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

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2026-03-16T13:46:42Z

Join split $$ formula lines for correct rendering

← Older revision		Revision as of 15:46, 16 March 2026
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	⇒   This results in the following assignments on symbol or bit level:		⇒   This results in the following assignments on symbol or bit level:

	:$$\underline {u} =(\alpha^1, \ \alpha^2)\hspace{0.57cm}\leftrightarrow\hspace{0.3cm}		:$$\underline {u} =(\alpha^1, \ \alpha^2)\hspace{0.57cm}\leftrightarrow\hspace{0.3cm}\underline {c} = (1, \ \alpha^2, \ 0)\hspace{0.05cm},$$
	\underline {c} = (1, \ \alpha^2, \ 0)\hspace{0.05cm},$$		:$$\underline {u}_{\rm bin} =(1,\ 0,\ 1,\ 1)\hspace{0.3cm}\leftrightarrow\hspace{0.3cm}\underline {c}_{\rm bin} = (0,\ 1,\ 1,\ 1,\ 0,\ 0)\hspace{0.05cm}.$$}}<br>
	:$$\underline {u}_{\rm bin} =(1,\ 0,\ 1,\ 1)\hspace{0.3cm}\leftrightarrow\hspace{0.3cm}
	\underline {c}_{\rm bin} = (0,\ 1,\ 1,\ 1,\ 0,\ 0)\hspace{0.05cm}.$$}}<br>

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
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	:$$g_{12} = u_{2}(\alpha^{2}) = \alpha^{2}\hspace{0.05cm}.$$		:$$g_{12} = u_{2}(\alpha^{2}) = \alpha^{2}\hspace{0.05cm}.$$
	* So the complete generator matrix is:		* So the complete generator matrix is:
	:$$ { \boldsymbol{\rm G} } =		:$$ { \boldsymbol{\rm G} } =\begin{pmatrix}1 & 1 & 1\\1 & \alpha & \alpha^2\end{pmatrix}\hspace{0.05cm}.$$
	\begin{pmatrix}
	1 & 1 & 1\\
	1 & \alpha & \alpha^2
	\end{pmatrix}\hspace{0.05cm}.$$
	[[File:P ID2519 KC T 2 3 S2a v1.png\|right\|frame\|Code table of the  $\text{RSC (3, 2, 2)}_4$  at symbol level\|class=fit]]		[[File:P ID2519 KC T 2 3 S2a v1.png\|right\|frame\|Code table of the  $\text{RSC (3, 2, 2)}_4$  at symbol level\|class=fit]]

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	:$$\Rightarrow\hspace{0.3cm}W(X) = 1 + 9 \cdot X^2 + 6 \cdot X^3\hspace{0.05cm}.$$		:$$\Rightarrow\hspace{0.3cm}W(X) = 1 + 9 \cdot X^2 + 6 \cdot X^3\hspace{0.05cm}.$$

	:$$\text{RSC (6, 4, 2)}_2\text{:} \hspace{0.3cm} W_0 = 1\hspace{0.05cm}, \hspace{0.2cm}W_2 = 3\hspace{0.05cm}, \hspace{0.2cm}W_3 = 8		:$$\text{RSC (6, 4, 2)}_2\text{:} \hspace{0.3cm} W_0 = 1\hspace{0.05cm}, \hspace{0.2cm}W_2 = 3\hspace{0.05cm}, \hspace{0.2cm}W_3 = 8\hspace{0.05cm}, \hspace{0.2cm}W_4 = 3\hspace{0.05cm}, \hspace{0.2cm}W_6 = 1.$$
	\hspace{0.05cm}, \hspace{0.2cm}W_4 = 3\hspace{0.05cm}, \hspace{0.2cm}W_6 = 1.$$
	:$$\Rightarrow\hspace{0.3cm}W(X) = 1 + 3 \cdot X^2 + 8 \cdot X^3 + 4 \cdot X^4 + X^6 \hspace{0.05cm}.$$}}		:$$\Rightarrow\hspace{0.3cm}W(X) = 1 + 3 \cdot X^2 + 8 \cdot X^3 + 4 \cdot X^4 + X^6 \hspace{0.05cm}.$$}}

Maintenance script: Add German interlanguage link

2026-03-16T12:27:34Z

Add German interlanguage link

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	{{Display}}		{{Display}}
			[[de:Kanalcodierung/Definition_und_Eigenschaften_von_Reed–Solomon–Codes]]

Hwang at 19:46, 23 November 2022

2022-11-23T19:46:02Z

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

Hwang at 19:32, 23 November 2022

2022-11-23T19:32:46Z

← Older revision		Revision as of 21:32, 23 November 2022
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	<br>		<br>
	An important parameter of any block code is the  [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding\|$\text{minimum distance}$]]  between any two code words  $\underline {c}_i$  and  $\underline {c}_j$.		An important parameter of any block code is the  [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding\|$\text{minimum distance}$]]  between any two code words  $\underline {c}_i$  and  $\underline {c}_j$.
	*The Reed–Solomon–codes belong to the class of  '''linear  and  cyclic codes'''.  For these,  one can start from the zero word  $\underline {c}_0 = (0, 0, \hspace{0.05cm} \text{...} \hspace{0.05cm}, 0)$  as the reference.		*The Reed–Solomon–codes belong to the class of  »'''linear  and  cyclic codes'''«.  For these,  one can start from the zero word  $\underline {c}_0 = (0, 0, \hspace{0.05cm} \text{...} \hspace{0.05cm}, 0)$  as the reference.

	*From the number of zeros in the other code words  $\underline {c}_j ≠ \underline {c}_0$  the  [[Channel_Coding/Bounds_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code\|$\text{distance spectrum}$]]  $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$  can be specified.<br>		*From the number of zeros in the other code words  $\underline {c}_j ≠ \underline {c}_0$  the  [[Channel_Coding/Bounds_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code\|$\text{distance spectrum}$]]  $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$  can be specified.<br>

Hwang at 16:22, 23 November 2022

2022-11-23T16:22:12Z

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Hwang at 23:28, 12 November 2022

2022-11-12T23:28:48Z

Guenter at 14:32, 8 October 2022

2022-10-08T14:32:20Z

Guenter at 16:48, 5 October 2022

2022-10-05T16:48:58Z

← Older revision		Revision as of 18:48, 5 October 2022
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	\hspace{0.05cm}.</math>		\hspace{0.05cm}.</math>

	As with any linear  $(n, k)$  block code,  the generator matrix consists of		As with any linear  $(n,\ k)$  block code,  the generator matrix consists of
	*$k$  rows and		*$k$  rows and
	*$n$  columns.		*$n$  columns.

Guenter at 16:20, 5 October 2022

2022-10-05T16:20:06Z

← Older revision		Revision as of 18:20, 5 October 2022
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$  An   $(n, k)\text{ Reed–Solomon code}$  for the Galois field  ${\rm GF}(2^m)$  is defined by		$\text{Definition:}$  An   $(n,\ k)\text{ Reed–Solomon code}$  for the Galois field  ${\rm GF}(2^m)$  is defined by
	*the  $n= 2^{m-1}$  elements of  ${\rm GF}(2^m) \hspace{-0.05cm}\setminus \hspace{-0.05cm} \{0\} =\{ \alpha^0, \alpha^1, \, \text{...} \, , \alpha^{n-1}\}$,  where  $\alpha$  denotes a  [[Channel_Coding/Extension_Field#Binary_extension_fields_.E2.80.93_Primitive_polynomials\|"primitive element"]]  of  ${\rm GF}(2^m)$,		*the  $n= 2^{m-1}$  elements of  ${\rm GF}(2^m) \hspace{-0.05cm}\setminus \hspace{-0.05cm} \{0\} =\{ \alpha^0, \alpha^1, \, \text{...} \, , \alpha^{n-1}\}$,  where  $\alpha$  denotes a  [[Channel_Coding/Extension_Field#Binary_extension_fields_.E2.80.93_Primitive_polynomials\|"primitive element"]]  of  ${\rm GF}(2^m)$,

Guenter at 15:00, 5 October 2022

2022-10-05T15:00:17Z

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


	Beside this:		Beside this:

← Older revision		Revision as of 01:28, 13 November 2022
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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 3:}$  We consider as in  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Coding_principle_and_code_parameters\|$\text{Example 2}$]]  (see previous ~~page~~)  the  $\text{RSC (3, 2, 2)}_4$  whose generator matrix has the following form:		$\text{Example 3:}$  We consider as in  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Coding_principle_and_code_parameters\|$\text{Example 2}$]]  (see previous section)  the  $\text{RSC (3, 2, 2)}_4$  whose generator matrix has the following form:

	::<math> { \boldsymbol{\rm G} } =		::<math> { \boldsymbol{\rm G} } =
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	#Of course,  with the same generator matrix you get exactly the same code table  $\underline {u}   ↔   \underline {c}$  as after the calculation via the function  $u(x)$.		#Of course,  with the same generator matrix you get exactly the same code table  $\underline {u}   ↔   \underline {c}$  as after the calculation via the function  $u(x)$.
	#The corresponding code table on bit level  $($see  $\text{Example 2}$  on the previous ~~page~~$)$  results again,  if the elements are not given in exponent representation,  but in coefficient form:		#The corresponding code table on bit level  $($see  $\text{Example 2}$  in the previous section$)$  results again,  if the elements are not given in exponent representation,  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)		:::<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)
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	== Generator matrix and parity-check matrix ==		== Generator matrix and parity-check matrix ==
	<br>		<br>
	We now generalize the result of the last ~~page~~ for a Reed–Solomon code with		We now generalize the result of the last section for a Reed–Solomon code with
	*dimension  $k$  $($number of symbols per information block$)$,<br>		*dimension  $k$  $($number of symbols per information block$)$,<br>

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 6:}$  As in  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Singleton_bound_and_minimum_distance\|$\text{Example 5}$]]  $($previous ~~page~~$)$,  we consider the  $\text{RSC (3, 2, 2)}_4$.		$\text{Example 6:}$  As in  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Singleton_bound_and_minimum_distance\|$\text{Example 5}$]]  $($previous section$)$,  we consider the  $\text{RSC (3, 2, 2)}_4$.
	[[File:EN_KC_T_2_3_S3.png\|right\|frame\|Derivation of the distance spectra of  $\text{RSC (3, 2, 2)}_4$  and  $\text{RSC (6, 4, 2)}_2$\|class=fit]]		[[File:EN_KC_T_2_3_S3.png\|right\|frame\|Derivation of the distance spectra of  $\text{RSC (3, 2, 2)}_4$  and  $\text{RSC (6, 4, 2)}_2$\|class=fit]]
	#The upper table shows again the  "distance spectrum"  $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$  of the  $\text{RSC (3, 2, 2)}_4$.		#The upper table shows again the  "distance spectrum"  $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$  of the  $\text{RSC (3, 2, 2)}_4$.

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	$\text{Example 1:}$  We consider the following code parameters:		$\text{Example 1:}$  We consider the following code parameters:
	[[File:~~P ID2517 KC T 2 3 S1b v3~~.png\|right\|frame\|$\rm GF(2^2)$  in three different representations: <br>Exponential, polynomial and coefficient form ~~'''Korrektur'''~~\|class=fit]]		[[File:KC_T_2_3_S1b_neu2.png\|right\|frame\|$\rm GF(2^2)$  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}		::<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>		\underline {c} = (c_0,\ c_1,\ c_2)\hspace{0.05cm},</math>
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	*The distance  $d$  between  $\underline {c}_j$  and the null word  $\underline {c}_0$  is identical to the  [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding\|"Hamming weight"]]  $w_{\rm H}(\underline {c}_j)$.<br>		*The distance  $d$  between  $\underline {c}_j$  and the null word  $\underline {c}_0$  is identical to the  [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding\|"Hamming weight"]]  $w_{\rm H}(\underline {c}_j)$.<br>

	[[File:~~P ID2520 KC T 2 3 S3 v1~~.png\|right\|frame\|To derive the distance spectrum for the  $\text{RSC (3, 2, 2)}_4$\|class=fit]]		[[File:EN_KC_T_2_3_S3.png\|right\|frame\|To derive the distance spectrum for the  $\text{RSC (3, 2, 2)}_4$\|class=fit]]
	<br>		<br>
	⇒   From the upper table can be read among others:		⇒   From the upper table can be read among others:
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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 6:}$  As in  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Singleton_bound_and_minimum_distance\|$\text{Example 5}$]]  $($previous page$)$,  we consider the  $\text{RSC (3, 2, 2)}_4$.		$\text{Example 6:}$  As in  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Singleton_bound_and_minimum_distance\|$\text{Example 5}$]]  $($previous page$)$,  we consider the  $\text{RSC (3, 2, 2)}_4$.
	[[File:~~P ID2521 KC T 2 3 S3 v1~~.png\|right\|frame\|Derivation of the distance spectra of  $\text{RSC (3, 2, 2)}_4$  and  $\text{RSC (6, 4, 2)}_2$\|class=fit]]		[[File:EN_KC_T_2_3_S3.png\|right\|frame\|Derivation of the distance spectra of  $\text{RSC (3, 2, 2)}_4$  and  $\text{RSC (6, 4, 2)}_2$\|class=fit]]
	#The upper table shows again the  "distance spectrum"  $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$  of the  $\text{RSC (3, 2, 2)}_4$.		#The upper table shows again the  "distance spectrum"  $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$  of the  $\text{RSC (3, 2, 2)}_4$.
	#The lower table applies to the equivalent binary Reed–Solomon code  $\text{RSC (6, 4, 2)}_2$.		#The lower table applies to the equivalent binary Reed–Solomon code  $\text{RSC (6, 4, 2)}_2$.