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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]]
:<math>F^n_2.</math>
Aside: ''F''<sub>2</sub> = {0,1} is the [[field (mathematics)|field]] with two [[element (mathematics)|element]]s and
==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''). Thus 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''.
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:
:<math>d(c_1, c_2)=d(c_1+c_2, 0)\,</math>
implying:
:<math>\min_{c_1, c_2\in C}d(c_1,c_2)=\min_{c_1, c_2\in C}d(c_1+c_2, 0)=\min_{c_1\in C}d(c_1, 0).</math>
The motivation behind creating linear codes is to allow for [[syndrome decoding]].
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