New Constructions of Codebooks Nearly Meeting the Welch Bound With Equality

An (N, K) codebook C is a set of N unit-norm complex vectors in BBC^K. Optimal codebooks meeting the Welch bound with equality are desirable in a number of areas. However, it is very difficult to construct such optimal codebooks. There have been a number of attempts to construct codebooks nearly meeting the Welch bound with equality, i.e., the maximal cross-correlation amplitude Imax(C) is slightly higher than the Welch bound equality, but asymptotically achieves it for large enough N. In this paper, using difference sets and the product of Abelian groups, we propose new constructions of codebooks nearly meeting the Welch bound with equality. Our methods yield many codebooks with new parameters. In some cases, our constructions are comparable to known constructions.