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The elements of the alphabet $W = (a,b,c,d)$ are uniformly distributed and will be decoded with the following code:

$C(a)= 00 $, $C(b)=01, C(c)=10, C(d)=11 $

The coded elements will be transmitted through a discrete channel.

Hereby the following error can occur: with probability 0.1 a 1 instead of a 0 is transmitted and with probability 0.05 a 0 instead of a 1 is transmitted. The received bit-pairs will be decoded according to $C^{-1}$.

I have to calculate the maximal probability of error and the average probability of error for the code. The problem is that for computing them I need the probability for the single codeword ($\lambda_i)$ and I am not understanding how to calculate that.

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Well, as far as i understand from your question you just need the error probability of the single codewords.

for the codeword C(a)=00 is just 0.1*0.1 which is the probability of transmitting the first bit wrong times the probability of transmitting the second one wrong, so:

  1. c(a) = 0.01
  2. c(b) = 0.1*0.05 = 0.005
  3. c(c) = 0.05*0.1 = 0.005
  4. c(d) = 0.05*0.05 = 0.0025
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  • $\begingroup$ I was thinking the same, then the maximal error probability, which from the definition is the max of those $ \lambda_i $, is 0.01. Is this right? I am not that sure and I have several doubts. $\endgroup$
    – dadadima
    Jun 13, 2017 at 15:15
  • $\begingroup$ which doubts? i think it is right. for example take the case you have to transmit a message made only by C(a), so you will have a possible error with a probability higher than the probability of error for a message made by only C(d) $\endgroup$
    – Simone Masiero
    Jun 14, 2017 at 7:35
  • $\begingroup$ please mark the solution $\endgroup$
    – Simone Masiero
    Jun 14, 2017 at 22:41

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