A Construction of Codebooks Asymptotically Meeting the Levenshtein Bound

Codebooks with small maximal cross-correlation amplitudes have important applications in code division multiple access (CDMA) communication, coding theory and compressed sensing. In this letter, we design a new codebook based on construction of Ramanujan graphs over finite abelian groups. We prove that the length K=q+1 size N=q2+2q+2 is asymptotically optimal nearly achieving Levenshtein bound when n=3, where q prime power. The parameters are new.