For an aperiodic invertible sequence of length L, the optimality criterion is chosen to be the minimum noise enhancement factor of the corresponding aperiodic inverse filter. The noise enhancement factor is defined as the ratio of the noise energies at the outputs of the aperiodic inverse and the normalized matched filters of the sequence due to white noise at the filter inputs. Optimal binary sequences of length L/spl les/32 and optimal binary skew symmetric sequences of length 33/spl les/L/spl les/59 were found by exhaustive search. The search for near-optimal sequences is shown to have strong connections to the search for sequences with large products of Golay (1977) merit factor times the minimum of the energy density spectrum. Based on Legendre sequences and on Kronecker product sequences, long near-optimal binary sequences are found that have associated noise enhancement factors close to 1 dB.