Hybrid character sums and near-optimal partial Hadamard codebooks

Abstract

In this paper, we first study exponential sums of the form∑x∈Fqψ(f(x))χ(g(x)) called hybrid character sums, where f,g are functions over finite field Fq, ψ is a nontrivial multiplicative character and χ is a nontrivial additive character of Fq. The absolute values or explicit values of them are determined under certain conditions. We then present several constructions of complex codebooks which are also called frames. The proposed codebooks are constructed from Hadamard matrices with binary row selection sequences. The hybrid character sums play an important role in determining the maximum magnitude of the inner products between vectors in these codebooks. It is proved that the magnitude of the inner products between distinct code vectors of each codebook is two-valued, and the maximum magnitude of each codebook asymptotically achieves the Welch bound equality. The constructed codebooks are equivalent to nearly equiangular tight frames and have nice application in compressed sensing matrices.