55 citations · 628 across the 77 of their papers we have counts for
56 papers · 1 filter
Triangle decompositions of PG(n-1,2)
Minjia Shi, Xiaoxiao Li, Denis S. Krotov
We define a triangle design as a partition of the set of lines of a projective space into triangles, where a triangle consists of three pairwise intersecting lines with no common p…
Quasi-cyclic perfect codes in Doob graphs and special partitions of Galois rings
Minjia Shi, Xiaoxiao Li, Denis S. Krotov +1
The Galois ring GR is the residue ring , where is a basic primitive polynomial of degree over . For any odd larger than , we construct…
Do -Free Latin Squares Exist?
Aleksandr D. Krotov, Denis S. Krotov
We discuss the problem of the existence of latin squares without a substructure consisting of six elements , , , , $(r_3…
An upper bound on the number of frequency hypercubes
Denis S. Krotov, Vladimir N. Potapov
A frequency -cube is an -dimensional -by-...-by- array, where , filled by numbers with the property that each…
Multifold 1-perfect codes
Denis S. Krotov
A multifold -perfect code (-perfect code for list decoding) in any graph is a set of vertices such that every vertex of the graph is at distance not more than from ex…
Projective tilings and full-rank perfect codes
Denis S. Krotov
A tiling of a vector space is the pair of its subsets such that every vector in is uniquely represented as the sum of a vector from and a vector from . A til…
On the coset graph construction of distance-regular graphs
Minjia Shi, Denis S. Krotov, Patrick Solé
We show that no more new distance-regular graphs in the tables of the book of (Brouwer, Cohen, Neumaier, 1989) can be produced by using the coset graph of additive completely regul…
Self-dual Hadamard bent sequences
Minjia Shi, Yaya Li, Wei Cheng +3
A new notion of bent sequence related to Hadamard matrices was introduced recently, motivated by a security application ( Solé et al, 2021). We study the self dual class in length…
An enumeration of 1-perfect ternary codes
Minjia Shi, Denis S. Krotov
We study codes with parameters of the ternary Hamming code, i.e., ternary -perfect codes. The rank of the code is defined to be the dimension of its af…
On -ary shortened--perfect-like codes
Minjia Shi, Rongsheng Wu, Denis S. Krotov
We study codes with parameters of -ary shortened Hamming codes, i.e., . Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such…
Zero sum sets in abelian groups
Minjia Shi, Denis S. Krotov, Xiaoxiao Li +1
The distribution of cardinalities of zero-sum sets in abelian groups is completely determined. A complex summation involving the Möbius function is given for the general abelian gr…
On extended 1-perfect bitrades
Evgeny A. Bespalov, Denis S. Krotov
Extended -perfect codes in the Hamming scheme can be equivalently defined as codes that turn to -perfect codes after puncturing in any coordinate, as completely regu…
Equitable [[2,10],[6,6]]-partitions of the 12-cube
Denis S. Krotov
We describe the computer-aided classification of equitable partitions of the -cube with quotient matrix , or, equivalently, simple orthogonal arrays OA$(1536,12…
On minimal subspace Zp-null designs
Denis S. Krotov
Let be a power of a prime , and let be an -dimensional space over the field GF. A -valued function on the set of -dimensional subspaces of is cal…
Perfect colorings of the infinite square grid: coverings and twin colors
Denis S. Krotov
A perfect coloring (equivalent concepts are equitable partition and partition design) of a graph is a function from the set of vertices onto some finite set (of colors) suc…
On the number of frequency hypercubes
Minjia Shi, Shukai Wang, Xiaoxiao Li +1
A frequency -cube is an -dimensional -by-...-by- array filled by s and s such that each line contains exactly two s. We classify the frequency …
Perfect 2-colorings of Hamming graphs
Evgeny A. Bespalov, Denis S. Krotov, Aleksandr A. Matiushev +2
We consider the problem of existence of perfect -colorings (equitable -partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs a…
On the number of resolvable Steiner triple systems of small 3-rank
Minjia Shi, Li Xu, Denis S. Krotov
In a recent work, Jungnickel, Magliveras, Tonchev, and Wassermann derived an overexponential lower bound on the number of nonisomorphic resolvable Steiner triple systems (STS) of o…
The Steiner triple systems of order 21 with a transversal subdesign TD(3,6)
Yue Guan, Minjia Shi, Denis S. Krotov
We prove several structural properties of Steiner triple systems (STS) of order 3w+3 that include one or more transversal subdesigns TD(3,w). Using an exhaustive search, we find th…
On the OA(1536,13,2,7) and related orthogonal arrays
Denis S. Krotov
With a computer-aided approach based on the connection with equitable partitions, we establish the uniqueness of the orthogonal array OA, constructed in [D.G.Fon-Der…
On the number of autotopies of an -ary qusigroup of order
Denis S. Krotov, Evgeny V. Gorkunov, Vladimir N. Potapov
An algebraic system from a finite set of cardinality and an -ary operation invertible in each argument is called an -ary quasigroup of order . An autotopy of a…
On -resilient -functions
Denis S. Krotov
A -valued function on the vertices of the -cube is called a -resilient -function if it has the same number of s, s, s and s among the…
On completely regular codes of covering radius 1 in the halved hypercubes
Denis S. Krotov, Ivan Yu. Mogilnykh, Anastasia Yu. Vasil'eva
We consider constructions of covering-radius-1 completely regular codes, or, equivalently, equitable 2-partitions (regular 2-partitions, perfect 2-colorings), of halved n-cubes. Ke…
On unbalanced Boolean functions with best correlation immunity
Denis S. Krotov, Konstantin V. Vorob'ev
It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in variables cannot exceed ; moreover, it is if and only if the…