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Vladimir N. Potapov

Publications and source records attributed to Vladimir N. Potapov.

At least 19 recordsLinked to original sources
arXiv2026

On the Fourier Entropy-Influence Conjecture for Boolean Plateaued Functions

We prove the following inequality for Boolean functions: $2\sum_{x\in F_2^n}f(x)wt(x)\geq wt(f)(n-deg(f))$. Using this inequality, we establish the Fourier Entropy-Influence (FEI) conjecture for Boolean plateaued functions. In particular, we show that the sharp FEI constant for the class of plateaued functions is 4. We also prove the FEI conjecture for partially bent functions and show that the corresponding sharp constant is 2. Finally, we derive several estimates for the p-biased distribution on the Boolean hypercube. Keywords: Fourier entropy, total influence, average sensitivity, plateaued function, algebraic degree, Reed-Muller code, p-biased distribution.

cs.IT↗
arXiv2026

Perfect Combinatorial Structures in Coding Theory and Cryptography

This book develops algebraic and combinatorial methods for studying discrete structures. It brings together graph theory, Boolean functions, Fourier analysis on finite groups, coding theory, perfect colorings and perfect codes, association schemes, Latin squares, and related topics. A central theme is the interaction between different representations of the same object: combinatorial, algebraic, spectral, and coding-theoretic. The main mathematical object studied in this book is a perfect coloring of a graph, or, equivalently, an equitable partition of a graph. The book is intended for advanced undergraduate and graduate students in mathematics and computer science, as well as for researchers in discrete mathematics, combinatorics, coding theory, and related fields.

math.HO↗
arXiv2026

On the reconstruction of $n$-quasigroups of order $4$ and upper bounds on their number

(This is a reprint of the thesis published in the Transactions of the Conference Devoted to the 90th Anniversary Of Alexei A. Lyapunov, Novosibirsk, October 8-12, 2001, pages 323-327). The number of ways for partial $n$-quasigroups of order $4$ to be extended to $n$-quasigroups is investigated. Upper bounds on the number of $n$-quasigroups of order $4$ are obtained. The asymptotic $3^{n+1}2^{2^n+1}$ of that number is established.

math.CO↗
arXiv2026

Multidimensional convolution matrices and perfect colorings of subspace hypergraphs applied for bent functions and related designs

The main aim of the present paper is to introduce new methods for the study of combinatorial designs related to bent functions. They are based on interpretations of convolution on finite abelian groups as multiplication by a multidimensional matrix and designs as perfect colorings of subspace hypergraphs of $\mathbb{F}_2^n$. We establish a correspondence between eigenfunctions of convolution matrices and perfect colorings of subspace hypergraphs, show that perfect colorings of subspace hypergraphs admit a characterization in terms of convolution and that two-valued eigenfunctions of subspace hypergraphs correspond to perfect colorings. As applications, we represent partial difference sets, bent and plateaued Boolean functions, spreads, and strong bent partitions of $\mathbb{F}_2^n$ as eigenfunctions of convolution matrices and as perfect colorings of subspace hypergraphs. We also find some eigenvalues of convolution matrices over $\mathbb{F}_2^n$ and $\mathbb{F}_3^n$.

math.CO↗
arXiv2025

Generalizing the Bierbrauer-Friedman bound for orthogonal arrays

We characterize mixed-level orthogonal arrays in terms of algebraic designs in a special multigraph. We prove a mixed-level analog of the Bierbrauer-Friedman (BF) bound for pure-level orthogonal arrays and show that arrays attaining it are radius-1 completely regular codes (equivalently, intriguing sets, equitable 2-partitions, perfect 2-colorings) in the corresponding multigraph. For the case when the numbers of levels are powers of the same prime number, we characterize, in terms of multispreads, additive mixed-level orthogonal arrays attaining the BF bound. For pure-level orthogonal arrays, we consider versions of the BF bound obtained by replacing the Hamming graph by its polynomial generalization and show that in some cases this gives a new bound. Keywords: orthogonal array, algebraic t-design, completely regular code, equitable partition, intriguing set, Hamming graph, Bierbrauer-Friedman bound, additive codes.

math.CO↗
arXiv2025

On extremal properties of perfect 2-colorings

A coloring of vertices of a graph is called perfect if, for every vertex, the collection of colors of its neighbors depends only on its own color. The correspondent color partition of vertices is called equitable. We note that a number of bounds (Hoffman bound, Cheeger bound, Bierbrauer--Friedman bound and other) is only reached on perfect $2$-colorings. We show that the Expander Mixing Lemma is another example of an inequality that generates a perfect $2$-coloring. We prove a new upper bound for the size of $S\subset V(G)$ with the fixed average internal degree for an amply regular graph $G$. This bound is reached on the set $S$ if and only if $\{S, V(G)\setminus S\}$ is an equitable partition.

math.CO↗
arXiv2024

Partitioning the hypercube into smaller hypercubes

Denote by Q_d the d-dimensional hypercube. Addressing a recent question we estimate the number of ways the vertex set of Q_d can be partitioned into vertex disjoint smaller cubes. Among other results, we prove that the asymptotic order of this function is not much larger than the number of perfect matchings of Q_d. We also describe several new (and old) questions.

math.CO↗
arXiv2024

Asymptotic bounds on the numbers of vertices of polytopes of polystochastic matrices

A multidimensional nonnegative matrix is called polystochastic if the sum of entries in each line is equal to $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $Ω_n^d$. In the present paper, we compare known bounds on the number of vertices of the polytope $Ω_n^d$ and prove that the number of vertices of $Ω_3^d$ is doubly exponential on $d$.

math.CO↗
arXiv2024

An upper bound on the number of frequency hypercubes

A frequency $n$-cube $F^n(q;l_0,...,l_{m-1})$ is an $n$-dimensional $q$-by-...-by-$q$ array, where $q = l_0+...+l_{m-1}$, filled by numbers $0,...,m-1$ with the property that each line contains exactly $l_i$ cells with symbol $i$, $i = 0,...,m-1$ (a line consists of $q$ cells of the array differing in one coordinate). The trivial upper bound on the number of frequency $n$-cubes is $m^{(q-1)^{n}}$. We improve that lower bound for $n>2$, replacing $q-1$ by a smaller value, by constructing a testing set of size $s^{n}$, $s<q-1$, for frequency $n$-cubes (a testing sets is a collection of cells of an array the values in which uniquely determine the array with given parameters). We also construct new testing sets for generalized frequency $n$-cubes, which are essentially correlation-immune functions in $n$ $q$-valued arguments; the cardinalities of new testing sets are smaller than for testing sets known before. Keywords: frequency hypercube, correlation-immune function, latin hypercube, testing set.

math.CO↗
arXiv2022

DP-colorings of uniform hypergraphs and splittings of Boolean hypercube into faces

We develop a connection between DP-colorings of $k$-uniform hypergraphs of order $n$ and coverings of $n$-dimensional Boolean hypercube by pairs of antipodal $(n-k)$-dimensional faces. Bernshteyn and Kostochka established that the lower bound on edges in a non-2-DP-colorable $k$-uniform hypergraph is equal to $2^{k-1}$ for odd $k$ and $2^{k-1}+1$ for even $k$. They proved that these bounds are tight for $k=3,4$. In this paper, we prove that the bound is achieved for all odd $k\geq 3$.

math.CO↗
arXiv2021

Embedding in MDS codes and Latin cubes

An embedding of a code is a mapping that preserves distances between codewords. We prove that any code with code distance $ρ$ and length $d$ can be embedded into an MDS code with the same code distance and length but under a larger alphabet. As a corollary we obtain embeddings of systems of partial mutually orthogonal Latin cubes and $n$-ary quasigroups.

math.CO↗
arXiv2021

On weight spectrum of linear codes

We study sequences of linear or affine codes with uniform weight spectrum, i.e., a part of codewords with any fixed weight tends to zero. It is proved that a sequence of linear codes has a uniform weight spectrum if the number of vectors from codes with weight $1$ grows to infinity. We find an example of a sequence of linear codes such that the dimension of the code is the half of the codelength but it has not a uniform weight spectrum. This example generates eigenfunctions of the Fourier transform with minimal support and partial covering sets. Moreover, we generalize some MacWilliams-type identity. Keywords: weight distribution of code, dual code, MacWilliams identity, Fourier transform, partial covering array

cs.IT↗
arXiv2020

On multifold packings of radius-1 balls in Hamming graphs

A $λ$-fold $r$-packing (multiple radius-$r$ covering) in a Hamming metric space is a code $C$ such that the radius-$r$ balls centered in $C$ cover each vertex of the space by not more (not less, respectively) than $λ$ times. The well-known $r$-error-correcting codes correspond to the case $λ=1$, while in general multifold $r$-packing are related with list decodable codes. We (a) propose asymptotic bounds for the maximum size of a $q$-ary $2$-fold $1$-packing as $q$ grows; (b) prove that a $q$-ary distance-$2$ MDS code of length $n$ is an optimal $n$-fold $1$-packing if $q\ge 2n$; (c) derive an upper bound for the size of a binary $λ$-fold $1$-packing and a lower bound for the size of a binary multiple radius-$1$ covering (the last bound allows to update the small-parameters table); (d) classify all optimal binary $2$-fold $1$-packings up to length $9$, in particular, establish the maximum size $96$ of a binary $2$-fold $1$-packing of length $9$; (e) prove some properties of $1$-perfect unitrades, which are a special case of $2$-fold $1$-packings. Keywords: Hamming graph, multifold ball packings, two-fold ball packings, list decodable codes, multiple coverings, completely regular codes, linear programming bound

cs.DM↗
arXiv2020

Constructions of Pairs of Orthogonal Latin Cubes

A pair of orthogonal latin cubes of order $q$ is equivalent to an MDS code with distance $3$ or to an ${\rm OA}_1(3,5,q)$ orthogonal array. We construct pairs of orthogonal latin cubes for a sequence of previously unknown orders $q_i=16(18i-1)+4$ and $q'_i=16(18i+5)+4$. The minimal new obtained parameters of orthogonal arrays are ${\rm OA}_1(3,5,84)$. Keywords: latin square, latin cube, MOLS, MDS code, block design, Steiner system, orthogonal array

math.CO↗
arXiv2019

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.

cs.IT↗