Shannon limit: minimum Eb/N0 dependent on code rate r

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Wolfson

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code rate

I want to plot the minimum Eb/N0 in dB dependent on the code rate r=k/n where k=binary length of the information word and n=binary length of the code word. In the attached paper formula 12 explains how to do that.

The problem is I don't know how to calculate M^-1(r).
In formula 10 C=M(sigma^2) which means that M(sigma^2)=h-h(y|x)
I can't really solve h as well.

Can anybody help me?
It would be great to have formula 12 without a subfunction M() which means Eb/N0=... dependent only on r (=C).

Thanks in advance for every help!!!
 

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