New construction for families of binary sequences with optimal correlation properties

In this correspondence, we present a construction, in a closed form, for an optimal family of 2^m binary sequences of period 2^2m - 1 with respect to Welch's bound, whenever there exists a balanced binary sequence of period 2^m - 1 with ideal autocorrelation property using the trace function. This construction enables us to reinterpret a small set of Kasami and No sequences as a family constructed from m-sequences. New optimal families of binary sequences are constructed from the Legendre sequences of Mersenne prime period, Hall's sextic residue sequences, and miscellaneous sequences of unknown type. In addition, we enumerate the number of distinct families of binary sequences, which are constructed from a given binary sequence by this method.