  Low autocorrelation binary sequences (LABS) are very important for
communication applications. And it is a notoriously difficult computational
problem to find binary sequences with low aperiodic autocorrelations. The
problem can also be stated in terms of finding binary sequences with minimum
energy levels or maximum merit factor defined by M.J.E. Golay, F=N^2/2E, N and
E being the sequence length and energy respectively. Conjectured asymptotic
value of F is 12.32 for very long sequences. In this paper, a theorem has been
proved to show that there are finite number of possible energy levels, spaced
at an equal interval of 4, for the binary sequence of a particular length. Two
more theorems are proved to derive the theoretical minimum energy level of a
binary sequence of even and odd length of N to be N/2, and N-1/2 respectively,
making the merit factor equal to N and N^2/N-1 respectively. The derived
theoretical minimum energy level successfully explains the case of N =13, for
which the merit factor (F =14.083) is higher than the conjectured value.
Sequence of lengths 4, 5, 7, 11, 13 are also found to be following the
theoretical minimum energy level. These sequences are exactly the Barker
sequences which are widely used in direct-sequence spread spectrum and pulse
compression radar systems because of their low autocorrelation properties.
Further analysis shows physical reasoning in support of the conjecture that
Barker sequences exists only when N <= 13 (this has been proven for all odd N).
