~N+1~ button presses are required just to confirm a length-~N~ name, regardless of its letters (an A button press upon selecting each letter, plus a final + button press at the end). Therefore, if ~K < N+1~, then no valid name exists. Otherwise, let ~B = K-(N+1)~ be the number of additional </> button presses required.

Let's consider how many such button presses each letter in a name contributes. Let ~m(p, x)~ be the number of button presses required to move the cursor to a letter ~x~ from the previous letter ~p~ (with the previous letter before the first one assumed to be A). We'll either repeatedly press < or >, whichever requires fewer presses. These two options require ~|x-p|~ and ~26-|x-p|~ button presses (in some order), giving us ~m(p, x) = \min(|x-p|, 26-|x-p|)~. We can observe that, given any previous letter ~p~, it's possible to choose some next letter ~x~ such that ~m(p, x)~ has any wanted value between ~0~ and ~13~, inclusive.

We can then observe that a valid name exists if and only if ~B \le 13N~. Furthermore, we can construct a name for a given ~B~ value by greedily choosing letters one by one such that ~m(p, x)~ is maximized each time (up to at most ~13~), but without exceeding the remaining allotment of required button presses.