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A Linear Algebra Point of View for GF(2)

The code we have examined is a Hamming Code. We will build up to it through linear algebra. We are still using GF(2). Linear codes have a great advantage in that we can reproduce work with them on computers by simply storing a basis for the code space, rather than the entire list of codewords.

A code C $ \epsilon$ V is called linear if C is a subspace of the vector space V. We will then denote C's dimension by the letter $ k$.

A generator matrix G of C is the $ k$ x $ n$ matrix whose $ k$ rows form a basis that spans C.

Thus, we can store $ k$ vectors, rather than the $ 2^{k}$ codewords that constitute C.

The minimum weight $ w$(C) of a code C is defined as the min$ \{w$(c$ \vert$c $ \epsilon$ C, c $ \neq 0\}$.

Let C be a linear code. Then $ d$(C) = $ w$(C).


Bill Cherowitzo 2001-12-11