# reed muller

Started by April 16, 2007
```Hi,

Given a generator matrix G for a non systematic reed muller
code ( first order under permutation ), how to compute the syndrome
matrix H ?

G = [
0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,
1, 0, 1, 0, 1, 0, 1, 0, 1;
0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1,
1, 0, 0, 1, 1, 0, 0, 1, 1;
0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1,
1, 0, 0, 0, 0, 1, 1, 1, 1;
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0,
0, 1, 1, 1, 1, 1, 1, 1, 1;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1;
]

H = ?

```
```mel said the following on 16/04/2007 20:01:
> Hi,
>
>          Given a generator matrix G for a non systematic reed muller
> code ( first order under permutation ), how to compute the syndrome
> matrix H ?
>
>
> G = [
> 	0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,
> 1, 0, 1, 0, 1, 0, 1, 0, 1;
> 	0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1,
> 1, 0, 0, 1, 1, 0, 0, 1, 1;
> 	0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1,
> 1, 0, 0, 0, 0, 1, 1, 1, 1;
> 	0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0,
> 0, 1, 1, 1, 1, 1, 1, 1, 1;
> 	0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1;
> 	1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1;
>     ]
>
> H = ?
>

Convert to systematic form (G => G'), then use the fact that if:

G' = [I Q]

then:

H' = [I Q^T]

Then perform the equivalent transformations (H' => H) to get back to the
original parity matrix.

--
Oli
```