hamming code using linear algebra

Question

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.

Answer 1

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.

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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.