• 대한수학회지
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• 제60권1호
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• pp.91-113
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• 2023

The polytope of tristochastic tensors of degree three, the Latin polytope, has two kinds of extreme points. Those that are at a maximum distance from the barycenter of the polytope correspond to Latin squares. The remaining extreme points are said to be short. The aim of the paper is to determine the geometry of these short extreme points, as they relate to the Latin squares. The paper adapts the Latin square notion of an intercalate to yield the new concept of a cross-intercalate between two Latin squares. Cross-intercalates of pairs of orthogonal Latin squares of degree three are used to produce the short extreme points of the degree three Latin polytope. The pairs of orthogonal Latin squares fall into two classes, described as parallel and reversed, each forming an orbit under the isotopy group. In the inverse direction, we show that each short extreme point of the Latin polytope determines four pairs of orthogonal Latin squares, two parallel and two reversed.

• Journal of applied mathematics & informatics
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• 제40권1_2호
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• pp.133-139
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• 2022

We offered a new method for constructing Latin squares. We introduce the concept of a standard form via example for Latin squares of order n and we also call it symmetric BCL algebras matrix, and thereby become BCL algebra representations of the picture of Latin squares. Our research shows that some new properties of the Latin squares with BCL algebras are in ℤ_n.

• 한국수학사학회지
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• 제23권3호
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• pp.21-31
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• 2010

2006년 이전까지도 유럽의 오일러가 직교라틴방진의 첫 연구자로서 인정을 받아왔다. 그러나 오일러 이전에 조선의 최석정이 오일러 이전에 이미 9차의 직교라틴 방진을 만들었다는 사실이 2006년 출판된 '조합론 디자인 편람' 에 소개됨으로써 우리만 알고 있던 사실이 세계적으로 공인되었다. 본 논문에서는 최석정과 양휘산법의 마방진을 비교하고 세계최초로 만들어진 최석정의 직교라틴방진과 오일러 가설의 역사를 설명한다.

• 대한수학회논문집
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• 제37권2호
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• pp.371-384
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• 2022

Fix an integer n ≥ 1. Then the simplex Π_n, Birkhoff polytope Ω_n, and Latin square polytope Λ_n each yield projective geometries obtained by identifying antipodal points on a sphere bounding a ball centered at the barycenter of the polytope. We investigate conditions for homogeneous coordinates of points in the projective geometries to locate exact vertices of the respective polytopes, namely crisp distributions, permutation matrices, and quasigroups or Latin squares respectively. In the latter case, the homogeneous conditions form a crucial part of a recent projective-geometrical approach to the study of orthogonality of Latin squares. Coordinates based on the barycenter of Ω_n are also suited to the analysis of generalized doubly stochastic matrices, observing that orthogonal matrices of this type form a subgroup of the orthogonal group.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.285-293
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• 2006

Let X be a n-set and let A = [aij] be a $n {\times} n$ matrix for which $aij {\subseteq} X$, for $1 {\le} i,\;j {\le} n$. A is called a generalized Latin square on X, if the following conditions is satisfied: $U^n_{i=1}\;aij = X = U^n_{j=1}\;aij$. In this paper, we prove that every generalized Latin square has an orthogonal mate and introduce a Hv-structure on a set of generalized Latin squares. Finally, we prove that every generalized Latin square of order n, has a transversal set.

• 한국수학교육학회지시리즈E:수학교육논문집
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• 제6권
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• pp.371-381
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• 1997

Let p be an odd prime number. We introduce simple and useful decoding algorithm for orthogonal Latin square codes of order p. Let H be the parity check matrix of orthogonal Latin square code. For any x ${\in}$ GF(p)$^{n}$, we call xH$^{T}$ the syndrome of x. This method is based on the syndrome decoding for linear codes. In L$_{p}$, we need to find the first and the second coordinates of codeword in order to correct the errored received vector.

• Asian-Australasian Journal of Animal Sciences
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• 제26권1호
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• pp.65-71
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• 2013

Two digestion trials, one with sheep and another with goats, were conducted to evaluate the long-term bias (LTB) of the indigestible dry matter (iDM), indigestible neutral detergent fiber (iNDF) and indigestible acid detergent fiber (iADF) internal markers. The study used eight Santa In$\hat{e}$s castrated male sheep (average body weight of 16.6 kg) distributed in two $4{\times}4$ Latin squares and eight Saanen castrated male goats (average body weight of 22.6 kg) distributed in two $4{\times}4$ Latin squares. The experiments were conducted simultaneously, and the animals were housed in 1.2 $m^2$ individual pens with wood-battened floors equipped with individual feeders and drinkers. The animals received isonitrogenous diets that were offered ad libitum and contained 14% crude protein and 70% sugar cane (with 0, 0.75, 1.5 or 2.25% CaO, in natural matter percentage), corrected with 1% urea and 30% concentrate. The experiment consisted of four experimental periods of 14 d each, with the feed, leftovers and feces sampled on the last four days of each period. The marker concentrations in the feed, leftovers and fecal samples were estimated by an in situ ruminal incubation procedure with a duration 240 h. The relationship between the intake and excretion of the markers was obtained by adjusting a simple linear regression model, independently from the treatment (diets) fixed effects and Latin squares. For both the sheep and goats, a complete recovery of the iDM and iNDF markers was observed (p>0.05), indicating the absence of LTB for these markers. However, the iADF was not completely recovered, exhibiting an LTB of -9.12% (p<0.05) in the sheep evaluation and -3.02% (p<0.05) in the goat evaluation.

• 대한수학회보
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• 제59권3호
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• pp.617-642
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• 2022

For positive integers n and d with d < n, an n × n array A based on 𝒳 = {0, 1, …, nd} is called a sparse anti-magic square of order n with density d, denoted by SAMS(n, d), if each non-zero element of X occurs exactly once in A, and its row-sums, column-sums and two main diagonal-sums constitute a set of 2n + 2 consecutive integers. An SAMS(n, d) is called regular if there are exactly d non-zero elements in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order n ≡ 1, 5 (mod 6), and prove that there exists a regular SAMS(n, d) for any n ≥ 5, n ≡ 1, 5 (mod 6) and d with 2 ≤ d ≤ n - 1.

• Korean Journal of Mathematics
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• 제9권2호
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• pp.129-131
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• 2001

Let $q=p^{e1}_1{\cdots}p^{em}_m$ be the product of distinct prime elements. In this short paper, we show that the largest value of M such that there exists an ($n$, M, $n-1$) $q$-ary code is $q^2$ if $n-1{\leq}p^{ei}_i$ for all $i$.

• Journal of applied mathematics & informatics
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• 제19권1_2호
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• pp.57-75
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• 2005

Quasigroups are algebraic structures closely related to Latin squares which have many different applications. There are several classifications of quasigroups based on their algebraic properties. In this paper we propose another classification based on the properties of strings obtained by specific quasigroup transformations. More precisely, in our research we identified some quasigroup transformations which can be applied to arbitrary strings to produce pseudo random sequences. We performed tests for randomness of the obtained pseudo-random sequences by random walks on torus. The randomness tests provided an empirical classification of quasigroups.