Inverse Hadamard transforms of two-level autocorrelation sequences

It is well-known that a balanced binary sequence {a_k} of period 2ⁿ-1 with two-level autocorrelation is constant on cyclotomic cosets, i.e. {a_2k}={a_k+r} for all k and some fixed value of r. Moreover, there is a cyclic shift of the original sequence for which r=0. Such two-level autocorrelation sequences are in one-to-one correspondence with cyclic Hadamard difference sets with parameters (2ⁿ-1,2^n-1-1,2^n-2-1). Perhaps best known among such sequences are the m-sequences, which correspond to Singer difference sets. For any primitive element α in GF(2ⁿ), the set of m-sequences is given by S_q={Tr(α^qk)}, (q, 2ⁿ-1)=1, where S_q and S_q´ are distinct m-sequences iff q and q´ belong to different cyclotomic cosets