  One of the fundamental ways to construct De Bruijn sequences is by using a
shift-rule. A shift-rule receives a word as an argument and computes the symbol
that appears after it in the sequence. An optimal shift-rule for an $(n,k)$-De
Bruijn sequence runs in time $O(n)$. We propose an extended notion we name a
generalized-shift-rule, which receives a word, $w$, and an integer, $c$, and
outputs the $c$ symbols that comes after $w$. An optimal generalized-shift-rule
for an $(n,k)$-De Bruijn sequence runs in time $O(n+c)$. We show that, unlike
in the case of a shift-rule, a time optimal generalized-shift-rule allows to
construct the entire sequence efficiently. We provide a time optimal
generalized-shift-rule for the well-known prefer-max and prefer-min De Bruijn
sequences.
