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> .
where ''F'' is the [[field (mathematics)|field]] with two elements {0,1}. The name arises from the use of these codes as [[error-correcting code]]s. Occasionally ''F'' is taken to be some other [[finite field]]. If the field ''F'' contains ''q'' [[element (mathematics)|element]]s then the code ''C'' is said to be a q-ary code (special exceptions are ''binary'' codes and ''ternary'' codes, corresponding to ''q=2'' and ''q=3'' respectively).▼
▲whereAside: ''F''<sub>2</sub> = {0,1} is the [[field (mathematics)|field]] with two elements[[element {0,1}.(mathematics)|element]]s Theand name''F''<sup>''n''</sup><sub>2</sub> arises fromis the useset of these codes asall [[errorn-correcting codetuple]]s.Occasionallyof length ''Fn'' isover taken''F''<sub>2</sub>. to beOccasionally some other [[finite field]]. If the field ''F''<sub>q</sub> containscontaining ''q'' [[element> (mathematics)|element]]s2 thenelements is used, in which case the code ''C'' is said to be a q-ary code (specialrather than a binary code). Special exceptions to the [[adjective]] 'q-ary' are ''binary'' codes and ''ternary'' codes, (corresponding to ''q=2'' and ''q=3'' respectively).
==Properties==
By virtue of the fact that the code is a [[subspace]] of ''F''<sup>''n''</sup>, the sum ''c<sub>1</sub> + c<sub>2</sub>'' of two codewords in ''C'' is also a codeword (ie an [[element (mathematics)|element]] of the subspace ''C''). This definition also allowsThus the entire code (which may be very large) to be represented as the [[span (linear algebra)|span]] of a minimal set of codewords (known as a [[basis (linear algebra)|basis]] in [[linear algebra]] terms). These basis codewords are often collated in the rows of a matrix known as a ''generating matrix'' for the code ''C''.
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The mostmotivation importantbehind property ofcreating linear codes is that the linearity of the codeto allowsallow for [[syndrome decoding]].
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'''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''.
