66792

Construction of the generator matrix of a RM(r,p) codc

StcpO.

The first row of G is the all-1 codeword: G° = g, = [1 — 1].

Step 1.

The next p

rows, the 1^y order codewords. form submatrix

G' =

8:		0 1 ••• 0 f
83		0 0 - • I 1
8,		0 0 ••• 1 1
8/>+i		0 0 ••• 1 1

₂P~2

2p~^l

0 I    n-1

The /-th column (/ = 0,1.....n-1) is the intcgcr / in the binary codc:

l = G'_ltl2^r-' +-~+Gl,2°.

Step 2.

The next ^ | rows. the 2¹*¹ order codewords. form submatrixG². Its codewords arc logie

(Boolean) products of all pairs of the l^s‘ order words. The product of two binary /ł-tuplcs a = \a_x a₂ and b = [/>, b₂ •••/>„) isdcfincdas:

c = ab = [c, Cj—c,],

where c, = a_tb₍\ i = 1.....n, c_{ = 1 if and only if a, =b, = 1.

Step r.

The last

rows, the r-th order codewords. form G^r. Its codewords are products of all

combinations of r words of the I*¹ order.

RM codes in nonsystematic form can be decoded in different ways. Polynomial notation based, a cyclic codę decoding (7.1.20-7.1.25) is one of them. Nonsystematic codę can be always converted into systematic codę. by mod-2 additions of nonsystematic codę generator matrix rows (codewords). and then. matrix algebra based decoding (5.2.25, 5.2.26) can be applied.