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

Given the alphabet the problem investigated is the construction of the largest set of mutually orthogonal sequences of length under the following constraint on the form of the sequences. Each sequence is a concatenation of elements from but not all concatenations are allowed. Rather, the sign of the th element is negative if and only if the st element is negative with an odd subscript, or the 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 mutually orthogonal sequences under the phase constraint. If is the order of a Hadamard matrix, then , for even, and , for odd.