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Answer: Yes, there are several other codes with n=7 and k=4, like the one given here. G = [1 1 0 1 0 0 0; 0 1 1 0 1 0 0; 1 1 1 0 0 1 0; 1 0 1 0 0 0 1]; The above happens to be a generator matrix for the (7,4,3) Hamming code, which is a very famous code historically. It is also a useful code in practice for small lengths. Also, if you want a linear code with n=7, k=4 and d=3, you will not get anything very different from the Hamming code. For the above reasons, it is given as an example in many texts. You could definitely work with other examples, particularly for other values of n and k and d.

Answer: In practice, one can simulate for different possible maximum number of iterations and pick the number beyond which there is no incremental gain. If you are implementing the decoder on a chip, you will also have hardware and system-level restrictions on the possible number of iterations. There has been some limited work on fixing a maximum number of iterations by analytical means, but simulations are a good solution in practice.

Answer: While there are some simple estimation methods for calculating the variance, a useful alternative is the max-log-map version of soft decoding. In this version, the variance is not required. See Viterbi, A.J.; , "An intuitive justification and a simplified implementation of the MAP decoder for convolutional codes," Selected Areas in Communications, IEEE Journal on , vol.16, no.2, pp.260-264, Feb 1998 doi: 10.1109/49.661114 URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=661114&isnumber=14424

Answer: I am unable to suggest anything more concrete than a simple Google search. All the best.

Answer: Depends on what you want to do with the generator matrix. Could elaborate on your question and make it more clear? What is your goal in choosing the generator matrix?

Answer: Yes, there are several other codes with n=7 and k=4, like the one given here. G = [1 1 0 1 0 0 0; 0 1 1 0 1 0 0; 1 1 1 0 0 1 0; 1 0 1 0 0 0 1]; The above happens to be a generator matrix for the (7,4,3) Hamming code, which is a very famous code historically. It is also a useful code in practice for small lengths. Also, if you want a linear code with n=7, k=4 and d=3, you will not get anything very different from the Hamming code. For the above reasons, it is given as an example in many texts. You could definitely work with other examples, particularly for other values of n and k and d.

Answer: Yes, one typically uses the parity-check matrix of the component code in the decoding. Here are a couple of comments based on the wording of your question: -You dont have to only use Hamming code as the component code in a product code. Other codes are possible, as well. -There is no such thing as a parity-check matrix for a codeword - you should say a parity-check matrix of the code.

Answer: Encoders are classified as systematic or non-systematic depending on whether or not the message appears as part of the codeword. Every linear code has a systematic encoder. So, an LDPC code, which is a linear code, has a systematic encoder as well. Actually, systematic code is technically wrong usage - the correct question to ask is whether a code has a systematic encoder. For any code, the error-correcting capability is (d-1)/2, where d is the minimum distance. This property is definitely true for LDPC codes as well. However, the reason for the good performance of LDPC codes goes beyond merely minimum distance. The message-passing decoder plays an important role, and its performance is not strongly tied to the minimum distance alone.