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#### Proof

Consider the distances between the vectors of C and 0 (who is a codeword since it can be generated by G). The smallest of these distances will be the minimum weight of C since this distance is a count of nonzero positions. Thus the minimum weight of C will be greater than or equal to the minimum distance of C (since it's still possible for a smaller distance to exist between two other vectors of C).

To finish the proof, we need to show that there exists a codeword whose weight equals the mininum distance of C. Choose c and c' from C such that . Set (a linear combination). Then we have .

We define the dual code as . Thus the dual code consists of the vectors orthogonal to all the codewords.

If C is a linear code of dimension , then is a subspace of V of dimension . Proof on page 187 of book.

Let C be a linear code. Then the matrix H whose rows form a basis of the dual code is called a parity check matrix of C.

Next: Constructing G (and hence, Up: A Linear Algebra Point Previous: A Linear Algebra Point
Bill Cherowitzo 2001-12-11