A Barker code is a string of digits of length such that
for all .
Barker codes are used for pulse compression of radar signals. There are Barker codes
of lengths 2, 3, 4, 5, 7, 11, and 13, and it is conjectured that no longer Barker
codes exist. A list of known Barker codes up to reversal of digits and negation is
given below.
length
code
2
,
3
4
,
5
7
11
13
The number of candidate codes of length is therefore equal to the number of -bead black-white reversible strings 1, 2, 3, 6, 10, 20, 36,
72, ... (OEIS A005418), while the numbers of
Barker codes of order ,
3, ... are 2, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, and no further known codes at higher
lengths (OEIS A091704).
OpenAI (2026) announced that a real circulant matrix that is a Hadamard matrix has order 1 or 4. The
manuscript claims as a consequence that Barker codes of length greater than 1 exist
exactly at lengths 2, 3, 4, 5, 7, 11, and 13. This is an announced resolution of
the longer-length conjecture.
of length 
such that
.
Barker codes are used for pulse compression of radar signals. There are Barker codes
of lengths 2, 3, 4, 5, 7, 11, and 13, and it is conjectured that no longer Barker
codes exist. A list of known Barker codes up to reversal of digits and negation is
given below.
, 
, 
is therefore equal to the number of 
-bead black-white reversible strings 1, 2, 3, 6, 10, 20, 36,
72, ... (OEIS A005418), while the numbers of
Barker codes of order 
,
3, ... are 2, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, and no further known codes at higher
lengths (OEIS A091704).
