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Constructing G (and hence, C) for Generalize GF($ q$)

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($ q$) 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 $ q^{n}-1$ and dividing this by $ q-1$ 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 $ k$ x $ \frac{q^{n}-1}{q-1}$. It makes things really nice if we choose these vectors such that their first nonzero entry is 1 and such that we place them into H$ ^{*}$ so that the first columns of H$ ^{*}$ form the identity matrix.

Then H$ ^{*}$ will be of the form: H$ ^{*}$=[I -P$ ^{\top}$]

Then one valid generator matrix G$ ^{*}$ can be formed by the equation G$ ^{*}=$[P I].



Subsections
next up previous
Next: Proof Up: Projective Geometry and Hamming Previous: Proof
Bill Cherowitzo 2001-12-11