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You have received the message 1110111 coded by (7,4)-Hamming. What 4-bit word did the sender want to convey?

I know hamming code but I need to solve it using linear algebra. I know its a linear code but not getting the exact way to solve it.

Any help would be truly appreciated.

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This question is currently not fully unanswerable, because you need to specify the encoding method. Or, the generator matrix. May be a systematic generator matrix was specified, but we cannot tell, you have give us that bit. See the discussion in this thread. Dilip Sarwate comments there about practical reasons for preferring a certain check matrix over others. His comments make perfect sense. But even assuming that you have not told us the method of mapping a 4-bit message to a 7-bit codeword, so we cannot invert that for you.


A few things OTOH are clear.

  • An error has occured, because that code has no words of weight six.
  • It has no words of five either, but it does have a unique word of weight seven. You are expected to do an error correcting step using these bits of information.
  • After error-correction you can do the rest with linear algebra using the generator matrix that you chose to keep a secret.
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  • $\begingroup$ This "answer" is close to being a comment (with a hint in the end) given that half of it is ranting about the question being badly formulated. If somebody thinks the question should be closed, I understand them. Do take it into account that a newcomer to coding theory may easily miss the need to specify the generator matrix (or some other method of encoding). That is not immediately clear to them. Also their teachers emphasize the error-correcting aspect at this point in their lectures, so it is natural that this step is only glossed over even though it is central to actually using a code :-) $\endgroup$ Sep 13, 2015 at 5:38

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