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

中文翻译：

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通过 Walsh-Hadamard 变换表征基本 5 值谱函数

第一和第三作者最近介绍了平稳和 5 值谱函数的谱构造。特别是后一类的设计需要整数的规范 $\{W(u):u\in \mathbb {F}^{n}_{2}\}$ ， 在哪里 $W(u)\in \left\{{0, \pm 2^{\frac {n+s_{1}}{2}}, \pm 2^{\frac {n+s_{2}}{ 2}}}\右\}$ ，使得序列 $\{W(u):u\in \mathbb {F}^{n}_{2}\}$ 是布尔函数的有效谱（使用逆沃尔什变换恢复）。从技术上讲，这是通过分配合适的 Walsh 支持来完成的 $S=S^{[{1}]}\cup S^{[{2}]}\subset \mathbb {F}^{n}_{2}$ ， 在哪里 $S^{[i]}$ 对应于那些 $u \in \mathbb {F} _{2}^{n}$ 为此 $W(u)=\pm 2^{\frac {n+s_{i}}{2}}$ . 此外，两双 职能 $g_{[i]}:S^{[i]}\rightarrow \mathbb {F}_{2}$ （和 $\#S^{[i]}=2^{\lambda _{i}}$ ) 用于指定符号通过 $W(u)=2^{\frac {n+s_{i}}{2}}(-1)^{g_{[i]}(u)}$ 为了 $u\in S^{[i]}$ 然而 $W(u)=0$ 为了 $u\not \in S$ . 在这项工作中，考虑了两个密切相关的问题。一、平台函数的规范（对偶） $g_{[i]}$ ，另外满足所谓的完全不相交的光谱特性，被完全表征（使得 $W(u)$ 是布尔函数的谱），当 Walsh 支持 $S$ 作为两个不相交的仿射子空间的并集给出 $S^{[i]}$ . 尤其是，当双重函数趋于平稳时 $g_{[i]}$ 本身具有仿射沃尔什支持，这是一种利用任意弯曲函数的有效谱设计（作为对偶 $g_{[i]}$ ) 在相应的环境空间上给出。还解决了指定仿射不等价 5 值谱函数的问题，并导出了一种确保不等价性质的有效构造方法（充分条件是选择仿射不等价对偶）。在这项工作的第二部分，我们研究了具有仿射 Walsh 支持的平台函数的对偶。对于给定的这种平稳函数，我们表明，采用 Sylvester-Hadamard 递归的 Walsh 支持的不同排序实际上会导致仿射等效的弯曲对偶。