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This is my understanding: the statistics of the source described in the paper depend on which character is produced first. If the first character is $a$, then one of the source properties is that letters in odd positions are always $a$. However, if the first letter is $b$ (in other words, a shift of $1$ in the circuit), then letters in odd positions can be ...

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\begin{align*}||Y - a_0||^2 &\overset{X = a_1}{\underset{X = a_0}{\gtrless}} ||Y - a_1||\\ \end{align*} can be written as: \begin{align*}(a_1 - a_0)^T Y &\overset{X = a_1}{\underset{X = a_0}{\gtrless}} \frac{||a_1||^2 - ||a_0||^2}{2}\\ \end{align*} Note that $g(Y) = \left(a_1 - a_0 \right)^T Y$ is a sufficient statistic and a scalar quantity. You ...

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Hint: do a change of variables so that the points $a_0$ and $a_1$ lie on one axis in $n$-space. This is a standard method that is used repeatedly in analysis of digital communication systems and it is good to get a handle on it right away.

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