Construction of the generator matrix of a RM( r , p ) code Step 0.
The first row of G is the all-1 codeword: .
The next rows, the 1 st order codewords, form submatrix
0 1 n 1
The l -th column is the integer l in the binary code:
The next rows, the 2 nd order codewords, form submatrix . Its codewords are logic (Boolean) products of all pairs of the 1 st order words. The product of two binary n -tuples and is defined as:
, where , if and only if .
Step r .
The last rows, the r -th order codewords, form . Its codewords are products of all combinations of r words of the 1 st order.
RM codes in nonsystematic form can be decoded in different ways. Polynomial notation based, a cyclic code decoding (7.1.20-7.1.25) is one of them. Nonsystematic code can be always converted into systematic code, by mod-2 additions of nonsystematic code generator matrix rows (codewords), and then, matrix algebra based decoding (5.2.25, 5.2.26) can be applied.
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