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In [[mathematics]], a '''(binary) linear code''' of length ''n'' > 0 and ''rank'' 0 < ''k'' < ''n'' + 1 is a [[linear subspace]] ''C'' with [[dimension (linear algebra)|dimension]] ''k'' of the [[vector space]]
:''F''<sup>''n''</sup><sub>2</sub>
Aside: ''F''<sub>2</sub> = {0,1} is the [[field (mathematics)|field]] with two [[element (mathematics)|element]]s and ''F''<sup>''n''</sup><sub>2</sub> is the set of all [[n-tuple]]s of length ''n'' over ''F''<sub>2</sub>. Occasionally some other [[finite field]] ''F''<sub>q</sub> containing ''q'' > 2 elements is used, in which case the code is said to be a ''q''-ary code (rather than a binary code). Special exceptions to the [[adjective]] 'q-ary' are ''binary'' and ''ternary'' codes (corresponding to ''q=2'' and ''q=3'' respectively).▼
▲Aside: ''F''<sub>2</sub> = {0,1} is the [[field (mathematics)|field]] with two [[element (mathematics)|element]]s and ''F''<sup>''n''</sup><sub>2</sub> is the set of all [[n-tuple]]s of length ''n'' over ''F''<sub>2</sub>. Occasionally some other [[finite field]] ''F''<sub>q</sub> containing ''q'' > 2 elements is used, in which case the code is said to be a q-ary code (rather than a binary code). Special exceptions to the [[adjective]] 'q-ary' are ''binary'' and ''ternary'' codes (corresponding to ''q=2'' and ''q=3'' respectively).
==Properties==
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The subspace definition also gives rise to the important property that the minimum [[Hamming distance]] between codewords is simply the minimum [[Hamming weight]] of all codewords since:
:''d'' (''c<sub>1</sub>'' , ''c<sub>2</sub>'' ) = ''d'' (''c<sub>1</sub> + c<sub>2</sub>'' , 0 )
implying:
:min<sub>'' c<sub>1</sub> , c<sub>2</sub>'' in ''C''</sub> { ''d'' (''c<sub>1</sub>'' , ''c<sub>2</sub>'' ) } = min<sub>'' c<sub>1</sub> , c<sub>2</sub>'' in ''C''</sub> { ''d'' (''c<sub>1</sub> + c<sub>2</sub>'' , 0 ) } = min<sub>'' c<sub>1</sub>'' in ''C''</sub> { ''d'' ( ''c<sub>1</sub>'' , 0 ) }
The motivation behind creating linear codes is to allow for [[syndrome decoding]].
==Popular notation==
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[[Code]]s in general are often denoted by the letter ''C''. A linear code of length ''n'', [[dimension (linear algebra)|rank]] ''k'' (ie having ''k'' codewords in its basis and ''k'' rows in its ''generating matrix'') and minimum Hamming weight ''d'' is referred to as an (''n''.''k'',''d'') code.
'''Remark
▲'''Remark:''' this is not to be confused with the notation [''n'',''r'',''d''] to denote a non-linear code of length ''n'', size ''r'' (ie having ''r'' codewords) and minimum Hamming distance ''d''.
==See also==
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*[[Code]]
*[[Syndrome decoding]]
[[Category:Linear algebra]]
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