2025/06/29 by Ziling Heng, Peng Wang, Heng, Ziling +3
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #graph theory and CDMA systems #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2506.23198
Hybrid character sums are an important class of exponential sums which have nice applications in coding theory and sequence design. Let \gfpm be the finite field with pm elements for a prime p and a positive integer m. Let Vn(p) be an n-dimensional vector space over \gfp for a prime p. In this paper, we study the hybrid character sums of the form ∑x ∈ Vn(p)ψ(F(x))χ1(a x), where F is a function from Vn(p) to \gfpm and a ∈ Vn(p), ψ is a nontrivial multiplicative character of \gfpm and χ1 is the canonical additive character of Vn(p). If F(x) is a vectorial dual-bent function and a ∈ Vn(p)∖ \0\, we determine their complex modulus or explicit values under certain conditions, which generalizes some known results as special cases. It is concluded that the hybrid character sums from vectorial dual-bent functions have very small complex modulus. As applications, three families of asymptotically optimal complex codebooks are constructed from vectorial dual-bent functions and their maximal cross-correlation amplitude are determined based on the hybrid character sums. The constructed codebooks have very small alphabet sizes, which enhances their appeal for implementation. Besides, all of the three families of codebooks have only two-valued or three-valued cross-correlation amplitudes.