I just discovered the term "de Bruijn sequence", but don't quite follow what it means exactly (or how de Bruijn is pronounced :), "brown" I guess).

There are two good resources I am looking at for better understanding (it seems like they have a good model of them):

Looking first at the website, they say that $k$ is the number of symbols in the alphabet (so for binary it's 2), and $n$ is the length of each substring in an overall string (I don't know what the string is labeled as, maybe "DB sequence" I guess, de Bruijn sequence, of substrings). They show this as a $k = 2, n = 4$ system:

0000 0001 0010 0101 1011 0110 1101 1010
0100 1001 0011 0111 1111 1110 1100 1000

So I see, $n = 4$ (length of substring), and $k = 2$ ($1$ and $0$).

However, then I get lost at the sentence:

0000101101001111 is a DB sequence where the 16 unique substrings of length 4 visited in order are:

I get there are 16 unique substrings, but where does 0000101101001111 come from? Breaking it into 4, it is:


That's the 1, 5, 9, 13 elements of that sequence. Why did they just use that to describe this DB sequence?

Then in the linked paper, they show:

enter image description here

They say each of these lines is a DB sequence with $k = 2$ and $n = 6$. So from all that I conclude that the substring is length 6 in this case. So I break down the first line:


It doesn't even break into equal length substrings all of length $6$, what am I missing? I thought this should be substrings of length 6 which cover all possibilities of the substring combinations...

So going back to wikipedia, they have some examples, let's see.

Taking A = {0, 1}, there are two distinct B(2, 3): 00010111 and 11101000, one being the reverse or negation of the other.

where $B(k, n)$. Breaking the first down into 3 chunks, we have:

000 101 11

Also not chunks of 3. What am I misinterpreting?

  • 2
    $\begingroup$ The substrings overlap inside the bigger string. E.g.: within "0000101101001111", look at substring(0,3), substring(1,4), substring(2,5), etc... $\endgroup$
    – mhum
    Commented Apr 20, 2022 at 19:28
  • $\begingroup$ That answers it, thanks. $\endgroup$
    – Lance
    Commented Apr 21, 2022 at 4:54
  • $\begingroup$ Also, I believe the "sequence" in "DeBruijn sequence" refers to the sequence of symbols themselves (i.e.: in this case the ones and zeroes) rather than a sequence of substrings. $\endgroup$
    – mhum
    Commented Apr 21, 2022 at 18:59

1 Answer 1


A deBruijn sequence $B(k,n)$ has length $N=k^n$ and is cyclic i.e. it can be extended by repeating the sequence.

If it is $x_0,\ldots,x_{N-1}$ then the subsequences $$ x_0\cdots x_{k-1}\\ x_1 \cdots x_{k}\\ ~\\ \vdots\\ ~\\ x_{N-1} \cdots x_{N+k-1} $$ are all distinct and thus cover each $n-$tuple over an alphabet of size $k.$


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