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What types of basic variations of the Hamming code are there and what are their objectives? I was taught the following version:

$$ L = n + k $$ $$ n \geq \log_2M $$ $$ k \ge \log_2(n+k+1) $$

where $M$ - number of alphabet symbols, $L$ - length of a codeword, $n$ - number of information bits, $k$ - number of parity bits. How does such a variation change the analysis and the algorithms involved as compared to the canonical way of coding?

Edit summary:

  • I switched the notation to the one I was provided with originally.
  • I switched the logarithm base from implicit $a$ to explicit 2.
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    Whomsoever is teaching you is using very idiosyncratic notation: $d$ is very commonly used to denote Hamming distance, or in some cases, minimum Hamming distance between codewords. Also, what is $a$ in $\log_a$?2012-08-31
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    @DilipSarwate I think $a$ indicates if it is binary or triple etc.. if binary then $a=2$ i guess. I agree that $d$ should me minimum hamming distance.2012-08-31
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    @DilipSarwate I am Sorry, I recklessly altered the notation. I will correct that in a minute.2012-08-31
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    @Seyhmus Nonbinary Hamming codes (meaning single error correcting codes) are different from binary Hamming codes. For example, a _canonical_ single error correcting Hamming code over $\mathbb F_q$, $q > 2$, does not have length $q^m-1$ while the binary Hamming code does have length $2^m-1$. So, if $a > 2$, we need to be more careful about the parameters.2012-08-31
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    @DilipSarwate if it is so, I will explicitly make my $a$ equal 2 because that is the case I am interested in.2012-08-31
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    Oh good grief! The notation has taken a turn for the worse. Now we have a $[L,n]$ binary code with $k$ denoting the number of _parity_ bits instead of the very commonly used $[n,k]$ linear code with $2^k$ codewords or $(n,M)$ code with $M$ codewords of length $n$. Most people would say the _alphabet_ is binary, not $M$-ary, reserving $M$ for the number of _messages_ and so on.2012-08-31
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    @DilipSarwate That's why I initially provided the alternative notation - the original one seems like nothing I have seen in the textbooks. Please edit my question if you think switching the notation might help improve its readability. Do you have any idea what sort of Hamming code variation is it? The notation alteration might be due to the fact that my lecturer comes from Russia.2012-08-31

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