On the construction ofN-ary orthogonal sequences under a continuous-phase constraint

Given the alphabet
A = \{ \pm a_0, \cdots, \pm a_i, \cdots, \pm a_{N-1} \}
the problem investigated is the construction of the largest set of mutually orthogonal sequences of length
n
under the following constraint on the form of the sequences. Each sequence is a concatenation of
n
elements from
A
but not all concatenations are allowed. Rather, the sign of the
j
th element is negative if and only if the
(j - 1)
st element is negative with an odd subscript, or the
(j - 1)
st element is positive with an even subscript. Such sequences have application in continuous-phase frequency-shift-keyed communication. The principal result is the construction of optimal sets of
\nu (N,n)
mutually orthogonal sequences under the phase constraint. If
n
is the order of a Hadamard matrix, then
\nu = nN/2
, for
N
even, and
n(N - 1)/2 + 1
, for
N
odd.