On q-ary Bent and Plateaued Functions

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.Comment: 14 pages, the results are partialy reported on XV and XVI International Symposia "Problems of Redundancy in Information and Control Systems

On the cardinality spectrum and the number of latin bitrades of order 3

By a (latin) unitrade, we call a set of vertices of the Hamming graph that is intersects with every maximal clique in $0$ or $2$ vertices. A bitrade is a bipartite unitrade, that is, a unitrade splittable into two independent sets. We study the cardinality spectrum of the bitrades in the Hamming graph $H(n,k)$ with $k=3$ (ternary hypercube) and the growth of the number of such bitrades as $n$ grows. In particular, we determine all possible (up to $2.5\cdot 2^n$) and large (from $14\cdot 3^{n-3}$) cardinatities of bitrades and prove that the cardinality of a bitrade is compartible to $0$ or $2^n$ modulo $3$ (this result has a treatment in terms of a ternary code of Reed--Muller type). A part of the results is valid for any $k$. We prove that the number of nonequivalent bitrades is not less than $2^{(2/3-o(1))n}$ and is not greater than $2^{\alpha^n}$, $\alpha<2$, as $n\to\infty$.Comment: 18 pp. In Russia