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Finding the decoding transformation for a hamming code

  1. Jun 13, 2007 #1
    I have the following linear transformation

    [​IMG]

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

    [tex]\delta \cdot\gamma(\upsilon) = \upsilon[/tex] for all [tex]\upsilon \in Z^4_2[/tex]

    is true for the transformation

    [tex]\delta := \varphi_B: Z^7_2 \longrightarrow Z^4_2, c \longmapsto Bc[/tex]


    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 [tex]Z^4_2[/tex] to [tex]Z^7_2[/tex] and the second one goes the other way round?

    If I can't just invert G, how would I go about this then?
     
    Last edited: Jun 13, 2007
  2. jcsd
  3. Jun 13, 2007 #2

    chroot

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    Staff Emeritus
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    Gold Member

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

    - Warren
     
  4. Jun 13, 2007 #3
    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.
     
  5. Jun 13, 2007 #4

    NateTG

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    Can you produce a 4x7 matrix so that the product with G is:
    Code (Text):

    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
     
     
  6. Jun 13, 2007 #5
    Wouldn't the resulting matrix be of type 4x4?

    Here's what I found for that scenario:

    Code (Text):
    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
     
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