Characterization of Basic 5-Value Spectrum Functions Through Walsh-Hadamard Transform

The first and the third authors recently introduced a spectral construction of plateaued 5-value spectrum functions. In particular, design latter class requires specification integers $\{W(u):u\in \mathbb {F}^{n}_{2}\}$ , where notation="LaTeX">$W(u)\in \left\{{0, \pm 2^{\frac {n+s_{1}}{2}}, {n+s_{2}}{2}}}\right\}$ so that sequence is valid Boolean function (recovered using inverse Walsh transform). Technically, this done by allocating suitable support notation="LaTeX">$S=S^{[{1}]}\cup S^{[{2}]}\subset {F}^{n}_{2}$ notation="LaTeX">$S^{[i]}$ corresponds to those notation="LaTeX">$u \in {F} _{2}^{n}$ for which notation="LaTeX">$W(u)=\pm {n+s_{i}}{2}}$ . addition, two dual functions notation="LaTeX">$g_{[i]}:S^{[i]}\rightarrow {F}_{2}$ (with notation="LaTeX">$\#S^{[i]}=2^{\lambda _{i}}$ ) are employed specify signs through notation="LaTeX">$W(u)=2^{\frac {n+s_{i}}{2}}(-1)^{g_{[i]}(u)}$ notation="LaTeX">$u\in S^{[i]}$ whereas notation="LaTeX">$W(u)=0$ notation="LaTeX">$u\not S$ work, closely related problems considered. Firstly, (duals) notation="LaTeX">$g_{[i]}$ additionally satisfy so-called totally disjoint spectra property, fully characterized (so notation="LaTeX">$W(u)$ function) when notation="LaTeX">$S$ given as union affine subspaces Especially, dual themselves have supports, an efficient utilizes arbitrary bent (as duals on corresponding ambient spaces given. problem specifying inequivalent also addressed method ensures inequivalence property derived (sufficient condition being selection duals). second part we investigate with supports. For such function, show different orderings its employing Sylvester-Hadamard recursion actually induce equivalent.