I know some procedures for decoding of Hamming Code (syndrome table, ...). But There is an easy way for decoding of Hamming codes for example here a very nice method for decoding of Hamming(7,4) code is suggested. I'm looking for similar decoding for other Hamming codes such as Hamming(15,11) , Hamming(31,26).

If you know similar decoding schemes, please let me know.

  • $\begingroup$ These are the general Hamming codes. They also have parity check matrices that do the decoding. However, I'm not entirely sure that they decode as nicely as the $[7,4]$-Hamming code (where the pc matrix actually identifies the precise position(s) of errors as a binary vector). This would depend on whether the matrix whose consecutive columns (from 1 up to $2^n-1$ in binary) is a legit pc matrix. $\endgroup$
    – Doc
    Dec 3, 2013 at 17:58
  • $\begingroup$ All hamming codes are arranged in a way that it is possible to easily find the position of error. But By decoding I mean the whole process (also finding the message corresponding to the codeword decoded). So after decoding the transmitted codeword, it is not computationally easy to look for the message corresponding to the codeword decoded. While the method mentioned directly decode the transmitted message. $\endgroup$ Dec 3, 2013 at 18:00
  • $\begingroup$ They're the same thing. If you can find the position of the errors then you have the correct decoding. Just complement each bit in error. $\endgroup$
    – Doc
    Dec 3, 2013 at 18:03
  • $\begingroup$ Finding the transmitted codeword is easy. But I think finding the corresponding message is not. Is it? $\endgroup$ Dec 3, 2013 at 18:04
  • $\begingroup$ If you know the information bits, the tasks are equivalent. $\endgroup$
    – Doc
    Dec 3, 2013 at 18:07

1 Answer 1


The wikipedia link you have given shows the very last step in the decoding process, where you go from the transmitted bits (7 bit vector) to the information bits(4 bit vector). The matrix $R$ is actually unnecessary because all it is doing is picking out bits 3,5,6 and 7 from vector $r$, which is possible because the code is a systematic code to begin with.

To summarize, if the Hamming code is systematic, all you need is to correct errors in the received vector by using the syndrome table and you can then pick out the appropriate information bits.

Edit: I'm adding the wikipedia link to systematic codes for completeness. http://en.wikipedia.org/wiki/Systematic_code

  • $\begingroup$ Thanks. But what determines the appropriate information bits? $\endgroup$ Dec 3, 2013 at 18:05
  • $\begingroup$ If the generator matrix has an identity submatrix, you can easily find out which bits to pull out of the corrected received vector, as per en.wikipedia.org/wiki/Parity-check_matrix $\endgroup$
    – svenkatr
    Dec 3, 2013 at 18:08
  • $\begingroup$ Any set of linear independent columns of a generator matrix can serve as the positions of the information bits. This is where the message is carried, and its usually the inital $k$ positions. $\endgroup$
    – Doc
    Dec 3, 2013 at 18:10
  • $\begingroup$ If we build the parity-check matrix in lexicographic order, how to find the position of information bits? $\endgroup$ Dec 3, 2013 at 19:47

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