Achieving the Welch bound with difference sets [optimal complex codebook design applications]

Consider a codebook containing N unit-norm complex vectors in a K-dimensional space. In a number of applications, the codebook that minimizes the maximal cross-correlation amplitude (I_max) is often desirable. Relying on tools from combinatorial design theory, we construct analytically optimal codebooks meeting, in certain cases, Welch´s lower bound. When analytical constructions are not available, we develop an efficient numerical search method based on Lloyd´s algorithm, which leads to considerable improvement on the achieved I_max over existing alternatives. We also derive a composite lower bound on the minimum achievable I_max that is effective for any N.