New construction for families of binary sequences with optimal correlation properties

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 (1974) 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 (1988) 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