Linear code: Difference between revisions

Revision as of 18:36, 11 May 2006

Revision as of 11:26, 13 May 2006

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==Properties==

By virtue of the fact that the code is a [[subspace]] of <math>\mathbb{F}^n_2</math>, the sum <math>c_1 + c_2</math> of two codewords in <math>C</math> is also a codeword (ie an [[element (mathematics)|element]] of the subspace <math>C</math>). Thus the entire code (which may be very large) tomay 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 <math>C</math>.

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: