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I am confused by one remark on the average probability of error for $(M, n)$ code given in Cover and Thomas' book "Elements of Information Theory".

They said $P_e^{(n)}$ is a probability of error only if the message is chosen uniformly over the message set $\{1, 2, ..., 2^M\}$. Isn't the message set $\{1, 2, ..., M\}$? Where does this $2^M$ come from? Is this an errata?

Thanks to everyone who's trying to help out.

Here are the definitions of $(M, n)$ code and the average probability of error $P_e^{(n)}$ given in the book. Also the exact statement I have a problem with (highlighted):

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    $\begingroup$ Agreed, this is very probably a typo. $2^M$ does not make any sense in this context. Probably the authors already had the rate in mind, which is defined next. $\endgroup$ Jan 28, 2022 at 7:55
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    $\begingroup$ I’m voting to close this question because it is too local (typo induced confusion) and has no lasting value to the site. $\endgroup$ Jan 28, 2022 at 9:45
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    $\begingroup$ Please delete the question now that you know it is due to a typo. Leaving it will only serve to obfuscate search results on this topic. $\endgroup$ Jan 28, 2022 at 9:49
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    $\begingroup$ Please do not delete the question as there is no reason to do so. Incidentally, you might want to know your question has sparked a discussion: math.meta.stackexchange.com/questions/34519 $\endgroup$
    – YiFan Tey
    Jan 29, 2022 at 5:12

1 Answer 1

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Yes, this is an error. Well spotted.

The correct paragraph would be:

over the message set $\{1,2,...,M\}$

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