Consider now the new non$-$standard Hamming $(7,4)$ code that uses encoding matrix

H$=110010001100101111011$

$,$

and produces an encoded message with check digits at positions $2$

$,3$

$,$ and $4$

$.$

When the encoding matrix is not the standard one, the message is encoded differently, though it is still done in such a way that H$$xT$=0$

over Z$2$

for any encoded message x

$.$ We can find the encoding of a message $($a$,$b$,$c$,$d$)$

by setting x$=$

$($a$,$x$,$y$,$z$,$b$,$c$,$d$)$

and evaluating H$$xT

$,$ deducing the values of x$,$y$,$z

needed to ensure H$$xT$=0$

over Z$2$

$.$

When the positions of the check digits correspond with the positions of the basis vectors in the encoding matrix, finding the encoding is fairly straightforward.

The decoding method is exactly the same as before, though we should again be careful to ignore the check digits when extracting the original message.

c$)$

Using the Hamming $(7,4)$ code described above, encode the message $(1,1,0,1)$

$.$