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.

New construction of mutually unbiased bases in square dimensions

2005 ◽  
Vol 5 (2) ◽  
pp. 93-101
Author(s):  
P. Wocjan ◽  
T. Beth
Keyword(s):  
Lower Bounds ◽  
Prime Power ◽  
Hadamard Matrices ◽  
Latin Squares ◽  
Mutually Unbiased Bases ◽  
Asymptotic Growth ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
New Construction

We show that k=w+2 mutually unbiased bases can be constructed in any square dimension d=s^2 provided that there are w mutually orthogonal Latin squares of order s. The construction combines the design-theoretic objects (s,k)-nets (which can be constructed from w mutually orthogonal Latin squares of order s and vice versa) and generalized Hadamard matrices of size s. Using known lower bounds on the asymptotic growth of the number of mutually orthogonal Latin squares (based on number theoretic sieving techniques), we obtain that the number of mutually unbiased bases in dimensions d=s^2 is greater than s^{1/14.8} for all s but finitely many exceptions. Furthermore, our construction gives more mutually unbiased bases in many non-prime-power dimensions than the construction that reduces the problem to prime power dimensions.

scholarly journals Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture

1960 ◽  
Vol 12 ◽  
pp. 189-203 ◽  
Author(s):  
R. C. Bose ◽  
S. S. Shrikhande ◽  
E. T. Parker
Keyword(s):  
Prime Power ◽  
Arithmetic Function ◽  
Latin Squares ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Image Position

Ifis the prime power decomposition of an integer v, and we define the arithmetic function n(v) bythen it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. We shall denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v. Then the Mann-MacNeish theorem can be stated asMacNeish conjectured that the actual value of N(v) is n(v).

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.

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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