Channel Coding/The Basics of Product Codes - Revision history

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	⇒   The  »<b>row-by-row syndrome decoding</b>«  is done via the syndrome  $\underline{s} = \underline{y} \cdot \mathbf{H}_1^{\rm T}$  with		⇒   The  »<b>row-by-row syndrome decoding</b>«  is done via the syndrome  $\underline{s} = \underline{y} \cdot \mathbf{H}_1^{\rm T}$  with
	:$$\boldsymbol{\rm H}_1^{\rm T} =		:$$\boldsymbol{\rm H}_1^{\rm T} =\begin{pmatrix}1 &0 &1 \\1 &1 &0 \\0 &1 &1 \\1 &1 &1 \\1 &0 &0 \\0 &1 &0 \\0 &0 &1\end{pmatrix} \hspace{0.05cm}. $$
	\begin{pmatrix}
	1 &0 &1 \\
	1 &1 &0 \\
	0 &1 &1 \\
	1 &1 &1 \\
	1 &0 &0 \\
	0 &1 &0 \\
	0 &0 &1
	\end{pmatrix} \hspace{0.05cm}. $$

	In particular:		In particular:

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	{{Display}}		{{Display}}
			[[de:Kanalcodierung/Grundlegendes_zu_den_Produktcodes]]

Guenter at 16:26, 16 January 2023

2023-01-16T16:26:38Z

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	[[File:~~EN_KC_T_4_2_S3_v1~~.png\|right\|frame\|Bit and block error probability of a  $(1024, 676)$  product code at AWGN ~~'''KORREKTUR: exakt'''~~\|class=fit]]		[[File:EN_KC_T_4_2_S3_v3.png\|right\|frame\|Bit and block error probability of a  $(1024, 676)$  product code at AWGN \|class=fit]]
	The graph shows for a  $(1024, 676)$  product code, based on the  extended Hamming code  ${\rm eHC} \ (32, 26)$  as component codes,		The graph shows for a  $(1024, 676)$  product code, based on the  extended Hamming code  ${\rm eHC} \ (32, 26)$  as component codes,
	*on the left,  the bit error probability as function of the AWGN parameter  $10 \cdot {\rm lg} \, (E_{\rm B}/N_0)$  the number of iterations  $(I)$,		*on the left,  the bit error probability as function of the AWGN parameter  $10 \cdot {\rm lg} \, (E_{\rm B}/N_0)$  the number of iterations  $(I)$,

Hwang at 21:34, 1 January 2023

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	== Exercises for the chapter==		== Exercises for the chapter==
	<br>		<br>
	[[Aufgaben:Exercise_4.6:_Product_Code_Generation\|"Exercise 4.6: Product Code Generation"]]		[[Aufgaben:Exercise_4.6:_Product_Code_Generation\|Exercise 4.6: Product Code Generation]]

	[[Aufgaben:Exercise_4.6Z:_Basics_of_Product_Codes\|"Exercise 4.6Z: Basics of Product Codes"]]		[[Aufgaben:Exercise_4.6Z:_Basics_of_Product_Codes\|Exercise 4.6Z: Basics of Product Codes]]

	[[Aufgaben:Exercise_4.7:_Product_Code_Decoding\|"Exercise 4.7: Product Code Decoding"]]		[[Aufgaben:Exercise_4.7:_Product_Code_Decoding\|Exercise 4.7: Product Code Decoding]]

	[[Aufgaben:Exercise_4.7Z:_Principle_of_Syndrome_Decoding\|"Exercise 4.7Z: Principle of Syndrome Decoding"]]		[[Aufgaben:Exercise_4.7Z:_Principle_of_Syndrome_Decoding\|Exercise 4.7Z: Principle of Syndrome Decoding]]

	==References==		==References==

Hwang at 21:33, 1 January 2023

2023-01-01T21:33:13Z

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	[[File:EN_KC_T_4_2_S3_v1.png\|right\|frame\|Bit and block error probability of a  $(1024, 676)$  product code at AWGN\|class=fit]]		[[File:EN_KC_T_4_2_S3_v1.png\|right\|frame\|Bit and block error probability of a  $(1024, 676)$  product code at AWGN '''KORREKTUR: exakt'''\|class=fit]]
	The graph shows for a  $(1024, 676)$  product code, based on the  extended Hamming code  ${\rm eHC} \ (32, 26)$  as component codes,		The graph shows for a  $(1024, 676)$  product code, based on the  extended Hamming code  ${\rm eHC} \ (32, 26)$  as component codes,
	*on the left,  the bit error probability as function of the AWGN parameter  $10 \cdot {\rm lg} \, (E_{\rm B}/N_0)$  the number of iterations  $(I)$,		*on the left,  the bit error probability as function of the AWGN parameter  $10 \cdot {\rm lg} \, (E_{\rm B}/N_0)$  the number of iterations  $(I)$,

Hwang at 21:25, 1 January 2023

2023-01-01T21:25:51Z

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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusions:}$  '''All product codes'''  according to the above graph have the following properties:		$\text{Conclusions:}$  »'''All product codes'''«  according to the above graph have the following properties:
	*For linear component codes  $\mathcal{C}_1$  and  $\mathcal{C}_2$  the product code  $\mathcal{C} = \mathcal{C}_1 × \mathcal{C}_2$  is also linear.<br>		*For linear component codes  $\mathcal{C}_1$  and  $\mathcal{C}_2$  the product code  $\mathcal{C} = \mathcal{C}_1 × \mathcal{C}_2$  is also linear.<br>

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	⇒   The  <b>row-by-row syndrome decoding</b>  is done via the syndrome  $\underline{s} = \underline{y} \cdot \mathbf{H}_1^{\rm T}$  with		⇒   The  »<b>row-by-row syndrome decoding</b>«  is done via the syndrome  $\underline{s} = \underline{y} \cdot \mathbf{H}_1^{\rm T}$  with
	:$$\boldsymbol{\rm H}_1^{\rm T} =		:$$\boldsymbol{\rm H}_1^{\rm T} =
	\begin{pmatrix}		\begin{pmatrix}
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	\hspace{0.05cm}.</math>		\hspace{0.05cm}.</math>

	⇒   The  <b>column-by-column syndrome decoding</b>  removes all single errors in columns  1,  2,  3,  4  and  7.		⇒   The  »<b>column-by-column syndrome decoding</b>«  removes all single errors in columns  1,  2,  3,  4  and  7.
	*<b>Column 5</b>  $($contains two errors$)$   ⇒   Error correction concerning bit  '''4''':		*<b>Column 5</b>  $($contains two errors$)$   ⇒   Error correction concerning bit  '''4''':

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	\hspace{0.05cm}.</math>		\hspace{0.05cm}.</math>

	⇒   The remaining three errors are corrected by decoding the  <b>second row iteration loop</b>  $($line-by-line$)$.<br>		⇒   The remaining three errors are corrected by decoding the  »<b>second row iteration loop</b>«  $($line-by-line$)$.<br>

	Whether all errors of a block are correctable depends on the error pattern.  Here we refer to  [[Aufgaben:Aufgabe_4.7:_Decodierung_von_Produktcodes\|"Exercise 4.7"]].}}<br>		Whether all errors of a block are correctable depends on the error pattern.  Here we refer to  [[Aufgaben:Aufgabe_4.7:_Decodierung_von_Produktcodes\|"Exercise 4.7"]].}}<br>

Hwang at 21:12, 1 January 2023

2023-01-01T21:12:56Z

Guenter at 13:51, 20 December 2022

2022-12-20T13:51:26Z

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	*the code matrix  $\mathbf{X}$  was chosen to be a  $6 × 7$ zero matrix,		*the code matrix  $\mathbf{X}$  was chosen to be a  $6 × 7$ zero matrix,

	*so ~~that~~ the eight  "ones"  in  $\mathbf{Y}$  represent transmission errors<br>		*so the eight  "ones"  in  $\mathbf{Y}$  represent transmission errors.<br>

Guenter at 13:49, 20 December 2022

2022-12-20T13:49:38Z

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	*the code matrix  $\mathbf{X}$  was chosen to be a  $6 × 7$ zero matrix,		*the code matrix  $\mathbf{X}$  was chosen to be a  $6 × 7$ zero matrix,

	*so that the ~~nine~~  "ones"  in  $\mathbf{Y}$  represent transmission errors<br>		*so that the eight  "ones"  in  $\mathbf{Y}$  represent transmission errors<br>

Guenter at 16:50, 19 December 2022

2022-12-19T16:50:51Z

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	[[File:EN_KC_T_4_2_S2a_v1.png\|right\|frame\|Syndrome decoding of the  $(42, 12)$  product code\|class=fit]]		[[File:EN_KC_T_4_2_S2a_v1.png\|right\|frame\|Syndrome decoding of the  $(42, 12)$  product code\|class=fit]]
	[[File:~~EN_KC_T_4_2_S2b_v1~~.png\|right\|frame\|Syndrome table for code $\mathcal{C}_1$]]		[[File:EN KC T 4 2 S2b v2 neu.png\|right\|frame\|Syndrome table for code $\mathcal{C}_1$]]
	[[File:EN_KC_T_4_2_S2c_v2.png\|right\|frame\|Syndrome table for the code $\mathcal{C}_2$]]		[[File:EN_KC_T_4_2_S2c_v2.png\|right\|frame\|Syndrome table for the code $\mathcal{C}_2$]]

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	[[File:EN_KC_T_4_2_S2a_v1.png\|right\|frame\|Syndrome decoding of the  $(42, 12)$  product code\|class=fit]]		[[File:EN_KC_T_4_2_S2a_v1.png\|right\|frame\|Syndrome decoding of the  $(42, 12)$  product code\|class=fit]]
	[[File:~~EN_KC_T_4_2_S2b_v1~~.png\|right\|frame\|Syndrome table for code $\mathcal{C}_1$]]		[[File:EN KC T 4 2 S2b v2 neu.png\|right\|frame\|Syndrome table for code $\mathcal{C}_1$]]
	[[File:EN_KC_T_4_2_S2c_v2.png\|right\|frame\|Syndrome table for the code $\mathcal{C}_2$]]		[[File:EN_KC_T_4_2_S2c_v2.png\|right\|frame\|Syndrome table for the code $\mathcal{C}_2$]]

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	<br>		<br>
	The graphic shows the principle structure of  »product codes«,  which were already introduced in 1954 by  [https://en.wikipedia.org/wiki/Peter_Elias $\text{Peter Elias}$].		The graphic shows the principle structure of  »product codes«,  which were already introduced in 1954 by  [https://en.wikipedia.org/wiki/Peter_Elias $\text{Peter Elias}$].
	*The  '''two-dimensional product code'''  $\mathcal{C} = \mathcal{C}_1 × \mathcal{C}_2$  shown here is based on the two linear and binary block codes with parameters  $(n_1, \ k_1)$  and  $(n_2, \ k_2)$ respectively.		*The  »'''two-dimensional product code'''«  $\mathcal{C} = \mathcal{C}_1 × \mathcal{C}_2$  shown here is based on the two linear and binary block codes with parameters  $(n_1, \ k_1)$  and  $(n_2, \ k_2)$ respectively.

	*The code word length is  $n = n_1 \cdot n_2$.		*The code word length is  $n = n_1 \cdot n_2$.
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	*The upper right matrix  $\mathbf{P}^{(1)}$  with dimension  $k_2 × m_1$  contains the parity bits with respect to the code  $\mathcal{C}_1$.		*The upper right matrix  $\mathbf{P}^{(1)}$  with dimension  $k_2 × m_1$  contains the parity bits with respect to the code  $\mathcal{C}_1$.

	*In each of the  $k_2$  rows,  $m_1 = n_1 - k_1$  check bits are added to the  $k_1$  information bits as described in an earlier chapter using the example of  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|"Hamming codes"]] .<br>		*In each of the  $k_2$  rows,  $m_1 = n_1 - k_1$  check bits are added to the  $k_1$  information bits as described in an earlier chapter using the example of  [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes\|$\text{Hamming codes}$]] .<br>

	*The lower left matrix  $\mathbf{P}^{(2)}$  of dimension  $m_2 × k_1$  contains the check bits for the second component code  $\mathcal{C}_2$.  Here the encoding  $($and also the decoding$)$  is done line by line:   In each of the  $k_1$  columns,  the  $k_2$  information bits are still supplemented by  $m_2 = n_2 -k_2$  check bits.<br>		*The lower left matrix  $\mathbf{P}^{(2)}$  of dimension  $m_2 × k_1$  contains the check bits for the second component code  $\mathcal{C}_2$.  Here the encoding  $($and also the decoding$)$  is done line by line:   In each of the  $k_1$  columns,  the  $k_2$  information bits are still supplemented by  $m_2 = n_2 -k_2$  check bits.<br>
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	⇒   The remaining three errors are corrected by decoding the  <b>second row iteration loop</b>  $($line-by-line$)$.<br>		⇒   The remaining three errors are corrected by decoding the  <b>second row iteration loop</b>  $($line-by-line$)$.<br>

	Whether all errors of a block are correctable depends on the error pattern.  Here we refer to  [[Aufgaben:Aufgabe_4.7:_Decodierung_von_Produktcodes\|~~$\text{~~Exercise 4.7}$]].}}<br>		Whether all errors of a block are correctable depends on the error pattern.  Here we refer to  [[Aufgaben:Aufgabe_4.7:_Decodierung_von_Produktcodes\|"Exercise 4.7"]].}}<br>

	== Performance of product codes ==		== Performance of product codes ==
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	Usually one uses very long product codes   $($large  $n = n_1 \cdot n_2)$   with the following consequence:		Usually one uses very long product codes   $($large  $n = n_1 \cdot n_2)$   with the following consequence:
	*For effort reasons,  the  [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB\|"maximum likelihood decoding at block level"]]  is not applicable for the component codes  $\mathcal{C}_1$  and  $\mathcal{C}_2$  nor the  [[Channel_Coding/Decoding_of_Linear_Block_Codes#Principle_of_syndrome_decoding\| "syndrome decoding"]],  which is after all a realization form of maximum likelihood decoding.		*For effort reasons,  the  [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB\|$\text{maximum likelihood decoding at block level}$]]  is not applicable for the component codes  $\mathcal{C}_1$  and  $\mathcal{C}_2$  nor the  [[Channel_Coding/Decoding_of_Linear_Block_Codes#Principle_of_syndrome_decoding\|$\text{syndrome decoding}$]],  which is after all a realization form of maximum likelihood decoding.

	*Applicable,  on the other hand,  even with large  $n$  is the  [[Channel_Coding/Soft-in_Soft-Out_Decoder#Bit-wise_soft-in_soft-out_decoding\|"iterative symbol-wise MAP decoding"]].  The exchange of extrinsic and a-priori–information happens here between the two component codes.  More details on this can be found in  [Liv15]<ref name='Liv15'>Liva, G.:  Channels Codes for Iterative Decoding.  Lecture manuscript, Department of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2015.</ref>.<br>		*Applicable,  on the other hand,  even with large  $n$  is the  [[Channel_Coding/Soft-in_Soft-Out_Decoder#Bit-wise_soft-in_soft-out_decoding\|$\text{iterative symbol-wise MAP decoding}$]].  The exchange of extrinsic and a-priori–information happens here between the two component codes.  More details on this can be found in  [Liv15]<ref name='Liv15'>Liva, G.:  Channels Codes for Iterative Decoding.  Lecture manuscript, Department of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2015.</ref>.<br>