Achieving the Welch bound with difference sets

Pengfei Xia, Shengli Zhou, Georgios B. Giannakis

Research output: Contribution to journalArticlepeer-review

323 Scopus citations


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 (Imax) is often desirable. Relying on tools from combinatorial number theory, we construct analytically optimal codebooks meeting, in certain cases, the Welch lower bound. When analytical constructions are not available, we develop an efficient numerical search method based on a generalized Lloyd algorithm, which leads to considerable improvement on the achieved Imax over existing alternatives. We also derive a composite lower bound on the minimum achievable Imax that is effective for any codebook size N.

Original languageEnglish (US)
Pages (from-to)1900-1907
Number of pages8
JournalIEEE Transactions on Information Theory
Issue number5
StatePublished - May 2005


  • Difference sets
  • Generalized Lloyd algorithm
  • Grassmannian line packing
  • Welch bound


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