We obtain the following results. For any prime $q$ the minimal Hamming distance between distinct regular $q$-ary bent functions of $2n$ variables is equal to $q^n$. The number of $q$-ary regular bent functions at the distance $q^n$ from the quadratic bent function $Q_n=x_1x_2+\dots+x_{2n-1}x_{2n}$ is equal to $q^n(q^{n-1}+1)\cdots(q+1)(q-1)$ for $q>2$. The Hamming distance between distinct binary $s$-plateaued functions of $n$ variables is not less than $2^{\frac{s+n-2}{2}}$ and the Hamming distance between distinctternary $s$-plateaued functions of $n$ variables is not less than $3^{\frac{s+n-1}{2}}$. These bounds are tight. For $q=3$ we prove an upper bound on nonlinearity of ternary functions in terms of their correlation immunity. Moreover, functions reaching this bound are plateaued. For $q=2$ analogous result are well known but for large $q$ it seems impossible. Constructions and some properties of $q$-ary plateaued functions are discussed.

中文翻译：

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关于 q 元弯曲和平台函数

我们得到以下结果。对于任何素数 $q$，$2n$ 变量的不同正则 $q$-ary 弯曲函数之间的最小汉明距离等于 $q^n$。距离二次弯曲函数 $Q_n=x_1x_2+\dots+x_{2n-1}x_{2n}$ 距离 $q^n$ 处的 $q$-ary 正则弯曲函数的数量等于 $q^n( q^{n-1}+1)\cdots(q+1)(q-1)$ 对于 $q>2$。$n$变量的不同二元$s$-plateaued函数之间的汉明距离不小于$2^{\frac{s+n-2}{2}}$和不同三元$s$-plateaued函数之间的汉明距离$n$ 个变量不少于 $3^{\frac{s+n-1}{2}}$。这些界限很紧。对于 $q=3$，我们证明了三元函数在相关免疫方面的非线性上限。此外，达到此界限的函数处于稳定状态。对于 $q=2$ 类似的结果是众所周知的，但对于大 $q$ 似乎是不可能的。讨论了 $q$-ary 平台函数的构造和一些性质。