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Maths - linear Hamming codes

I have attached both the questions and the 'answers'.
I don't see how finding a check matrix shows 3a) and I have no clue about 3b onwards.

In Q4 I thought you did n=(2^r)-1 and r=3 therefore n=7 but in the answers they have n as 14. I am v confused.
codes.png141.8KB
Codes answers.png157.6KB
Reply 1
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Original post by CammieInfinity
I have attached both the questions and the 'answers'.
I don't see how finding a check matrix shows 3a) and I have no clue about 3b onwards.

In Q4 I thought you did n=(2^r)-1 and r=3 therefore n=7 but in the answers they have n as 14. I am v confused.


I'm just learning this myself so I could be wrong..

Since there is a weight 7 codeword, the existence of weight 1 and 2 codewords implies the existence of weight 6 and 5 codewords respectively.

A weight 1 or 2 codeword is not possible since minimum weight for Hamming code is 3.

As for question 4, this one isn't binary. You need n=qr−1q−1n=\frac{q^r-1}{q-1}

Edit:...and shouldn't it be 13?
(edited 9 years ago)
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Your parity check matrix for Ham(3,3) can have as its columns all non-zero vectors in V(3,3) which have 1 as their first non-zero entry.

001, 010, 011, 012, 100, etc..
Yes you are right I meant 13. Apologies. I see it now. I'll just ask my tutor when I get back about the other bit because I do not see the relevance of the check matrix.

I am also struggling with 1c) (new attachment)

I don't know how to find the magnitude of C? the base included 3 words if that helps. and for S(c,1) why is 6 choose 1 multiplied by 1? As in why 1? What does the 1 represent? if it was S(c,2) would it be S(c,1) + 6 choose 2 *1?
codes.png146.8KB
1c.jpg94.7KB
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Original post by CammieInfinity
I'll just ask my tutor when I get back about the other bit because I do not see the relevance of the check matrix.


Which other bit do you mean?

Wasn't the check matrix required?
As in you said use check matrix to show you can't have a weight of 1
Reply 6
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Original post by CammieInfinity
As in you said use check matrix to show you can't have a weight of 1


The check matrix was used to show that 1111111 is a codeword.

You can use the check matrix to show that no codeword has weight one but when I mentioned that, I neglected to use the fact that the minimum weight of non-zero codewords of a Hamming code is 3.

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