Systematic code

<h2 id="properties">Properties</h2>
<p>Every non-systematic linear code can be transformed into a systematic code with essentially the same properties (i.e., minimum distance).<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
Because of the advantages cited above, <a href="/facts/Linear_code/HoFI3eyF">linear</a> error-correcting codes are therefore generally implemented as systematic codes. However, for certain decoding algorithms such as sequential decoding or maximum-likelihood decoding, a non-systematic structure can increase performance in terms of undetected decoding error probability when the minimum <i>free</i> distance of the code is larger.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p><p>For a systematic <a href="/facts/Linear_code/HoFI3eyF">linear code</a>, the <a href="/facts/Generator_matrix/V76ugWLN">generator matrix</a>, 
  
    
      
        G
      
    
    {\displaystyle G}
  
, can always be written as 
  
    
      
        G
        =
        [
        
          I
          
            k
          
        
        
          |
        
        P
        ]
      
    
    {\displaystyle G=[I_{k}|P]}
  
, where 
  
    
      
        
          I
          
            k
          
        
      
    
    {\displaystyle I_{k}}
  
 is the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a> of size 
  
    
      
        k
      
    
    {\displaystyle k}
  
.
</p>
<h2 id="examples">Examples</h2>
<ul><li><a href="/facts/Checksum/AqxQ0I8g">Checksums</a> and <a href="/facts/Hash_function/rk6MlCt3">hash functions</a>, combined with the input data, can be viewed as systematic error-detecting codes.</li>
<li>Linear codes are usually implemented as systematic error-correcting codes (e.g., Reed-Solomon codes in <a href="/facts/Compact_Disc/g51GRloc">CDs</a>).</li>
<li><a href="/facts/Convolutional_code/HMCDdEGx">Convolutional codes</a> are implemented as either systematic or non-systematic codes. Non-systematic convolutional codes can provide better performance under maximum-likelihood (<a href="/facts/Viterbi_decoder/7Vu7USwt">Viterbi</a>) decoding.</li>
<li>In <a href="/facts/DVB-H/uRGX039M">DVB-H</a>, for additional error protection and power efficiency for mobile receivers, a systematic <a href="/facts/Reed%E2%80%93Solomon_error_correction/HPc3Qj6N">Reed-Solomon code</a> is employed as an erasure code over packets within a <a href="/facts/Burst_transmission/ep9vYM2N">data burst</a>, where each packet is protected with a <a href="/facts/Cyclic_redundancy_check/u2exo4sv">CRC</a>: data in verified packets count as correctly received symbols, and if all are received correctly, evaluation of the additional parity data can be omitted, and receiver devices can switch off reception until the start of the next burst.</li>
<li><a href="/facts/Fountain_code/st95SSf8">Fountain codes</a> may be either systematic or non-systematic: as they do not exhibit a fixed <a href="/facts/Code_rate/xPrLLTwt">code rate</a>, the set of source symbols is diminishing among the possible output set.</li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Shu Lin; Daniel J. Costello, Jr. (1983). <a href="https://archive.org/details/errorcontrolcodi00lins_929"><i>Error Control Coding: Fundamentals and Applications</i></a>. <a href="/facts/Prentice_Hall/bynWAuat">Prentice Hall</a>. pp. <a href="https://archive.org/details/errorcontrolcodi00lins_929/page/n296">278</a>–280. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-13-283796-X.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>James L. Massey, Daniel J. Costello, Jr. (1971). "Nonsystematic convolutional codes for sequential decoding in space applications". IEEE Transactions on Communication Technology. 19 (5): 806–813. doi:10.1109/TCOM.1971.1090720. S2CID 51650729.{{cite journal}}:  CS1 maint: multiple names: authors list (link) <a href="/wiki/James_L._Massey" target="_blank">/wiki/James_L._Massey</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>James L. Massey, Daniel J. Costello, Jr. (1971). "Nonsystematic convolutional codes for sequential decoding in space applications". IEEE Transactions on Communication Technology. 19 (5): 806–813. doi:10.1109/TCOM.1971.1090720. S2CID 51650729.{{cite journal}}:  CS1 maint: multiple names: authors list (link) <a href="/wiki/James_L._Massey" target="_blank">/wiki/James_L._Massey</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Richard E. Blahut (2003). Algebraic codes for data transmission (2nd ed.). Cambridge. Univ. Press. pp. 53–54. ISBN 978-0-521-55374-2. <a href="978-0-521-55374-2" target="_blank">978-0-521-55374-2</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>James L. Massey, Daniel J. Costello, Jr. (1971). "Nonsystematic convolutional codes for sequential decoding in space applications". IEEE Transactions on Communication Technology. 19 (5): 806–813. doi:10.1109/TCOM.1971.1090720. S2CID 51650729.{{cite journal}}:  CS1 maint: multiple names: authors list (link) <a href="/wiki/James_L._Massey" target="_blank">/wiki/James_L._Massey</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Shu Lin; Daniel J. Costello, Jr. (1983). Error Control Coding: Fundamentals and Applications. Prentice Hall. pp. 278–280. ISBN 0-13-283796-X. <a href="0-13-283796-X" target="_blank">0-13-283796-X</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Systematic code open-in-new

Systematic code