Finding the decoding transformation for a hamming code

[Icheb]

I have the following linear transformation

http://img162.imageshack.us/img162/3306/hammingcodeex4.gif

with G being a generating matrix for a hamming code and I have to find a matrix B so that the following:

$$\delta \cdot\gamma(\upsilon) = \upsilon$$ for all $$\upsilon \in Z^4_2$$

is true for the transformation

$$\delta := \varphi_B: Z^7_2 \longrightarrow Z^4_2, c \longmapsto Bc$$The way I understand this is that I have to reverse the initial transformation by finding the correct B. I figure it would be sufficient to invert G (since G * G^-1 * v = 1 * v = v and then B = G^-1), but how would that comply with the requirement that the first transformation goes from $$Z^4_2$$ to $$Z^7_2$$ and the second one goes the other way round?

If I can't just invert G, how would I go about this then?

 

[chroot]

You cannot take the inverse of non-square matrices. Let me think about this one a bit.

- Warren

 

[Icheb]

I figure I'd have to "invent" a solution and then find a B that's based on that? I just have no idea how that would work.

 

[NateTG]

Can you produce a 4x7 matrix so that the product with G is:

1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
0 0 0 0 0 0 0

 

[Icheb]

Wouldn't the resulting matrix be of type 4x4?

Here's what I found for that scenario:

1 0 0 0 0 0 0
1 0 0 0 0 0 0
0 1 0 0 0 0 0
1 0 0 0 0 0 0