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# Constructing G (and hence, C) for Generalize GF()

The preceding section came from our book and will be helpful for what follows, but most of the papers I read approach the topic a little differently. Here goes.

We define two vectors in V over GF() to be equivalent if they are non-zero scalar multiples of each other. We count the number of equivalence classes by taking the number of possible non-zero vectors and dividing this by since this is the number of non-zero elements of our field.

We then choose a representative vector from each of these equivalence classes (they will all be pairwise linearly independent) and use them as columns in a parity check matrix H. Thus H will be x . It makes things really nice if we choose these vectors such that their first nonzero entry is 1 and such that we place them into H so that the first columns of H form the identity matrix.

Then H will be of the form: H=[I -P]

Then one valid generator matrix G can be formed by the equation G[P I].

Subsections

Next: Proof Up: Projective Geometry and Hamming Previous: Proof
Bill Cherowitzo 2001-12-11