Three New Constructions of 5-valued Spectrum Functions with Totally Disjoint Spectra Duals

A function $f:\mathbb{F}_2^n \to {\mathbb{F}_2}$ is called a 5-valued spectrum function if the Walsh transform Wftakes the values $0, \pm {2^{\frac{{n + {s_1}}}{2}}}, \pm {2^{\frac{{n + {s_2}}}{2}}}$ for some different non-negative integers si,i = 1,2 with n + sieven. In [IEEE Transactions on Information Theory, 67 (2), 2021], by spectral method, Hodžić et al. characterized the so-called basic 5-valued spectrum functions f whose duals $f_{[i]}^{\ast}(i = 1,2)$ are totally disjoint spectra functions. For a special case that the corresponding functions $\overline {f_{[i]}^{\ast}} (i = 1,2)$ are basic plateaued functions, Hodžić et al. gave a construction of basic 5-valued spectrum functions. They left open problems to provide constructions of 5-valued spectrum functions f such that f are basic and the corresponding functions $\overline {f_{[i]}^{\ast}} (i = 1,2)$ are non-basic plateaued functions, or f are non-basic. In this paper, we provide three new constructions of 5-valued spectrum functions f with totally disjoint spectra duals $f_{[i]}^{\ast}(i = 1,2)$, including the case that f are basic and the corresponding functions $\overline {f_{[i]}^{\ast}} (i = 1,2)$ are non-basic plateaued functions, and the case that f are non-basic, which provide answers to the problems proposed by Hodžić et al.. Since non-basic 5-valued spectrum functions are EA-inequivalent to basic ones, two constructions in this paper can produce 5-valued spectrum functions which are EA-inequivalent to ones in [IEEE Transactions on Information Theory, 67 (2), 2021].