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Denis Krotov

Publications and source records attributed to Denis Krotov.

At least 19 recordsLinked to original sources
arXiv2026

Closed Expressions for the Weight Distributions of Codes Associated with Perfect Codes

Perfect codes are arguably the most fascinating structures in combinatorial coding theory, and their classification and weight distribution are of considerable interest. This classification also involves the analysis of some related structures. This paper considers five closely related structures, but all of them have never been tied together before. These structures are 1-perfect codes, extended 1-perfect codes, nearly perfect 1-covering codes, extended nearly perfect 1-covering codes, and one family of completely regular codes (to be called diamond codes). The current work concentrates on the weight distributions of these five families of codes. In the past, some of these weight distributions were not computed, some required heavy tools, and for some only the weight enumerator was presented. We provide complete weight distributions for all five families using some methods that do not require any heavy tools.

math.CO↗
arXiv2026

The punctured dodecacode is unique

The punctured dodecacode is an additive $4$-ary code of length $11$ and distance $5$ which is uniformly packed. We show that a code with the same weight distribution is equivalent to it. This code is also shown to be nonlinear. We also establish the nonexistence of analogues of the dodecacode and the punctured dodecacode in Doob graphs. To that end, we classify two-weight codes of weights $6$ and $8$ in Doob and $4$-ary Hamming graphs of diameter $9$ and the corresponding strongly regular graphs. Keywords: dodecacode, additive code, trace Hermitian duality, uniformly packed code, completely regular code, Doob graph, strongly regular graph

math.CO↗
arXiv2025

The spectral Tur\'{a}n problem: Characterizing spectral-consistent graphs

Let ${\rm EX}(n,H)$ and ${\rm SPEX}(n,H)$ denote the families of $n$-vertex $H$-free graphs with the maximum size and the maximum spectral radius, respectively. A graph $H$ is said to be spectral-consistent if ${\rm SPEX}(n,H)\subseteq {\rm EX}(n,H)$ for sufficiently large $n$. A fundamental problem in spectral extremal graph theory is to determine which graphs are spectral-consistent. Cioab\u{a}, Desai and Tait [European J. Combin. 99 (2022) 103420] proposed the following conjecture: Let $H$ be any graph such that the graphs in ${\rm EX}(n,H)$ are Tur\'{a}n graph plus $O(1)$ edges. Then $H$ is spectral-consistent. Wang, Kang and Xue [J. Combin. Theory Ser. B 159 (2023) 20--41] confirmed this conjecture, along with a stronger result. Recently, Liu and Ning raised a general problem in spectral extremal graph theory: Characterize all graphs that are spectral-consistent. In this paper, we establish that for any finite graph \(H\), if its decomposition family is matching-good, then \(H\) is necessarily spectral-consistent. Notably, this structural condition is strictly weaker than the condition for spectral-consistency established by Wang, Kang, and Xue in their earlier work, thereby broadening the class of graphs known to satisfy the spectral-consistency property. Our main result enables us to fully characterize the spectral-consistency for several important families of forbidden graphs \(H\), including generalized color-critical graphs, odd-ballooning of trees and complete bipartite graphs, as well as edge blow-up of non-bipartite graphs and certain special bipartite graphs. Furthermore, we present a streamlined proof for an existing spectral-consistency result due to Chen, Lei, and Li, simplifying their original argument. Finally, we propose several open problems to motivate future research in this area.

math.CO↗
arXiv2025

Maximum Size $t$-Intersecting Families and Anticodes

The maximum size of $t$-intersecting families is one of the most celebrated topics in combinatorics, and its size is known as the Erd\H{o}s-Ko-Rado theorem. Such intersecting families, also known as constant-weight anticodes in coding theory, were considered in a generalization of the well-known sphere-packing bound. In this work we consider the maximum size of $t$-intersecting families and their associated maximum size constant-weight anticodes over alphabet of size $q >2$. It is proved that the structure of the maximum size constant-weight anticodes with the same length, weight, and diameter, depends on the alphabet size. This structure implies some hierarchy of constant-weight anticodes.

math.CO↗
arXiv2024

Completely regular codes in graphs covered by a Hamming graph

In Cayley graphs on the additive group of a small vector space over GF$(q)$, $q=2,3$, we look for completely regular (CR) codes whose parameters are new in Hamming graphs over the same field. The existence of a CR code in such Cayley graph $G$ implies the existence of a CR code with the same parameters in the corresponding Hamming graph that covers $G$. In such a way, we find several completely regular codes with new parameters in Hamming graphs over GF$(3)$. The most interesting findings are two new CR-$1$ (with covering radius~$1$) codes that are independent sets (such CR are equivalent to optimal orthogonal arrays attaining the Bierbrauer--Friedman bound) and one new CR-$2$. By recursive constructions, every knew CR code induces an infinite sequence of CR codes (in particular, optimal orthogonal arrays if the original code was CR-$1$ and independent). In between, we classify feasible parameters of CR codes in several strongly regular graphs.

math.CO↗
arXiv2022

Self-dual Hadamard bent sequences

A new notion of bent sequence related to Hadamard matrices was introduced recently, motivated by a security application ( Sol\'e et al, 2021). We study the self dual class in length at most $196.$ We use three competing methods of generation: Exhaustion, Linear Algebra and Groebner bases. Regular Hadamard matrices and Bush-type Hadamard matrices provide many examples. We conjecture that if $v$ is an even perfect square, a self-dual bent sequence of length $v$ always exist. We introduce the strong automorphism group of Hadamard matrices, which acts on their associated self-dual bent sequences. We give an efficient algorithm to compute that group.

math.CO↗
arXiv2019

Constructions of transitive latin hypercubes

A function $f:\{0,...,q-1\}^n\to\{0,...,q-1\}$ invertible in each argument is called a latin hypercube. A collection $(π_0,π_1,...,π_n)$ of permutations of $\{0,...,q-1\}$ is called an autotopism of a latin hypercube $f$ if $π_0f(x_1,...,x_n)=f(π_1x_1,...,π_n x_n)$ for all $x_1$, ..., $x_n$. We call a latin hypercube isotopically transitive (topolinear) if its group of autotopisms acts transitively (regularly) on all $q^n$ collections of argument values. We prove that the number of nonequivalent topolinear latin hypercubes grows exponentially with respect to $\sqrt{n}$ if $q$ is even and exponentially with respect to $n^2$ if $q$ is divisible by a square. We show a connection of the class of isotopically transitive latin squares with the class of G-loops, known in noncommutative algebra, and establish the existence of a topolinear latin square that is not a group isotope. We characterize the class of isotopically transitive latin hypercubes of orders $q=4$ and $q=5$. Keywords: transitive code, propelinear code, latin square, latin hypercube, autotopism, G-loop.

cs.IT↗
arXiv2019

Distance-2 MDS codes and latin colorings in the Doob graphs

The maximum independent sets in the Doob graphs D(m,n) are analogs of the distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove the characterization of these sets stating that every such set is semilinear or reducible. As related objects, we study vertex sets with maximum cut (edge boundary) in D(m,n) and prove some facts on their structure. We show that the considered two classes (the maximum independent sets and the maximum-cut sets) can be defined as classes of completely regular sets with specified 2-by-2 quotient matrices. It is notable that for a set from the considered classes, the eigenvalues of the quotient matrix are the maximum and the minimum eigenvalues of the graph. For D(m,0), we show the existence of a third, intermediate, class of completely regular sets with the same property.

math.CO↗
arXiv2019

The minimum volume of subspace trades

A subspace bitrade of type $T_q(t,k,v)$ is a pair $(T_0,T_1)$ of two disjoint nonempty collections of $k$-dimensional subspaces of a $v$-dimensional space $V$ over the finite field of order $q$ such that every $t$-dimensional subspace of $V$ is covered by the same number of subspaces from $T_0$ and $T_1$. In a previous paper, the minimum cardinality of a subspace $T_q(t,t+1,v)$ bitrade was established. We generalize that result by showing that for admissible $v$, $t$, and $k$, the minimum cardinality of a subspace $T_q(t,k,v)$ bitrade does not depend on $k$. An example of a minimum bitrade is represented using generator matrices in the reduced echelon form. For $t=1$, the uniqueness of a minimum bitrade is proved.

cs.DM↗
arXiv2019

Classification of the Z2Z4-linear Hadamard codes and their automorphism groups

A $Z_2Z_4$-linear Hadamard code of length $α+2β=2^t$ is a binary Hadamard code which is the Gray map image of a $Z_2Z_4$-additive code with $α$ binary coordinates and $β$ quaternary coordinates. It is known that there are exactly $[(t-1)/2]$ and $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$, with $α=0$ and $α\not=0$, respectively, for all $t\geq 3$. In this paper, it is shown that each $Z_2Z_4$-linear Hadamard code with $α=0$ is equivalent to a $Z_2Z_4$-linear Hadamard code with $α\not=0$; so there are only $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$. Moreover, the order of the monomial automorphism group for the $Z_2Z_4$-additive Hadamard codes and the permutation automorphism group of the corresponding $Z_2Z_4$-linear Hadamard codes are given.

cs.IT↗
arXiv2018

A new distance-regular graph of diameter 3 on 1024 vertices

The dodecacode is a nonlinear additive quaternary code of length $12$. By puncturing it at any of the twelve coordinates, we obtain a uniformly packed code of distance $5$. In particular, this latter code is completely regular but not completely transitive. Its coset graph is distance-regular of diameter three on $2^{10}$ vertices, with new intersection array $\{33,30,15;1,2,15\}$. The automorphism groups of the code, and of the graph, are determined. Connecting the vertices at distance two gives a strongly regular graph of (previously known) parameters $(2^{10},495,238,240)$. Another strongly regular graph with the same parameters is constructed on the codewords of the dual code. A non trivial completely regular binary code of length $33$ is constructed.

math.CO↗
arXiv2018

A new approach to the Kasami codes of type 2

The dual of the Kasami code of length $q^2-1$, with $q$ a power of $2$, is constructed by concatenating a cyclic MDS code of length $q+1$ over $F_q$ with a Simplex code of length $q-1$. This yields a new derivation of the weight distribution of the Kasami code, a new description of its coset graph, and a new proof that the Kasami code is completely regular. The automorphism groups of the Kasami code and the related $q$-ary MDS code are determined. New cyclic completely regular codes over finite fields a power of $2$, generalized Kasami codes, are constructed; they have coset graphs isomorphic to that of the Kasami codes. Another wide class of completely regular codes, including additive codes, as well as unrestricted codes, is obtained by combining cosets of the Kasami or generalized Kasami code.

cs.IT↗
arXiv2017

The extended 1-perfect trades in small hypercubes

An extended $1$-perfect trade is a pair $(T_0,T_1)$ of two disjoint binary distance-$4$ even-weight codes such that the set of words at distance $1$ from $T_0$ coincides with the set of words at distance $1$ from $T_1$. Such trade is called primary if any pair of proper subsets of $T_0$ and $T_1$ is not a trade. Using a computer-aided approach, we classify nonequivalent primary extended $1$-perfect trades of length $10$, constant-weight extended $1$-perfect trades of length $12$, and Steiner trades derived from them. In particular, all Steiner trades with parameters $(5,6,12)$ are classified.

math.CO↗
arXiv2016

On the number of maximum independent sets in Doob graphs

The Doob graph $D(m,n)$ is a distance-regular graph with the same parameters as the Hamming graph $H(2m+n,4)$. The maximum independent sets in the Doob graphs are analogs of the distance-$2$ MDS codes in the Hamming graphs. We prove that the logarithm of the number of the maximum independent sets in $D(m,n)$ grows as $2^{2m+n-1}(1+o(1))$. The main tool for the upper estimation is constructing an injective map from the class of maximum independent sets in $D(m,n)$ to the class of distance-$2$ MDS codes in $H(2m+n,4)$.

math.CO↗
arXiv2016

A partition of the hypercube into maximally nonparallel Hamming codes

By using the Gold map, we construct a partition of the hypercube into cosets of Hamming codes such that for every two cosets the corresponding Hamming codes are maximally nonparallel, that is, their intersection cardinality is as small as possible to admit nonintersecting cosets.

cs.IT↗
arXiv2016

On the number of n-ary quasigroups of finite order

Let $Q(n,k)$ be the number of $n$-ary quasigroups of order $k$. We derive a recurrent formula for Q(n,4). We prove that for all $n\geq 2$ and $k\geq 5$ the following inequalities hold: $({k-3}/2)^{n/2}(\frac{k-1}2)^{n/2} < log_2 Q(n,k) \leq {c_k(k-2)^{n}} $, where $c_k$ does not depend on $n$. So, the upper asymptotic bound for $Q(n,k)$ is improved for any $k\geq 5$ and the lower bound is improved for odd $k\geq 7$. Keywords: n-ary quasigroup, latin cube, loop, asymptotic estimate, component, latin trade.

math.CO↗
arXiv2016

On the Automorphism Groups of the Z2Z4-Linear 1-Perfect and Preparata-Like Codes

We consider the symmetry group of a $Z_2Z_4$-linear code with parameters of a $1$-perfect, extended $1$-perfect, or Preparata-like code. We show that, provided the code length is greater than $16$, this group consists only of symmetries that preserve the $Z_2Z_4$ structure. We find the orders of the symmetry groups of the $Z_2Z_4$-linear (extended) $1$-perfect codes. Keywords: additive codes, $Z_2Z_4$-linear codes, $1$-perfect codes, Preparata-like codes, automorphism group, symmetry group.

cs.IT↗