Code division multiplexing performance of minimum total-squared-correlation binary signature sets

The total-squared-correlation (TSC), maximum squared correlation (MSC), total asymptotic efficiency (TAE), and sum capacity (C_sum) of underloaded signature sets as well as the TSC and C_sum of overloaded signature sets are metrics that are optimized simultaneously over the real/complex field. We derive closed-form expressions for the MSC, TAE, and C_sum of minimum-TSC binary signature sets. Our expressions disprove the general equivalence of these performance metrics over the binary field and establish conditions on the number of signatures and signature length under which simultaneous optimization can or cannot be possible. We comment that the sum capacity loss of the recently designed minimum-TSC binary sets is found to be rather negligible in comparison with minimum-TSC real/complex-valued (Welch-bound-equality) sets.