arXiv subjects

Denys Shcherbak

Publications and source records attributed to Denys Shcherbak.

5 recordsLinked to original sources

arXiv2024

Enumeration of Row-Column Designs

We computationally completely enumerate a number of types of row-column designs up to isotopism, including double, sesqui and triple arrays as known from the literature, and two newly introduced types that we call mono arrays and AO-arrays. We calculate autotopism group sizes for the designs we generate. For larger parameter values, where complete enumeration is not feasible, we generate examples of some of the designs, and generate exhaustive lists of admissible parameters. For some admissible parameter sets, we prove non-existence results. We also give some explicit constructions of sesqui arrays, mono arrays and AO-arrays, and investigate connections to Youden rectangles and binary pseud Youden designs.

math.CO↗

arXiv2024

Enumeration of Sets of Mutually Orthogonal Latin Rectangles

We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares and MOLR, namely that each Latin rectangle in a set of MOLR is isotopic to each other rectangle in the set. We call such a set of MOLR \emph{homogeneous}. In the course of doing this, we perform a complete enumeration of non-isotopic sets of $t$ mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$, for all $t < n$. Specifically, we keep track of homogeneous sets of MOLR, as well as sets of MOLR where the autotopism group acts transitively on the rectangles, and we call such sets of MOLR \emph{transitive}. We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are homogeneous and/or transitive in each step of the construction process. We use the prefix \emph{stepwise} to refer to sets of MOLR with this property. Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular graphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.

math.CO↗

arXiv2023

Geometry on Optimal Problem

We introduce an algorithm which can be directly used to feasible and optimum search in linear programming. Starting from an initial point the algorithm iteratively moves a point in a direction to resolve the violated constraints. At the same time, it ensures that previously fulfilled constraints are not breached during this process. The method is based on geometrical properties of n-dimensional space and can be used on any type of linear constraints (>, =, $\geq$), moreover it can be used when the feasible region is non-full-dimensional.

math.OC↗

arXiv2023

Small Youden Rectangles, Near Youden Rectangles, and Their Connections to Other Row-Column Designs

In this paper we first study $k \times n$ Youden rectangles of small orders. We have enumerated all Youden rectangles for a range of small parameter values, excluding the almost square cases where $k = n-1$, in a large scale computer search. In particular, we verify the previous counts for $(n,k) = (7,3), (7,4)$, and extend this to the cases $(11,5), (11,6), (13,4)$ and $(21,5)$. For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values, differing by 1, which we call \emph{near Youden rectangles}. For all the designs we generate, we calculate the order of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays. Finally, we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.

math.CO↗

arXiv2018

Triples of Orthogonal Latin and Youden Rectangles For Small Orders

We have performed a complete enumeration of non-isotopic triples of mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$. Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of $k \times 8$ rectangles which are formed by extending mutually orthogonal triples with non-trivial autotopisms one row at a time, and requiring that the autotopism group is non-trivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of $4 \times 8$ rectangles, formed by juxtaposing two selected copies of complete sets of MOLS of order 4.

math.CO↗