scholarly journals The geometric reason for the non-existence of a MOL(6)

2022 ◽  
Author(s):  
Dieter Betten
Keyword(s):  
Design Theory ◽  
Latin Squares ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Geometric Reason

AbstractThe problem of Euler concerning the 36 officers, (Euler, in Leonardi Euleri Opera Ser I 7:291–392, 1782), was first solved by Tarry (Comptes rendus Ass Franc Sci Nat 1 (1900) 2:170–203, 1901). Short proofs for the non-existence were given in Betten (Unterricht 36:449–453, 1983), Beth et al. (Design Theory, Bibl. Inst. Mannheim, Wien, Zürich, 1985), Stinson (J Comb Theory A 36:373–376, 1984). This problem is equivalent to the existence of a MOL(6), i. e., a pair of mutually orthogonal latin squares of order 6. Therefore in Betten (Mitt Math Ges Hamburg 39:59–76, 2019; Res Math 76:9, 2021; Algebra Geom 62:815–821, 2021) the structure of a (hypothetical) MOL(6) was studied. Now we combine the old proofs and the new studies and filter out a simple way for the proof of non-existence. The aim is not only to give still other short proofs, but to analyse the problem and reveal the geometric reason for the non-existence of a MOL(6)- and the non-solvability of Euler’s problem.

scholarly journals SOME CONSTRUCTIONS OF MUTUALLY ORTHOGONAL LATIN SQUARES AND SUPERIMPOSED CODES

2012 ◽  
Vol 04 (03) ◽  
pp. 1250022
Author(s):  
JENNIFER SEBERRY ◽  
DONGVU TONIEN
Keyword(s):  
Design Theory ◽  
Data Communication ◽  
Explicit Construction ◽  
Latin Squares ◽  
Combinatorial Structure ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Superimposed Codes ◽  
Alternative Approach ◽  
New Construction

Superimposed codes is a special combinatorial structure that has many applications in information theory, data communication and cryptography. On the other hand, mutually orthogonal latin squares is a beautiful combinatorial object that has deep connection with design theory. In this paper, we draw a connection between these two structures. We give explicit construction of mutually orthogonal latin squares and we show a method of generating new larger superimposed codes from an existing one by using mutually orthogonal latin squares. If n denotes the number of codewords in the existing code then the new code contains n2codewords. Recursively, using this method, we can construct a very large superimposed code from a small simple code. Well-known constructions of superimposed codes are based on algebraic Reed–Solomon codes and our new construction gives a combinatorial alternative approach.

Projective planes of infinite but isolic order

1976 ◽  
Vol 41 (2) ◽  
pp. 391-404 ◽  
Author(s):  
J. C. E. Dekker
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Latin Squares ◽  
Combinatorial Theory ◽  
Ternary Ring ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Planar Ternary Ring ◽  
Partial Recursive Functions

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.(I) For every finite projective plane Π there is a unique number n such that Π has exactly n2 + n + 1 points and exactly n2 + n + 1 lines.(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

New bounds of Mutually unbiased maximally entangled bases in C^dxC^(kd)

2018 ◽  
Vol 18 (13&14) ◽  
pp. 1152-1164
Author(s):  
Xiaoya Cheng ◽  
Yun Shang
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Prime Power ◽  
Latin Squares ◽  
Mutually Unbiased Bases ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Bipartite System

Mutually unbiased bases which is also maximally entangled bases is called mutually unbiased maximally entangled bases (MUMEBs). We study the construction of MUMEBs in bipartite system. In detail, we construct 2(p^a-1) MUMEBs in \cd by properties of Guss sums for arbitrary odd d. It improves the known lower bound p^a-1 for odd d. Certainly, it also generalizes the lower bound 2(p^a-1) for d being a single prime power. Furthermore, we construct MUMEBs in \ckd for general k\geq 2 and odd d. We get the similar lower bounds as k,b are both single prime powers. Particularly, when k is a square number, by using mutually orthogonal Latin squares, we can construct more MUMEBs in \ckd, and obtain greater lower bounds than reducing the problem into prime power dimension in some cases.

A Partial Generalization of Mann's Theorem Concerning Orthogonal Latin Squares

1988 ◽  
Vol 31 (4) ◽  
pp. 409-413 ◽  
Author(s):  
E. T. Parker ◽  
Lawrence Somer
Keyword(s):  
Latin Square ◽  
Latin Squares ◽  
Necessary Condition ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Complete Set

AbstractLetn = 4t+- 2, where the integert ≧ 2. A necessary condition is given for a particular Latin squareLof ordernto have a complete set ofn — 2mutually orthogonal Latin squares, each orthogonal toL.This condition extends constraints due to Mann concerning the existence of a Latin square orthogonal to a given Latin square.

scholarly journals On Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1895 ◽  
Author(s):  
M. Higazy ◽  
A. El-Mesady ◽  
M. S. Mohamed
Keyword(s):  
Computer Science ◽  
Finite Fields ◽  
Orthogonal Array ◽  
Orthogonal Arrays ◽  
Latin Squares ◽  
Error Correcting Codes ◽  
Recursive Construction ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Definition Of

During the last two centuries, after the question asked by Euler concerning mutually orthogonal Latin squares (MOLS), essential advances have been made. MOLS are considered as a construction tool for orthogonal arrays. Although Latin squares have numerous helpful properties, for some factual applications these structures are excessively prohibitive. The more general concepts of graph squares and mutually orthogonal graph squares (MOGS) offer more flexibility. MOGS generalize MOLS in an interesting way. As such, the topic is attractive. Orthogonal arrays are essential in statistics and are related to finite fields, geometry, combinatorics and error-correcting codes. Furthermore, they are used in cryptography and computer science. In this paper, our current efforts have concentrated on the definition of the graph-orthogonal arrays and on proving that if there are k MOGS of order n, then there is a graph-orthogonal array, and we denote this array by G-OA(n2,k,n,2). In addition, several new results for the orthogonal arrays obtained from the MOGS are given. Furthermore, we introduce a recursive construction method for constructing the graph-orthogonal arrays.

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