generator matrix

Question

Regarding the linear codes:

Is it important to be able to determine the generator matrix from a given code or would it be given in a task?

Can the control matrix only detect bit errors in the redundancy bit range? The script says that H*e belongs to a subspace generated by the column vectors of H. For example, if there are 2 errors in the redundancy bits, I think you can determine the matrix columns (because of the unit matrix columns as a kind of basis?). But is this also the case if more than 2 errors occur, which do not only affect the redundancy bits?

Best regards!

Answer 1

So far we never had exam problem about linear codes, but that does not mean that it is unimportant. I would say that a generator matrix should be given, and you should be able to encode and decode words with it. You may even assume a linear code that is separated so that the construction of the control matrix would be simple. it is then not much more than matrix multiplication, right?

Comments

I was looking at the example on slide 99: There we can locate the error but if the first bit is also inverted then I get the vector (0,1,0) from the control matrix so the 6th bit could be wrong but also the first and the 5th bit could have caused the error. How can I be sure how many errors have occured?

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It is quite unlikely that a bit error occurs, and yet even much more unlikely it is that two occur at the same time. We therefore speculate (!) that there is just a single bit flip, and not two. At the end, that is just a wild guess you may say, but if the probability for a bit flip would be, say 2^{-20} then for two at once it is 2^{-40} which is really unlikely.

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Remark: If we should introduce new kinds of exercises, we will do that mildly, i.e., not starting with the most difficult ones when doing that the first time. So, better focus on the so far seen problems which may occur again in full detail, and understand the rest at some level of detail only.

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Oh, thanks! Also for the advice! :)