# On the Cardinality Spectrum and the Number of Latin Bitrades of Order 3

@article{Krotov2019OnTC,
title={On the Cardinality Spectrum and the Number of Latin Bitrades of Order 3},
author={Denis S. Krotov and Vladimir N. Potapov},
journal={Probl. Inf. Transm.},
year={2019},
volume={55},
pages={343-365}
}
• Published 2 December 2018
• Mathematics, Computer Science
• Probl. Inf. Transm.
By a (latin) unitrade, we call a set of vertices of the Hamming graph that is intersects with every maximal clique in $0$ or $2$ vertices. A bitrade is a bipartite unitrade, that is, a unitrade splittable into two independent sets. We study the cardinality spectrum of the bitrades in the Hamming graph $H(n,k)$ with $k=3$ (ternary hypercube) and the growth of the number of such bitrades as $n$ grows. In particular, we determine all possible (up to $2.5\cdot 2^n$) and large (from $14\cdot 3^{n-3… 2 Citations Splitting of hypercube into k-faces and DP-colorings of hypergraphs We develop a connection between DP-colorings of$k$-uniform hypergraphs of order$n$and coverings of$n$-dimensional hypercube by pairs of antipodal$(n-k)$-dimensional faces. Bernshteyn and On the Number of Frequency Hypercubes F$ {}^{n}(4;2,2) $• Mathematics Siberian Mathematical Journal • 2021 A frequency hypercube that cannot be refined to a latin hypercube, while each of its sub-$F^{n-1}(4;2,2)$can. ## References SHOWING 1-10 OF 39 REFERENCES Multidimensional Latin bitrades The sizes of the components of M DS codes and the possibility of obtaining Latin bitrades of a size given from MDS codes are studied and each MDS code is shown to embed in a Hamiltonian 2-fold M DS code. On the number of n-ary quasigroups of finite order • Mathematics • 2009 Let$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$the following inequalities hold: Cardinality spectra of components of correlation immune functions, bent functions, perfect colorings, and codes For bent functions, it is shown that for any of these combinatorial objects the component cardinality in the interval from 2k to 2k+1 can only take values of the form 2k-1 − 2p, where p ∈ {0, ..., k} and 2k is the minimum component cardinalities for a combinatorsial object with the same parameters. On large subsets of$F_q^n$with no three-term arithmetic progression • Mathematics • 2016 In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of$F_q^n$with no three terms in arithmetic progression by$c^n$with$c < q$. For$q=3$, the n-Ary Quasigroups of Order 4 • Mathematics SIAM J. Discret. Math. • 2009 Every n-ary quasigroups of order 4 is permutably reducible or semilinear, which means that an$n$-aryQuasigroup can be represented as a composition of$k-ary and $(n-k+1)$-aries for some $k$ from 2 to $n-1$, where the order of arguments in the representation can differ from the original order.
On the Volumes and Affine Types of Trades
• Mathematics
Electron. J. Comb.
• 2020
By considering the affine rank, it is proved that $[t]$-trades of Type\,(B) do not exist and the spectrum of volumes of simple trades up to $2.5\cdot 2^t$ is derived, extending the known result for volumes less than 1.5.
On the weight structure of Reed-Muller codes
• Computer Science
IEEE Trans. Inf. Theory
• 1970
This theorem completely characterizes the codewords of the \nu th-order Reed-Muller code whose weights are less than twice the minimum weight and leads to the weight enumerators for thosecodewords.
On the gaps of the spectrum of volumes of trades
Using the weight distribution of the Reed–Muller code, it is proved the conjecture that for every i from 2 to t, there are no t-trades of volume greater than 2t+1−2i and less than 1 and restrictions on the t-trade volumes are derived.