Optimal sequence sets meeting Welch's lower bound

Welch's lower bounds on total periodic and odd correlation energy of an equi-energy set of sequences are presented. It is shown that both bounds are simultaneously achieved precisely when the sequence set forms an aperiodic complementary sequence set, which has been extensively studied and is of independent interests. Then a lower bound closely related to an approximate SNR formula of Pursley (1977) for asynchronous DS/SSMA is derived. Our results are an extension of the works of Massey and Mittelholzer (see Sequences II: Methods in Communication, Security, and Computer Science, R. Capocelli, et al., Eds. New York: Springer-Verlag, 1993) for synchronous DS/SSMA.