  A Barker sequence is a binary sequence for which all nontrivial aperiodic
autocorrelations are either 0, 1 or -1. The only known Barker sequences have
length 2, 3, 4, 5, 7, 11 or 13. It is an old conjecture that no longer Barker
sequences exist and in fact, there is an overwhelming evidence for this
conjecture. For binary sequences of odd length, this conjecture is known to be
true, whereas for even length it is still open, whether a Barker sequence of
even length greater 4 exists.
  Similar to the well-known fact that a Barker sequence of odd length is
necessarily skew-symmetric, we show that in the case of even length there is
also a form of symmetry albeit weaker. In order to exploit this symmetry, we
derive different formulas for the calculation of the aperiodic correlation. We
prove by using only elementary methods that there is no Barker sequence of even
length n>4 with $C_{1}=C_{3}=\cdots=C_{\frac{n}{2}-1}$, where $C_{k}$ denotes
the $k$th aperiodic autocorrelation of the sequence.
