Regeneration of a Binary Signal in a Uniform Transmission Line

A "square" pulse, positive or negative, is sent along a uniform transmission line from
\ell = 0
and is regenerated at
\ell = L
, with some probability of error. The pulse is possibly regenerated at an intermediate point. In all the most reliable channels and in several others that we have examined, the best intermediate point for regeneration is the midpoint,
\ell = L/2
, but it is sometimes better still not to regenerate at any intermediate point. Some of the theory might be suggestive for explaining why the "nodes of Ranvier" are approximately equally spaced in neural axons. For some of the less reliable channels the midpoint is locally worst or even absolutely worst, and the optimal intermediate regeneration point is elsewhere, at say
\ell = \ell_{1}
(or equally good at
L - ell_{1}
, by symmetry).