3 research outputs found
Investigating the discrepancy property of de Bruijn sequences
The discrepancy of a binary string refers to the maximum (absolute)
difference between the number of ones and the number of zeroes over all
possible substrings of the given binary string. We provide an investigation of
the discrepancy of known simple constructions of de Bruijn sequences.
Furthermore, we demonstrate constructions that attain the lower bound of
and a new construction that attains the previously known upper
bound of . This extends the work of Cooper and
Heitsch~[\emph{Discrete Mathematics}, 310 (2010)].Comment: 15 page
Constructing de Bruijn sequences by concatenating smaller universal cycles
We present sufficient conditions for when an ordering of universal cycles
for disjoint sets can be concatenated together to obtain a
universal cycle for . When is the set of all -ary strings of
length , the result of such a successful construction is a de Bruijn
sequence. Our conditions are applied to generalize two previously known de
Bruijn sequence constructions and then they are applied to develop three new de
Bruijn sequence constructions
An Efficient Generalized Shift-Rule for the Prefer-Max De Bruijn Sequence
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 -De
Bruijn sequence runs in time . We propose an extended notion we name a
generalized-shift-rule, which receives a word, , and an integer, , and
outputs the symbols that comes after . An optimal generalized-shift-rule
for an -De Bruijn sequence runs in time . 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