Channel Coding/Decoding of Linear Block Codes

1. Channel Coding
2. Binary Block Codes for Channel Coding
3. Decoding of Linear Block Codes

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Block diagram and requirements

We start from the block diagram already shown in the chapter  $\text{Channel Models and Decision Structures}$  where the digital channel model used is mostly the  $\text{Binary Symmetric Channel}$ $\rm (BSC)$. 

For code word estimation,  we use the  "Maximum Likelihood Decision"  $\rm (ML)$,  which for binary codes   ⇒   $\underline{x} \in {\rm GF}(2^n)$  at the block level gives the same result as the  $\text{MAP Receiver}$.

Block diagram for decoding block codes

The task of the channel decoder can be described as follows:

• The vector  $\underline{v}$  after decoding  (at the sink)  should match the information word  $\underline{u}$  as well as possible.

• That is:   The  »block error probability«  should be as small as possible:

\[{ \rm Pr(block\:error)} = { \rm Pr}( \underline{v} \ne \underline{u}) \stackrel{!}{=} { \rm minimum}\hspace{0.05cm}.\]

• Because of assignments  $\underline{x} = {\rm enc}(\underline{u})$  resp.  $\underline{v} = {\rm enc}^{-1}(\underline{z})$  also holds:

\[{ \rm Pr(block\:error)} = { \rm Pr}( \underline{z} \ne \underline{x}) \stackrel{!}{=} { \rm minimum}\hspace{0.05cm}.\]

• Sought is the most likely sent code word  $\underline{y} = \underline{x} +\underline{e}$  for the given received word  $\underline{x}_i$,  which is passed on as result  $\underline{z}$:

\[\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} {\rm Pr}( \underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm}|\hspace{0.05cm} \underline{y} ) \hspace{0.05cm}.\]

• For the BSC model,  both   $\underline{x}_i \in {\rm GF}(2^n)$   and   $\underline{y} \in {\rm GF}(2^n)$,  so the maximum likelihood decision rule can also be written using the  $\text{Hamming distance}$  $d_{\rm H}( \underline{y}, \, \underline{x}_i)$:

\[\underline{z} = {\rm arg} \min_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} d_{\rm H}(\underline{y} \hspace{0.05cm}, \hspace{0.1cm}\underline{x}_{\hspace{0.03cm}i})\hspace{0.05cm}.\]

Principle of syndrome decoding

Assumed here is a  $(n, \, k)$  block code with the parity-check matrix  $\boldsymbol{\rm H}$  and the systematic code words

\[\underline{x}\hspace{0.05cm} = (x_1, x_2, \hspace{0.05cm}\text{...} \hspace{0.05cm}, x_i, \hspace{0.05cm}\text{...} \hspace{0.05cm}, x_n) = (u_1, u_2, \hspace{0.05cm}\text{...} \hspace{0.05cm}, u_k, p_1, \hspace{0.05cm}\text{...} \hspace{0.05cm}, p_{n-k})\hspace{0.05cm}. \]

With the error vector  $\underline{e}$  then applies to the received word:

\[\underline{y} = \underline{x} + \underline{e} \hspace{0.05cm}, \hspace{0.4cm} \underline{y} \in \hspace{0.1cm} {\rm GF}(2^n) \hspace{0.05cm}, \hspace{0.4cm} \underline{x} \in \hspace{0.1cm} {\rm GF}(2^n) \hspace{0.05cm}, \hspace{0.4cm} \underline{e} \in \hspace{0.1cm} {\rm GF}(2^n) \hspace{0.05cm}.\]

A bit error at position  $i$   ⇒   $y_i ≠ x_i$  is expressed by the  »error coefficient«  $e_i = 1$.

$\text{Definition:}$  The  »syndrome«  $\underline{s} = (s_0, s_1, \hspace{0.05cm}\text{...} \hspace{0.05cm}, s_{m-1})$  is calculated  (as row resp. column vector)  from the received word  $\underline{y}$  and the parity-check matrix  $\boldsymbol{\rm H}$  as follows:

\[\underline{s} = \underline{y} \cdot { \boldsymbol{\rm H} }^{\rm T}\hspace{0.3cm}{\rm bzw.}\hspace{0.3cm} \underline{s}^{\rm T} = { \boldsymbol{\rm H} } \cdot \underline{y}^{\rm T}\hspace{0.05cm}.\]

• The vector length of  $\underline{s}$  is equal to  $m = n-k$  $($row number of  $\boldsymbol{\rm H})$.

The syndrome  $\underline{s}$  shows the following characteristics:

• Because of the equation  $\underline{x} \cdot { \boldsymbol{\rm H}}^{\rm T} = \underline{0}$   the syndrome  $\underline{s}$  does not depend on the code word  $\underline{x}$  but solely on the error vector  $\underline{e}$:

\[\underline{s} = \underline{y} \cdot { \boldsymbol{\rm H}}^{\rm T} = \hspace{0.05cm} \underline{x} \cdot { \boldsymbol{\rm H}}^{\rm T} + \hspace{0.05cm} \underline{e} \cdot { \boldsymbol{\rm H}}^{\rm T} = \hspace{0.05cm} \underline{e} \cdot { \boldsymbol{\rm H}}^{\rm T} \hspace{0.05cm}.\]

• For sufficiently few bit errors,  $\underline{s}$  provides a clear indication of the error locations,  allowing full error correction.
  
  

$\text{Example 1:}$  Starting from the systematic  $\text{(7, 4, 3)}$ Hamming code,  the following result is obtained for the received vector  $\underline{y} = (0, 1, 1, 1, 0, 0, 1)$:

\[{ \boldsymbol{\rm H} } \cdot \underline{y}^{\rm T} = \begin{pmatrix} 1 &1 &1 &0 &1 &0 &0\\ 0 &1 &1 &1 &0 &1 &0\\ 1 &1 &0 &1 &0 &0 &1 \end{pmatrix} \cdot \begin{pmatrix} 0 \\ 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \underline{s}^{\rm T} \hspace{0.05cm}.\]

Comparing the syndrome with the  $\text{parity-check equations}$  of the Hamming code,  we see that

• most likely the fourth symbol  $(x_4 = u_4)$  of the code word has been falsified,
  

• the code word estimator will thus yield the result  $\underline{z} = (0, 1, 1, 0, 0, 0, 1)$,
  

• the decision is correct only if only one bit was falsified during transmission.
  
  

Below are the required corrections for the  $\text{(7, 4, 3)}$  Hamming code resulting from the calculated syndrome  $\underline{s}$  corresponding to the columns of the parity-check matrix:

\[\underline{s} = (0, 0, 0) \hspace{0.10cm} \Rightarrow\hspace{0.10cm}{\rm no\hspace{0.15cm} correction}\hspace{0.05cm};\hspace{0.8cm}\underline{s} = (1, 0, 0)\hspace{0.10cm} \Rightarrow\hspace{0.10cm}{\rm invert}\hspace{0.15cm}p_1\hspace{0.05cm};\]

\[\underline{s} =(0, 0, 1)\hspace{0.10cm} \Rightarrow\hspace{0.10cm} {\rm invert}\hspace{0.15cm}p_3\hspace{0.05cm};\hspace{1.63cm}\underline{s} = (1, 0, 1)\hspace{0.10cm} \Rightarrow\hspace{0.10cm} {\rm invert}\hspace{0.15cm}u_1\hspace{0.05cm};\]

\[\underline{s} =(0, 1, 0)\hspace{0.10cm} \Rightarrow\hspace{0.10cm} {\rm invert}\hspace{0.15cm}p_2\hspace{0.05cm};\hspace{1.63cm}\underline{s} = (1, 1, 0)\hspace{0.10cm} \Rightarrow\hspace{0.10cm} {\rm invert}\hspace{0.15cm}u_3\hspace{0.05cm};\]

\[\underline{s} =(0, 1, 1)\hspace{0.10cm} \Rightarrow\hspace{0.10cm} {\rm invert}\hspace{0.15cm}u_4\hspace{0.05cm};\hspace{1.63cm}\underline{s} = (1, 1, 1)\hspace{0.10cm} \Rightarrow\hspace{0.10cm} {\rm invert}\hspace{0.15cm}u_2\hspace{0.05cm}. \]

Generalization of syndrome coding

We continue to assume the BSC channel model. This means:

• The received vector  $\underline{y}$   and the error vector  $\underline{e}$   are elements of  ${\rm GF}(2^n)$.

• The possible code words  $\underline{x}_i$  belong to the code  $\mathcal{C}$,  which spans a  $(n-k)$-dimensional subspace of  ${\rm GF}(2^n)$.

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  $\underline{y}$:

\[\underline{z} = {\rm arg} \min_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} d_{\rm H}(\underline{y} \hspace{0.05cm}, \hspace{0.1cm}\underline{x}_{\hspace{0.03cm}i})\hspace{0.05cm}.\]

• But the syndrome decoding is also the search for the most probable error vector  $\underline{e}$  that satisfies the condition  $\underline{e} \cdot { \boldsymbol{\rm H}}^{\rm T} = \underline{s}$.  The  "syndrome"  is thereby determined by the equation  

$$\underline{s} = \underline{y} \cdot { \boldsymbol{\rm H}}^{\rm T} .$$

• With the  $\text{Hamming weight}$  $w_{\rm H}(\underline{e})$  the second interpretation can also be mathematically formulated as follows:

\[\underline{z} = \underline{y} + {\rm arg} \min_{\underline{e}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} {\rm GF}(2^n)} \hspace{0.1cm} w_{\rm H}(\underline{e}_{\hspace{0.03cm}i})\hspace{0.05cm}.\]

$\text{Conclusion:}$  Note that the error vector  $\underline{e}$  as well as the received vector  $\underline{y}$  is an element of  ${\rm GF}(2^n)$  unlike the syndrome  $\underline{s} \in {\rm GF}(2^m)$  with number  $m = n-k$  of parity-check equations. This means,  that

• the association between the syndrome  $\underline{s}$  and the error vector  $\underline{e}$  is not unique,  but

• each  $2^k$  error vectors lead to the same syndrome  $\underline{s}$  which one groups together into a so-called   "coset".

$\text{Example 2:}$  The facts shall be illustrated by the example with parameters  $n = 5, \ k = 2$   ⇒   $m = n-k = 3$ .

Splitting the  $2^k$  error vectors into  "cosets"

You can see from this graph:

1. The  $2^n = 32$  possible error vectors  $\underline{e}$  are divided into  $2^m = 8$  cosets  ${\it \Psi}_0$,  ...  , ${\it \Psi}_7$.  Explicitly drawn here are only the cosets  ${\it \Psi}_0$  and  ${\it \Psi}_5$.
   
   
2. All  $2^k = 4$  error vectors of the coset  ${\it \Psi}_\mu$  lead to the syndrome  $\underline{s}_\mu$.
   
   
3. Each minor class  ${\it \Psi}_\mu$  has a  "leader"  $\underline{e}_\mu$, namely the one with the minimum Hamming weight.

$\text{Example 3:}$  Starting from the systematic  $\text{(5, 2, 3)}$  code  $\mathcal{C} = \big \{ (0, 0, 0, 0, 0) \hspace{0.05cm},\hspace{0.15cm}(0, 1, 0, 1, 1) \hspace{0.05cm},\hspace{0.15cm}(1, 0, 1, 1, 0) \hspace{0.05cm},\hspace{0.15cm}(1, 1, 1, 0, 1) \big \}$  the syndrome decoding procedure is now described in detail.

Example  $\text{(5, 2, 3)}$ syndrome table with cosets

• The generator matrix and the parity-check matrix are:

\[{ \boldsymbol{\rm G} } = \begin{pmatrix} 1 &0 &1 &1 &0 \\ 0 &1 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},\]

\[{ \boldsymbol{\rm H} } = \begin{pmatrix} 1 &0 &1 &0 &0 \\ 1 &1 &0 &1 &0 \\ 0 &1 &0 &0 &1 \end{pmatrix} \hspace{0.05cm}.\]

• The table summarizes the final result. Note:

1. The index  $\mu$  is not identical to the binary value of  $\underline{s}_\mu$.
2. The order is rather given by the number of ones in the minor class leader  $\underline{e}_\mu$.
3. For example, the syndrome  $\underline{s}_5 = (1, 1, 0)$  and the syndrome  $\underline{s}_6 = (1, 0, 1)$.
   

• To derive this table,  note:

1. The row 1 refers to the syndrome  $\underline{s}_0 = (0, 0, 0)$  and the associated cosets  ${\it \Psi}_0$. The most likely error sequence here is  $(0, 0, 0, 0, 0)$   ⇒   no bit error, which we call the  "coset leader"  $\underline{e}_0$.
2. The other entries in the first row,  namely  $(1, 0, 1, 1, 0 )$,  $(0, 1, 0, 1, 1)$  and  $(1, 1, 1, 0, 1 )$,  also each yield the syndrome  $\underline{s}_0 = (0, 0, 0)$,  but only result with at least three bit errors and are correspondingly unlikely.
   
3. In rows 2 to 6,  the respective coset leader  $\underline{e}_\mu$  contains exactly a single  "$1$"  $(\mu = 1$, ... , $5)$. Here  $\underline{e}_\mu$  is always the most likely error pattern of the class  ${\it \Psi}_\mu$. The other group members result only with at least two bit errors.
   
4. The syndrome  $\underline{s}_6 = (1, 0, 1)$  is not possible with only one bit error.  In creating the table,  we then considered all  "$5\text{ over }2 = 10$"  error patterns  $\underline{e}$  with weight  $w_{\rm H}(\underline{e}) = 2$.
5. The first found sequence with syndrome  $\underline{s}_6 = (1, 0, 1)$  was chosen as coset leader  $\underline{e}_6 = (1, 1, 0, 0)$ . With a different probing order,  the sequence  $(0, 0, 1, 0, 1)$  could also have resulted from  ${\it \Psi}_6$.
   
6. Similar procedure was followed in determining the leader  $\underline{e}_7 = (0, 1, 1, 0, 0)$  of the cosets class  ${\it \Psi}_7$  characterized by the uniform syndrome  $\underline{s}_7 = (1, 1, 1)$.  Also in the class  ${\it \Psi}_7$  there is another sequence with Hamming weight  $w_{\rm H}(\underline{e}) = 2$,  namely  $(1, 0, 0, 0, 1)$.
   
   

• The table only needs to be created once.  First,  the syndrome must be determined,  e.g. for the received vector  $\underline{y} = (0, 1, 0, 0, 1)$:

\[\underline{s} = \underline{y} \cdot { \boldsymbol{\rm H} }^{\rm T} = \begin{pmatrix} 0 &1 &0 &0 &1 \end{pmatrix} \cdot \begin{pmatrix} 1 &1 &0 \\ 0 &1 &1 \\ 1 &0 &0 \\ 0 &1 &0 \\ 0 &0 &1 \\ \end{pmatrix} = \begin{pmatrix} 0 &1 &0 \end{pmatrix}= \underline{s}_2 \hspace{0.05cm}.\]

• Using the coset leader  $\underline{e}_2 = (0, 0, 0, 1, 0)$  from the above table $($red entry for  $\mu =2)$  finally arrives at the decoding result:

\[\underline{z} = \underline{y} + \underline{e}_2 = (0,\hspace{0.05cm} 1,\hspace{0.05cm} 0,\hspace{0.05cm} 0,\hspace{0.05cm} 1) + (0,\hspace{0.05cm} 0,\hspace{0.05cm} 0,\hspace{0.05cm} 1,\hspace{0.05cm} 0) = (0,\hspace{0.05cm} 1,\hspace{0.05cm} 0,\hspace{0.05cm} 1,\hspace{0.05cm} 1) \hspace{0.05cm}.\]

$\text{Conclusion:}$  From these examples it is already clear that the  »syndrome decoding means a considerable effort«,  if one cannot use certain properties e.g. cyclic codes.

• For large block code lengths, this method fails completely. Thus, to decode a  $\text{BCH code}$  $($the abbreviation stands for their inventors Bose, Chaudhuri, Hocquenghem$)$  with code parameters  $n = 511$,  $k = 259$  and  $d_{\rm min} = 61$,  one has to evaluate and store exactly  $2^{511-259} \approx 10^{76}$ error patterns of length  $511$.
• Happily,  however,  there are special decoding algorithms for these and also for other codes of large block length,  which lead to success with less effort

.

Coding gain - bit error rate with AWGN

We now consider the  $\text{bit error rate}$  for the following constellation:

Bit error rate for the  $\text{(7, 4, 3)}$ Hamming code

1. Hamming code  $\text{HC (7, 4, 3)}$,
   
2. AWGN–channel, characterized by the quotient  $E_{\rm B}/N_0$  (in dB),
   
3. Maximum Likelihood Decoder  $\rm (ML)$  with 
   "Hard Decision"  $\rm (HD)$  and  "Soft Decision"  $\rm (SD)$,  resp.

It should be noted with regard to this graph:

• The black comparison curve applies e.g. to binary phase modulation  $\rm (BPSK)$  without coding.  For this one needs for the bit error rate   $10^{-5}$  about  $10 \cdot \lg \, E_{\rm B}/N_0 = 9.6 \, \rm dB$.
  

• The red circles apply to the  $\text{HC (7, 4, 3)}$  and hard decisions of the maximum likelihood decoder  $\text{(ML–HD)}$.  Syndrome decoding is a possible realization form for this.
  

• This configuration brings an improvement over the comparison system only for  $10 \cdot \lg \, E_{\rm B}/N_0 >6 \, \rm dB$.  For  $\rm BER =10^{-5}$  one only needs  $10 \cdot \lg \, E_{\rm B}/N_0 \approx 9.2 \, \rm dB$.
  

• The green crosses for  $\text{HC (7, 4, 3)}$  and  $\text{soft decision}$  $\text{(ML–SD)}$  are below the red curve throughout the range.  For  $\rm BER =10^{-5}$  this gives  $10 \cdot \lg \, E_{\rm B}/N_0 \approx 7.8 \, \rm dB$.
  
  

$\text{Definition:}$  We refer as  »coding gain«  of a considered system configuration, 

• characterized by its code and
• the way it is decoded,

the smaller  $10 \cdot \lg \, E_{\rm B}/N_0$  required for a given bit error rate  $\rm (BER)$  compared to the comparison system  $($without coding$)$:

\[G_{\rm Code} (\hspace{0.05cm}\text{considered system} \hspace{0.05cm}\vert\hspace{0.05cm}{\rm BER}\hspace{0.05cm}) =10 \cdot {\rm lg}\hspace{0.1cm}{E}_{\rm B}/N_0 \hspace{0.15cm}(\hspace{0.05cm}\text{comparison system}\hspace{0.05cm}\vert\hspace{0.05cm}{\rm BER}\hspace{0.05cm})- 10 \cdot {\rm lg}\hspace{0.1cm}{E}_{\rm B}/N_0 \hspace{0.15cm}(\hspace{0.05cm}\text{considered system}\hspace{0.05cm}\vert\hspace{0.05cm}{\rm BER}\hspace{0.05cm}) \hspace{0.05cm}. \]

Applied to the above graph, one obtains:

\[G_{\rm Code} (\hspace{0.05cm}{\rm Hamming \hspace{0.1cm}(7,\hspace{0.02cm}4,\hspace{0.02cm}3), ML-HD}\hspace{0.05cm}|\hspace{0.05cm}{\rm BER} = 10^{-5}\hspace{0.05cm}) = 0.4\ {\rm dB}\hspace{0.05cm},\]

\[G_{\rm Code} (\hspace{0.05cm}{\rm Hamming \hspace{0.1cm}(7,\hspace{0.02cm}4,\hspace{0.02cm}3), ML-SD}\hspace{0.05cm}|\hspace{0.05cm}{\rm BER} = 10^{-5}\hspace{0.05cm}) = 1.8\ {\rm dB}\hspace{0.05cm}.\]

Decoding at the Binary Erasure Channel

Finally,  it will be shown to what extent the decoder has to be modified

• if instead of the  $\text{BSC model}$  ("Binary Symmetric Channel") 
• the  $\text{BEC model}$  ("Binary Erasure Channel")  is used, 

which does not produce errors but marks uncertain bits as  "erasures".

$\text{Example 4:}$  We consider again as in  $\text{Example 3}$  the systematic  $\text{(5, 2, 3)}$  block code 

$$\mathcal{C} = \big \{ (0, 0, 0, 0, 0) \hspace{0.05cm},\hspace{0.3cm}(0, 1, 0, 1, 1) \hspace{0.05cm},\hspace{0.3cm}(1, 0, 1, 1, 0) \hspace{0.05cm},\hspace{0.3cm}(1, 1, 1, 0, 1) \big \}.$$

The graphic shows the system model and gives exemplary values for the individual vectors.

• The left part of the picture  (yellow background)  is valid for  "BSC"  with one bit error  $(0 → 1)$  at the third bit.
• The right part of the picture  (green background)  is for  "BEC"  and shows two  erasures  $(\rm 1 → E)$  at the second and the fourth bit.

Error correction with BSC and BEC

One recognizes:

• With BSC only  $t = 1$  bit error  (marked in red in the left part)  can be corrected due to  $d_{\rm min} = 3$.  If one restricts oneself to error detection, this works up to  $e= d_{\rm min} -1 = 2$  bit errors.
  

• For BEC,  error detection makes no sense, because already the channel locates an uncertain bit as an  "erasure"  $\rm E$.  The zeros and ones in the BEC received word  $\underline{y}$  are safe.  Therefore the error correction works here up to  $e = 2$  erasures  (marked in red in the right part)  with certainty.
  

• Also  $e = 3$  erasures are sometimes still correctable.  So  $\underline{y} \rm = (E, E, E, 1, 1)$  to  $\underline{z} \rm = (0, 1, 0, 1, 1)$  to be corrected since no second code word ends with two ones.  But  $\underline{y} \rm = (0, E, 0, E, E)$  is not correctable because of the all zero word allowed in the code.
  

• If it is ensured that there are no more than two erasures in any received word,  the BEC block error probability  ${\rm Pr}(\underline{z} \ne \underline{x}) = {\rm Pr}(\underline{v} \ne \underline{u}) \equiv 0$.  In contrast,  the corresponding block error probability in the BSC model always has a value greater than  $0$.

Since after the BEC each received word is either decoded correctly or not at all,  we call here the block  $\underline{y} → \underline{z}$  in the future  "code word finder".  An  "estimation"  takes place only in the BSC model.

But how does the decoding of a received word   $\underline{y}$   with erasures work algorithmically?

$\text{Example 5:}$  Starting from the received word  $\underline{y} \rm = (0, E, 0, E, 1)$  in  $\text{Example 4}$  we formally set the output of the code word finder to  $\underline{z} \rm = (0, z_2, 0, z_4, 1)$,  where the symbols  $z_2 \in \{0, \, 1\}$  and  $z_4 \in \{0, \, 1\}$  are to be determined according to the following equation:

\[\underline{z} \cdot { \boldsymbol{\rm H} }^{\rm T}= \underline{0} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} { \boldsymbol{\rm H} } \cdot \underline{z}^{\rm T}= \underline{0}^{\rm T} \hspace{0.05cm}.\]

The task is now to implement this determination equation as efficiently as possible.  The following calculation steps result:

• With the parity-check matrix  $\boldsymbol{\rm H}$  of the  $\text{(5, 2, 3)}$  block code and the vector  $\underline{z} \rm = (0, z_2, 0, z_4, 1)$  is the above determination equation:

\[{ \boldsymbol{\rm H} } \cdot \underline{z}^{\rm T} = \begin{pmatrix} 1 &0 &1 &0 &0\\ 1 &1 &0 &1 &0\\ 0 &1 &0 &0 &1 \end{pmatrix} \cdot \begin{pmatrix} 0 \\ z_2 \\ 0 \\ z_4 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \hspace{0.05cm}.\]

• We sum up the  "correct bits"  (German:  "korrekt"   ⇒   subscript "K")  to the vector  $\underline{z}_{\rm K}$  and the  "erased bits"  to the vector  $\underline{z}_{\rm E}$.  Then we split the parity-check matrix  $\boldsymbol{\rm H}$  into the corresponding submatrices  $\boldsymbol{\rm H}_{\rm K}$  and  $\boldsymbol{\rm H}_{\rm E}$:

\[\underline{z}_{\rm K} =(0, 0, 1)\hspace{0.05cm},\hspace{0.3cm} { \boldsymbol{\rm H} }_{\rm K}= \begin{pmatrix} 1 &1 &0\\ 1 &0 &0\\ 0 &0 &1 \end{pmatrix} \hspace{0.3cm}\Rightarrow\hspace{0.3cm} \text{Rows 1, 3 and 5 of the parity-check matrix} \hspace{0.05cm},\]

\[\underline{z}_{\rm E} = (z_2, z_4)\hspace{0.05cm},\hspace{0.35cm} { \boldsymbol{\rm H} }_{\rm E}= \begin{pmatrix} 0 &0\\ 1 &1\\ 1 &0 \end{pmatrix} \hspace{1.05cm}\Rightarrow\hspace{0.3cm} \text{Rows 2 and 4 of the parity-check matrix} \hspace{0.05cm}.\]

• Remembering that in  $\rm GF(2)$  subtraction equals addition,  the above equation can be represented as follows:

\[{ \boldsymbol{\rm H} }_{\rm K} \cdot \underline{z}_{\rm K}^{\rm T} + { \boldsymbol{\rm H} }_{\rm E} \cdot \underline{z}_{\rm E}^{\rm T}= \underline{0}^{\rm T} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} { \boldsymbol{\rm H} }_{\rm E} \cdot \underline{z}_{\rm E}^{\rm T}= { \boldsymbol{\rm H} }_{\rm K} \cdot \underline{z}_{\rm K}^{\rm T} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} \begin{pmatrix} 0 &0\\ 1 &1\\ 1 &0 \end{pmatrix} \cdot \begin{pmatrix} z_2 \\ z_4 \end{pmatrix} = \begin{pmatrix} 1 &1 &0\\ 1 &0 &0\\ 0 &0 &1 \end{pmatrix} \cdot \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \hspace{0.05cm}.\]

• This leads to a linear system of equations with two equations for the unknown  $z_2$  and  $z_4$  $($each  $0$  or  $1)$.

• From the last row follows  $z_2 = 1$  and from the second row  $z_2 + z_4 = 0$   ⇒   $z_4 = 1$.  This gives the allowed code word  $\underline{z} \rm = (0, 1, 0, 1, 1)$.

Exercises for the chapter

Exercise 1.11: Syndrome Decoding

Exercise 1.11Z: Syndrome Decoding again

Exercise 1.12: Hard Decision vs. Soft Decision

Exercise 1.12Z: Comparison of HC (7, 4, 3) and HC (8, 4, 4)

Exercise 1.13: Binary Erasure Channel Decoding

Exercise 1.13Z: Binary Erasure Channel Decoding again