'Why does Hammingcode decoder represent with gates and circuits?

we have this:

m=4 data bits

r=3 check bits

the check bits are at position 1,2 and 4, i.e. at the powers of two.

All other positions are data bits.

The test bits were chosen in such a way that red is for p1, green for p2 and blue for p3.

Now we had to represent this hammingcode with gates and circuits, but how can that be done?

hdl hamming-code

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