• Title/Summary/Keyword: Orthogonal Latin squares

Search Result 6, Processing Time 0.017 seconds

Orthogonal Latin squares of Choi Seok-Jeong (최석정의 직교라틴방진)

  • Kim, Sung-Sook;Khang, Mee-Kyung
    • Journal for History of Mathematics
    • /
    • v.23 no.3
    • /
    • pp.21-31
    • /
    • 2010
  • A latin square of order n is an $n{\times}n$ array with entries from a set of n numbers arrange in such a way that each number occurs exactly once in each row and exactly once in each column. Two latin squares of the same order are orthogonal latin square if the two latin squares are superimposed, then the $n^2$ cells contain each pair consisting of a number from the first square and a number from the second. In Europe, Orthogonal Latin squares are the mathematical concepts attributed to Euler. However, an Euler square of order nine was already in existence prior to Euler in Korea. It appeared in the monograph Koo-Soo-Ryak written by Choi Seok-Jeong(1646-1715). He construct a magic square by using two orthogonal latin squares for the first time in the world. In this paper, we explain Choi' s orthogonal latin squares and the history of the Orthogonal Latin squares.

CROSS-INTERCALATES AND GEOMETRY OF SHORT EXTREME POINTS IN THE LATIN POLYTOPE OF DEGREE 3

  • Bokhee Im;Jonathan D. H. Smith
    • Journal of the Korean Mathematical Society
    • /
    • v.60 no.1
    • /
    • pp.91-113
    • /
    • 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.

A Syndrome-distribution decoding MOLS L$_{p}$ codes

  • Hahn, S.;Kim, D.G.;Kim, Y.S.
    • Communications of Mathematical Education
    • /
    • v.6
    • /
    • pp.371-381
    • /
    • 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.

  • PDF

HOMOGENEOUS CONDITIONS FOR STOCHASTIC TENSORS

  • Im, Bokhee;Smith, Jonathan D.H.
    • Communications of the Korean Mathematical Society
    • /
    • v.37 no.2
    • /
    • pp.371-384
    • /
    • 2022
  • Fix an integer n ≥ 1. Then the simplex Πn, Birkhoff polytope Ωn, and Latin square polytope Λn 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 Ωn 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.

GENERALIZED LATIN SQUARE

  • Iranmanesh A.;Ashrafi A.R.
    • Journal of applied mathematics & informatics
    • /
    • v.22 no.1_2
    • /
    • pp.285-293
    • /
    • 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.

ON SOME MDS-CODES OVER ARBITRARY ALPHABET

  • Chang, Gyu Whan;Park, Young Ho
    • Korean Journal of Mathematics
    • /
    • v.9 no.2
    • /
    • pp.129-131
    • /
    • 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 $i$.

  • PDF