XIII. Orthogonal latin squares

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.

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An inequality for pseudo-subplanes of sets of orthogonal latin squares

Journal of Combinatorial Theory Series A ◽

10.1016/0097-3165(78)90034-1 ◽

1978 ◽

Vol 25 (1) ◽

pp. 76

Author(s):

E.T Parker

Keyword(s):

Latin Squares ◽

Orthogonal Latin Squares

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

Quantum Information and Computation ◽

10.26421/qic18.13-14-6 ◽

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.

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

Canadian Mathematical Bulletin ◽

10.4153/cmb-1988-059-7 ◽

1988 ◽

Vol 31 (4) ◽

pp. 409-413 ◽

Cited By ~ 2

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.

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On Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares

Symmetry ◽

10.3390/sym12111895 ◽

2020 ◽

Vol 12 (11) ◽

pp. 1895 ◽

Cited By ~ 1

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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