A class of balanced binary sequences with optimal autocorrelation properties

The construction of a class of balanced binary sequences with optimal autocorrelation properties is described. Given any odd prime
p
and any positive integer
m
, a balanced
( \pm 1)
binary sequence of length
p^{m} - 1
whose cyclic autocorrelation function
c (\tau)
satisfies
c (0) = p^{m} - 1
, and, for
\tau \neq 0, c (\tau) = +2
or
-2
when
(p^{m} - 1)/2
is odd, and
c(\tau) = 0
or
-4
when
(p^{m} - 1)/2
is even is constructed. Optimality is proved by showing that every balanced binary sequence has at least two distinct out-of-phase correlation values which are at least as large as those obtained here.