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Hamming codę 7 4 is the first codę in the class of Hamming codes. Error syndromcs. corresponding to single-error pattems. are written as binary words:

s_H =[5₃s₂5,]

that encode a position of nonzero bit in z. as presented in Table

i	Z|......*7	S^S2S|
1	1()(X)(XK)	001
2	0100000	010
3	0010000	011
4	0001000	100
5	0000100	101
6	0000010	110
7	(X)(X)(X)1	111
8	0000000	000

As alrcady observed (Problem 5.2.2), syndromc of the i-th single error is the i-lh column of the parity chcck matrix. Then. parity-check ma tri x:

	1	0	1	0	1 0
H =	0	1	1	0	0 1
	0	0	0	1	1 1

1 5,

1 s₂

Pi Pi P> h h U

As can be seen, the Hamming codę is the nonsystematic one. In such codę, it is convenient to select check-bit positions such, that each check-bil is designatcd by information bits only. Thus, colunins of parity-check matrix that contain only one nonzero entry designate check-bits positions. For the (7.4) Hamming codę. these positions are: j = 2° = 1. 2‘ = 2, 2² = 4. Then. the codcword:

w = [ p₂i_y PiW₄ ] = [w, w₂ łv,vw’₄vc₅w_bw₇ ] whcre. parity-check bits can be designatcd from cquations:

Py = /, ©i₂ 0/₄ p₂ =i, ©/j ®/₄

p_y = i₂ © i_y © i*₄.