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We think of a linear code as a linear map from

(Here we denote the linear map by both and . The later is
used in the subsequent section on linear codes)
__Ex__

The above is the linear map for adding a parity bit. The fact that
this is a linear map means that every entry of the vector on the
right is a linear function of the entries on the vectors on the
left.
For a linear code, which is a map
(or more generally
) we have a set of
codewords: Vector Space generated by

You may remember that a linear map L from one vector space V to W
satisfies:

where 0=
is the zero
vector.
If we know what happens to a set of basis vectors
of a
vector space V, then we know what happens to all vectors
because:

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Frederick Leitner
2004-09-01