In information theory, how do I calculate the probability of an erroneous transmission? Let's take for instance a binary symmetric channel with an error probability $ 1-d=0.25 $ and send codewords of length 6 coded in a Hamming code able to correct up to 1 error.

  • I don't know any Hamming codes of length 6? AFAIK the length of a Hamming code is one less than a power of two. No matter - the question is still answerable, if it is about a single-error-correcting code of length six. – Jyrki Lahtonen Aug 31 '12 at 10:20
  • @JyrkiLahtonen That's a good remark, addressed by a previous comment unfortunately already deleted. It must be one of the Hamming code's variations. Thats the version I was taught at my University and the fact it is a non-standard version indeed caused me some confusion. I would be thankful if someone told me what are the objectives and the theory behind this modification. – infoholic_anonymous Aug 31 '12 at 15:42
  • I was the one who deleted my comment asking which shortened Hamming code was under consideration. I have posted a detailed answer about how shortened Hamming codes work in an answer to another question by the OP. I don't think the question is fully answerable without knowing more about the code. – Dilip Sarwate Aug 31 '12 at 20:23
up vote 1 down vote accepted

We assume independence of bit errors. This is a somewhat dubious assumption, since errors often occur in bursts.

The probability of erroneous interpretation (or inability to decode) of a codeword of length $6$ is the probability that $2$ or more bits are incorrectly transmitted. The probability that $0$ bits are wrong is $(0.75)^6$. The probability that exactly $1$ bit is wrong is $6(0.25)(0.75)^5$. Add these two numbers, subtract the result from $1$.

  • Actually, you have found the probability $P(C)$ of correct decoding. For shortened Hamming codes, the complementary probability $1 - P(C)$ is the sum of the error probability $P(E)$ (meaning the output is the wrong codeword) and the failure probability $P(F)$ (meaning the decoder is unable to decode the received word into a valid codeword because a detectable but undecodable error pattern has occurred. My answer to this question describes how such decoder failures occur, and they do depend on how the shortening was done. – Dilip Sarwate Aug 31 '12 at 20:30

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