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We have 2 stations that communicate with each other, but we need to detect (or even correct) when something is wrong.

We use 8 binary words: each consisting of 3 bits enter image description here and to send it we code it as enter image description here where enter image description here is the complement of enter image description here and enter image description here is the even parity check bit of enter image description here .

We need to find the capabilities of this code (up to how many can we detect and how many can we correct). BUT, a proof is required.

This is how far I've reached so far:

First we find the hamming distance: If changes then changes, also changes. So we have a hamming distance of 3.

This means that we can detect two bit errors or correct a single error.

Can you help me write the proof for that?

(also the tags may need some refinement - comments about the downvote are welcome)

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    $\begingroup$ This really isn't very difficult. Why not show us what you've done, and which part is giving you trouble? $\endgroup$
    – Beta
    Feb 4, 2014 at 2:04
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    $\begingroup$ Why are your making up your own error-correcting code instead of just using one of the standard ones? $\endgroup$
    – David Richerby
    Feb 4, 2014 at 15:19

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Consider any two codewords based on $b_2b_1b_0$ and $c_2c_1c_0$. The Hamming distance between the two codewords is at least twice the Hamming distance between $b_2b_1b_0$ and $c_2c_1c_0$ (why?). Furthermore, if the Hamming distance between $b_2b_1b_0$ and $c_2c_1c_0$ is exactly $1$, then your argument shows that the Hamming distance between the codewords is $3$. We can conclude that the Hamming distance is always at least $3$.

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