsp; $k$&nbsp; $($number of symbols per information block$)$,<br>
@@ Line 333: / Line 333: @@
 {{GraueBox|TEXT=
-$\text{Example 6:}$&nbsp; As in&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Singleton_bound_and_minimum_distance|$\text{Example 5}$]]&nbsp; $($previous page$)$,&nbsp; we consider the&nbsp; $\text{RSC (3, 2, 2)}_4$.&nbsp;
+$\text{Example 6:}$&nbsp; As in&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Singleton_bound_and_minimum_distance|$\text{Example 5}$]]&nbsp; $($previous section$)$,&nbsp; we consider the&nbsp; $\text{RSC (3, 2, 2)}_4$.&nbsp;
 [[File:EN_KC_T_2_3_S3.png|right|frame|Derivation of the distance spectra of&nbsp; $\text{RSC (3, 2, 2)}_4$&nbsp; and &nbsp;$\text{RSC (6, 4, 2)}_2$|class=fit]]
 #The upper table shows again the&nbsp; "distance spectrum"&nbsp; $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$&nbsp; of the&nbsp; $\text{RSC (3, 2, 2)}_4$.

@@ Line 74: / Line 74: @@
 {{GraueBox|TEXT=
 $\text{Example 1:}$&nbsp; We consider the following code parameters:
-[[File:P ID2517 KC T 2 3 S1b v3.png|right|frame|$\rm GF(2^2)$&nbsp; in three different representations: <br>Exponential, polynomial and coefficient form '''Korrektur'''|class=fit]]
+[[File:KC_T_2_3_S1b_neu2.png|right|frame|$\rm GF(2^2)$&nbsp; in three different representations: <br>Exponential, polynomial and coefficient form|class=fit]]
 ::<math>k = 2 \hspace{0.05cm},\hspace{0.15cm}n = 3 \hspace{0.5cm} \Rightarrow  \hspace{0.5cm} \underline {u} = (u_0,\ u_1)\hspace{0.05cm},\hspace{0.15cm}
 \underline {c} = (c_0,\ c_1,\ c_2)\hspace{0.05cm},</math>
@@ Line 285: / Line 285: @@
 *The distance&nbsp; $d$&nbsp; between&nbsp; $\underline {c}_j$&nbsp; and the null word&nbsp; $\underline {c}_0$&nbsp; is identical to the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|"Hamming weight"]]&nbsp; $w_{\rm H}(\underline {c}_j)$.<br>
-[[File:P ID2520 KC T 2 3 S3 v1.png|right|frame|To derive the distance spectrum for the&nbsp; $\text{RSC (3, 2, 2)}_4$|class=fit]]
+[[File:EN_KC_T_2_3_S3.png|right|frame|To derive the distance spectrum for the&nbsp; $\text{RSC (3, 2, 2)}_4$|class=fit]]
 <br>
 &rArr; &nbsp; From the upper table can be read among others:
@@ Line 334: / Line 334: @@
 {{GraueBox|TEXT=
 $\text{Example 6:}$&nbsp; As in&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Singleton_bound_and_minimum_distance|$\text{Example 5}$]]&nbsp; $($previous page$)$,&nbsp; we consider the&nbsp; $\text{RSC (3, 2, 2)}_4$.&nbsp;
-[[File:P ID2521 KC T 2 3 S3 v1.png|right|frame|Derivation of the distance spectra of&nbsp; $\text{RSC (3, 2, 2)}_4$&nbsp; and &nbsp;$\text{RSC (6, 4, 2)}_2$|class=fit]]
+[[File:EN_KC_T_2_3_S3.png|right|frame|Derivation of the distance spectra of&nbsp; $\text{RSC (3, 2, 2)}_4$&nbsp; and &nbsp;$\text{RSC (6, 4, 2)}_2$|class=fit]]
 #The upper table shows again the&nbsp; "distance spectrum"&nbsp; $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$&nbsp; of the&nbsp; $\text{RSC (3, 2, 2)}_4$.
 #The lower table applies to the equivalent binary Reed&ndash;Solomon code&nbsp; $\text{RSC (6, 4, 2)}_2$.

@@ Line 140: / Line 140: @@
 \hspace{0.05cm}.</math>
-As with any linear&nbsp; $(n, k)$&nbsp; block code,&nbsp; the generator matrix consists of&nbsp;
+As with any linear&nbsp; $(n,\ k)$&nbsp; block code,&nbsp; the generator matrix consists of&nbsp;
 *$k$&nbsp; rows and&nbsp;
 *$n$&nbsp; columns.

@@ Line 161: / Line 161: @@
 	{\rm GF}(2^2) \hspace{-0.01cm}\setminus \hspace{-0.01cm} \{0\} = \big \{\hspace{0.05cm} \alpha^{0}  \hspace{0.05cm},\hspace{0.1cm}
 \alpha^{1}\hspace{0.05cm},\hspace{0.1cm} \alpha^{2} \hspace{0.05cm}\big \}\hspace{0.05cm}. </math>
 Beside this: