The Size of Optimal Sequence Sets for Synchronous CDMA Systems
  The sum capacity on a symbol-synchronous CDMA system having processing gain $N$ and supporting $K$ power constrained users is achieved by employing at most $2N-1$ sequences. Analogously, the minimum received power (energy-per-chip) on the symbol-synchronous CDMA system supporting $K$ users that demand specified data rates is attained by employing at most $2N-1$ sequences. If there are $L$ oversized users in the system, at most $2N-L-1$ sequences are needed. $2N-1$ is the minimum number of sequences needed to guarantee optimal allocation for single dimensional signaling. $N$ orthogonal sequences are sufficient if a few users (at most $N-1$) are allowed to signal in multiple dimensions. If there are no oversized users, these split users need to signal only in two dimensions each. The above results are shown by proving a converse to a well-known result of Weyl on the interlacing eigenvalues of the sum of two Hermitian matrices, one of which is of rank 1. The converse is analogous to Mirsky's converse to the interlacing eigenvalues theorem for bordering matrices. 
Rajesh Sundaresan and Arun Padakandla