Hamming matrices Hamming(7,4)

bit position of data , parity bits

as mentioned above, rows 1, 2, , 4 of g should familiar map data bits parity bits:

p1 covers d1, d2, d4
p2 covers d1, d3, d4
p3 covers d2, d3, d4

the remaining rows (3, 5, 6, 7) map data position in encoded form , there 1 in row identical copy. in fact, these 4 rows linearly independent , form identity matrix (by design, not coincidence).

also mentioned above, 3 rows of h should familiar. these rows used compute syndrome vector @ receiving end , if syndrome vector null vector (all zeros) received word error-free; if non-zero value indicates bit has been flipped.

the 4 data bits — assembled vector p — pre-multiplied g (i.e., gp) , taken modulo 2 yield encoded value transmitted. original 4 data bits converted 7 bits (hence name hamming(7,4) ) 3 parity bits added ensure parity using above data bit coverages. first table above shows mapping between each data , parity bit final bit position (1 through 7) can presented in venn diagram. first diagram in article shows 3 circles (one each parity bit) , encloses data bits each parity bit covers. second diagram (shown right) identical but, instead, bit positions marked.

for remainder of section, following 4 bits (shown column vector) used running example:

p

=


(




d

1







d

2







d

3







d

4





)


=


(



1




0




1




1



)




{\displaystyle \mathbf {p} ={\begin{pmatrix}d_{1}\\d_{2}\\d_{3}\\d_{4}\\\end{pmatrix}}={\begin{pmatrix}1\\0\\1\\1\end{pmatrix}}}