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 V is called linear if C is a subspace of the vector space V. We will then denote C's dimension by the letter .
A generator matrix G of C is the x matrix whose rows form a basis that spans C.
Thus, we can store vectors, rather than the codewords that constitute C.
The minimum weight (C) of a code C is defined as the min(cc C, c .
Let C be a linear code. Then (C) = (C).