Linear code: Difference between revisions

Revision as of 08:35, 31 December 2006

Revision as of 21:32, 15 January 2007

Line 7:

==Properties==

As a [[linear subspace]] of <math>{\mathbb{F}_q}^n_qn</math>, the entire code <math>C</math>(which may be very large) may be represented as the [[span (linear algebra)|span]] of a minimal set of codewords (known as a [[basis (linear algebra)|basis]] in [[linear algebra]]). These basis codewords are often collated in the rows of a matrix known as a ''[[Generator matrix|generating matrix]]'' for the code <math>C</math>.

The subspace definition also gives rise to the important property that the minimum [[Hamming distance]] between any given codeword <math>c_0</math> and the other codewords <math>c \neq c_0</math> is constant. Since the difference <math>c - c_0</math> of two codewords in <math>C</math> is also a codeword (ie an [[element (mathematics)|element]] of the subspace <math>C</math>) and <math>d(c, c_0)=d(c-c_0, 0)</math>