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Denis S. Krotov

Publications and source records attributed to Denis S. Krotov.

At least 37 records · Page 2Linked to original sources
arXiv2023

An enumeration of 1-perfect ternary codes

We study codes with parameters of the ternary Hamming $(n=(3^m-1)/2,3^{n-m},3)$ code, i.e., ternary $1$-perfect codes. The rank of the code is defined to be the dimension of its affine span. We characterize ternary $1$-perfect codes of rank $n-m+1$, count their number, and prove that all such codes can be obtained from each other by a sequence of two-coordinate switchings. We enumerate ternary $1$-perfect codes of length $13$ obtained by concatenation from codes of lengths $9$ and $4$; we find that there are $93241327$ equivalence classes of such codes. Keywords: perfect codes, ternary codes, concatenation, switching.

math.CO↗
arXiv2023

Perfect colorings of the infinite square grid: coverings and twin colors

A perfect coloring (equivalent concepts are equitable partition and partition design) of a graph $G$ is a function $f$ from the set of vertices onto some finite set (of colors) such that every node of color $i$ has exactly $S(i,j)$ neighbors of color $j$, where $S(i,j)$ are constants, forming the matrix $S$ called quotient. If $S$ is an adjacency matrix of some simple graph $T$ on the set of colors, then $f$ is called a covering of the target graph $T$ by the cover graph $G$. We characterize all coverings by the infinite square grid, proving that every such coloring is either orbit (that is, corresponds to the orbit partition under the action of some group of graph automorphisms) or has twin colors (that is, two colors such that unifying them keeps the coloring perfect). The case of twin colors is separately classified. Keywords: perfect coloring, equitable partition, partition design, square grid, rectangular grid, wallpaper group, twin colors, graph covering

math.CO↗
arXiv2022

Constructing MRD codes by switching

MRD codes are maximum codes in the rank-distance metric space on $m$-by-$n$ matrices over the finite field of order $q$. They are diameter perfect and have the cardinality $q^{m(n-d+1)}$ if $m\ge n$. We define switching in MRD codes as replacing special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting such switching, including punctured twisted Gabidulin codes and direct-product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in $m$ if the other parameters ($n$, $q$, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes. Keywords: MRD codes, rank distance, bilinear forms graph, switching, diameter perfect codes

cs.IT↗
arXiv2022

The existence of perfect codes in Doob graphs

We solve the problem of existence of perfect codes in the Doob graph. It is shown that 1-perfect codes in the Doob graph D(m,n) exist if and only if 6m+3n+1 is a power of 2; that is, if the size of a 1-ball divides the number of vertices. Keywords: perfect codes, distance-regular graphs, Doob graphs, Eisenstein-Jacobi integers.

cs.IT↗
arXiv2021

On the number of frequency hypercubes $F^n(4;2,2)$

A frequency $n$-cube $F^n(4;2,2)$ is an $n$-dimensional $4$-by-...-by-$4$ array filled by $0$s and $1$s such that each line contains exactly two $1$s. We classify the frequency $4$-cubes $F^4(4;2,2)$, find a testing set of size $25$ for $F^3(4;2,2)$, and derive an upper bound on the number of $F^n(4;2,2)$. Additionally, for any $n$ greater than $2$, we construct an $F^n(4;2,2)$ that cannot be refined to a latin hypercube, while each of its sub-$F^{n-1}(4;2,2)$ can. Keywords: frequency hypercube, frequency square, latin hypercube, testing set, MDS code

math.CO↗
arXiv2021

The number of the non-full-rank Steiner triple systems

The $p$-rank of a Steiner triple system $B$ is the dimension of the linear span of the set of characteristic vectors of blocks of $B$, over GF$(p)$. We derive a formula for the number of different Steiner triple systems of order $v$ and given $2$-rank $r_2$, $r_2<v$, and a formula for the number of Steiner triple systems of order $v$ and given $3$-rank $r_3$, $r_3<v-1$. Also, we prove that there are no Steiner triple systems of $2$-rank smaller than $v$ and, at the same time, $3$-rank smaller than $v-1$. Our results extend previous work on enumerating Steiner triple systems according to the rank of their codes, mainly by Tonchev, V.A.Zinoviev and D.V.Zinoviev for the binary case and by Jungnickel and Tonchev for the ternary case.

math.CO↗
arXiv2021

Zero sum sets in abelian groups

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 group, while in many special cases, including the case of elementary abelian groups, solved earlier by Li and Wan, it has a compact form. The proof involves two different Möbius transforms, on positive integers and on set partitions.

math.CO↗
arXiv2021

Perfect 2-colorings of Hamming graphs

We consider the problem of existence of perfect $2$-colorings (equitable $2$-partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we observe known constructions of perfect colorings and propose some new ones giving new parameters. At last, we deduce which parameters of colorings are covered by these constructions and give tables of admissible parameters of $2$-colorings in Hamming graphs $H(n,q)$ for small $n$ and $q$. Using the connection with perfect colorings, we construct an orthogonal array OA(2048,7,4,5).

math.CO↗
arXiv2020

On minimal subspace Zp-null designs

Let $q$ be a power of a prime $p$, and let $V$ be an $n$-dimensional space over the field GF$(q)$. A $Z_p$-valued function $C$ on the set of $k$-dimensional subspaces of $V$ is called a $k$-uniform $Z_p$-null design of strength $t$ if for every $t$-dimensional subspace $y$ of $V$ the sum of $C$ over the $k$-dimensional superspaces of $y$ equals $0$. For $q=p=2$ and $0\le t 2$, we give lower and upper bounds for that number.

math.CO↗
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

On unbalanced Boolean functions with best correlation immunity

It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in $n$ variables cannot exceed $2n/3-1$; moreover, it is $2n/3-1$ if and only if the function corresponds to an equitable $2$-partition of the $n$-cube with an eigenvalue $-n/3$ of the quotient matrix. The known series of such functions have proportion $1:3$, $3:5$, or $7:9$ of the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean function attains the correlation-immunity bound and has ratio $C:B$ of the number of ones and zeros, then $CB$ is divisible by $3$. In particular, this proves the nonexistence of equitable partitions for an infinite series of putative quotient matrices. We also establish that there are exactly $2$ equivalence classes of the equitable partitions of the $12$-cube with quotient matrix $[[3,9],[7,5]]$ and $16$ classes, with $[[0,12],[4,8]]$. These parameters correspond to the Boolean functions in $12$ variables with correlation immunity $7$ and proportion $7:9$ and $1:3$, respectively (the case $3:5$ remains unsolved). This also implies the characterization of the orthogonal arrays OA$(1024,12,2,7)$ and OA$(512,11,2,6)$.

math.CO↗
arXiv2019

On the OA(1536,13,2,7) and related orthogonal arrays

With a computer-aided approach based on the connection with equitable partitions, we establish the uniqueness of the orthogonal array OA$(1536,13,2,7)$, constructed in [D.G.Fon-Der-Flaass. Perfect $2$-Colorings of a Hypercube, Sib. Math. J. 48 (2007), 740-745] as an equitable partition of the $13$-cube with quotient matrix $[[0,13],[3,10]]$. By shortening the OA$(1536,13,2,7)$, we obtain $3$ inequivalent orthogonal arrays OA$(768,12,2,6)$, which is a complete classification for these parameters too. After our computing, the first parameters of unclassified binary orthogonal arrays OA$(N,n,2,t)$ attending the Friedman bound $N\ge 2^n(1-n/2(t+1))$ are OA$(2048,14,2,7)$. Such array can be obtained by puncturing any binary $1$-perfect code of length $15$. We construct orthogonal arrays with these and similar parameters OA$(N=2^{n-m+1},n=2^m-2,2,t=2^{m-1}-1)$, $m\ge 4$, that are not punctured $1$-perfect codes. Additionally, we prove that any orthogonal array OA$(N,n,2,t)$ with even $t$ attending the bound $N \ge 2^n(1-(n+1)/2(t+2))$ induces an equitable $3$-partition of the $n$-cube.

math.CO↗
arXiv2019

On the number of resolvable Steiner triple systems of small 3-rank

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 order $v$, where $v=3^k$, and $3$-rank $v-k$. We develop an approach to generalize this bound and estimate the number of isomorphism classes of STS$(v)$ of rank $v-k-1$ for an arbitrary $v$ of form $3^kT$.

math.CO↗
arXiv2019

The Steiner triple systems of order 21 with a transversal subdesign TD(3,6)

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 that there are 2004720 isomorphism classes of STS(21) including a subdesign TD(3,6), or, equivalently, a 6-by-6 latin square.

math.CO↗
arXiv2019

On the number of autotopies of an $n$-ary qusigroup of order $4$

An algebraic system from a finite set $Σ$ of cardinality $k$ and an $n$-ary operation $f$ invertible in each argument is called an $n$-ary quasigroup of order $k$. An autotopy of an $n$-ary quasigroup $(Σ,f)$ is a collection $(θ_0,θ_1,...,θ_n)$ of $n+1$ permutations of $Σ$ such that $f(θ_1(x_1),...,θ_n(x_n))\equiv θ_0(f(x_1,\ldots,x_n))$. We show that every $n$-ary quasigroup of order $4$ has at least $2^{[n/2]+2}$ and not more than $6\cdot 4^n$ autotopies. We characterize the $n$-ary quasigroups of order $4$ with $2^{(n+3)/2}$, $2\cdot 4^n$, and $6\cdot 4^n$ autotopies.

math.CO↗
arXiv2019

On dual codes in the Doob schemes

The Doob scheme $D(m,n'+n'')$ is a metric association scheme defined on $E_4^m \times F_4^{n'}\times Z_4^{n''}$, where $E_4=GR(4^2)$ or, alternatively, on $Z_4^{2m} \times Z_2^{2n'} \times Z_4^{n''}$. We prove the MacWilliams identities connecting the weight distributions of a linear or additive code and its dual. In particular, for each case, we determine the dual scheme, on the same set but with different metric, such that the weight distribution of an additive code $C$ in the Doob scheme $D(m,n'+n'')$ is related by the MacWilliams identities with the weight distribution of the dual code $C^\perp$ in the dual scheme. We note that in the case of a linear code $C$ in $E_4^m \times F_4^{n'}$, the weight distributions of $C$ and $C^\perp$ in the same scheme are also connected.

cs.IT↗