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jeffconner50@yahoo.com
science forum beginner
Joined: 11 Sep 2005
Posts: 12
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 Posted: Thu Jun 22, 2006 7:56 pm Post subject:
Hamming codes
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Normally when I work on problems, I'm always given the Generator for a
binary or nonbinary hamming code or a Parity Check matrix. I was
wondering if, given that I know the length and dimension of the various
Hamming Codes, if it's possible to (somewhat easily?) construct each
code JUST knowing length and dimension of the code, given nothing else.
I have not seen anything about this one way or another.
Jeff |
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William Rex Marshall
science forum beginner
Joined: 14 Jun 2005
Posts: 39
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| Posted: Thu Jun 22, 2006 9:34 pm Post subject:
Re: Hamming codes
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<jeffconner50@yahoo.com> wrote in message
news:1151006210.792110.66580@i40g2000cwc.googlegroups.com...
| Quote: | Normally when I work on problems, I'm always given the Generator for a
binary or nonbinary hamming code or a Parity Check matrix. I was
wondering if, given that I know the length and dimension of the various
Hamming Codes, if it's possible to (somewhat easily?) construct each
code JUST knowing length and dimension of the code, given nothing else.
I have not seen anything about this one way or another.
Jeff
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hamming codes are "gold" codes or "perfect" codes (has greatest minimum
distance) |
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perhje@gmail.com
science forum beginner
Joined: 22 Jun 2006
Posts: 1
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| Posted: Thu Jun 22, 2006 9:56 pm Post subject:
Re: Hamming codes
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jeffconner50@yahoo.com skrev:
| Quote: | I was wondering if, given that I know the length and dimension of the various
Hamming Codes, if it's possible to (somewhat easily?) construct each
code JUST knowing length and dimension of the code, given nothing else.
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Absolutely. For example, in order to construct the parity matrix H for
the (7,4) Hamming code, we simply set H = [a1, a2, a3, a4, a5, a6, a7]
where a1 is the binary representation of 1 (as a column vectory), a2 is
the binary representation of 2 etc. Following this procedure, we obtain
that
H = [[0 0 1]^T [0 1 0]^T [0 1 1]^T [1 0 0]^T [1 0 1]^T [1 1 0]^T [1 1
1]^T]
or (a less messy version)
H = [ 0 0 0 1 1 1 1
0 1 1 0 0 1 1
1 0 1 0 1 0 1]
HTH,
Per |
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jeffconner50@yahoo.com
science forum beginner
Joined: 11 Sep 2005
Posts: 12
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| Posted: Thu Jun 22, 2006 10:23 pm Post subject:
Re: Hamming codes
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perhje@gmail.com 작성:
| Quote: | jeffconner50@yahoo.com skrev:
I was wondering if, given that I know the length and dimension of the various
Hamming Codes, if it's possible to (somewhat easily?) construct each
code JUST knowing length and dimension of the code, given nothing else.
Absolutely. For example, in order to construct the parity matrix H for
the (7,4) Hamming code, we simply set H = [a1, a2, a3, a4, a5, a6, a7]
where a1 is the binary representation of 1 (as a column vectory), a2 is
the binary representation of 2 etc. Following this procedure, we obtain
that
H = [[0 0 1]^T [0 1 0]^T [0 1 1]^T [1 0 0]^T [1 0 1]^T [1 1 0]^T [1 1
1]^T]
or (a less messy version)
H = [ 0 0 0 1 1 1 1
0 1 1 0 0 1 1
1 0 1 0 1 0 1]
HTH,
Per
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Per,
That's really clever. How about for the non-binary hamming codes, like
the Ternary codes. I can't imagine there is anything as easy as that
but is there another way?
Jeff |
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hoz
science forum beginner
Joined: 23 Jun 2006
Posts: 1
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| Posted: Fri Jun 23, 2006 12:13 am Post subject:
Re: Hamming codes
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This is a generalisation of the binary case.
The Hamming code of length n=(q^r-1)/(q-1) over the field GF(q) has an
rxn parity check matrix whose columns are all the non-zero r-tuples
from GF(q) with first non-zero entry equal to 1.
It can also be defined as a cyclic code as in the binary case.
(See, for example, the book by MacWilliams and Sloane which has more of
this stuff than you can poke a stick at).
Cheers
Gaz
jeffconner50@yahoo.com wrote:
| Quote: |
That's really clever. How about for the non-binary hamming codes, like
the Ternary codes. I can't imagine there is anything as easy as that
but is there another way?
Jeff |
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