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.

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.

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).

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.

Partial λ-Geometries and Generalized Hadamard Matrices Over Groups

1979 ◽  
Vol 31 (3) ◽  
pp. 617-627 ◽  
Author(s):  
David A. Drake
Keyword(s):  
Existence Theorem ◽  
Hadamard Matrices ◽  
Natural Generalization ◽  
Latin Squares ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Transversal Designs ◽  
Complete Mappings

Section 1 of this paper contains all the work which deals exclusively with generalizations of Hadamard matrices. The non-existence theorem proven here (Theorem 1.10) generalizes a theorem of Hall and Paige [15] on the non-existence of complete mappings in certain groups.In Sections 2 and 3, we consider the duals of (Hanani) transversal designs; these dual structures, which we call (s, r, µ)-nets, are a natural generalization of the much studied (Bruck) nets which in turn are equivalent to sets of mutually orthogonal Latin squares. An (s, r, µ)-net is a set of s2µ points together with r parallel classes of blocks. Each class consists of s blocks of equal cardinality. Two non-parallel blocks meet in precisely µ points. It has been proven that r is always less than or equal to (s2µ – l) / (s – 1).

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