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y = mc + cos(θ)

Hamming codes work through extending the idea of parity by multiplying matrices called Hamming matrices. For the Hamming (7,4) code, we use two closely related matrices, call them

Firstly the data bits are inserted into their appropriate positions and the parity bits calculated in each case using even parity.

Calculation of Hamming code parity bits

	p1	p2	d1	p3	d2	d3	d4	p4	d5	d6	d7
Data word (without parity):			0		1	1	0		1	0	1
p1	1		0		1		0		1		1
p2		0	0			1	0			0	1
p3				0	1	1	0
p4								0	1	0	1
Data word (with parity):	1	0	0	0	1	1	0	0	1	0	1

The new data word (with parity bits) is now "10001100101". We now assume the final bit gets corrupted and turned from 1 to 0. Our new data word is "10001100100"; and this time when we analyse how the Hamming codes were created we flag each parity bit as 1 when the even parity check fails.