Construction
A generator matrix for a Reed–Muller code of length n = 2d can be constructed as follows. Let us write:
Note that each member of the set X is a point in . We define in n-dimensional space the indicator vectors
on subsets by:
together with, also in, the binary operation
referred to as the wedge product (this wedge product is not to be confused with the wedge product defined in exterior algebra). Here, and are points in, and the operation is the usual multiplication in the field .
is a d-dimensional vector space over the field, so it is possible to write
We define in n-dimensional space the following vectors with length n: v0 = (1, 1, 1, 1, 1, 1, 1, 1) and
where the Hi are hyperplanes in (with dimension d −1):
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