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Some results on construction of orthogonal Latin squares by the method of sum composition

1974
*
Journal of combinatorial theory. Series A
*

We also show that for any p > 2n and rz even

doi:10.1016/0097-3165(74)90048-x
fatcat:ttx5ecb6vvauvd3okdwdj5hcb4
*one*can*construct*an*orthogonal*pair*of**Latin**squares**of*size pa + n using*the**method**of**sum**composition*. ... A*method**of**sum**composition*for*construction**of*orthogona*Latin**squares*was introduced*by*A. Hedayat and E. Seiden [l]. ... However, our attempts to*construct*three mutually*orthogonal**Latin**squares*using*the**method**of**sum**composition*failed thus far. ...##
###
An application of sum composition: A self orthogonal latin square of order ten

1973
*
Journal of combinatorial theory. Series A
*

In this note we utilize

doi:10.1016/0097-3165(73)90027-7
fatcat:3vzuxvhq2nhhzhw23iybtkccq4
*the**sum**composition*technique, developed*by*Hedayat and Seiden, to produce a self*orthogonal**latin**square**of*order ten,*the*smallest unsettled order in*the*published literature. ... A*latin**square*is said to be self*orthogonal*if it is*orthogonal*to its own transpose. ... Sixteen is*the*smallest order that*one*can produce a self*orthogonal**latin**square**by*Horton's*result*. Horton's*result*also produces*some*orders which cannot be generated*by*Mendelsohn's*result*. ...##
###
A generalization of sum composition: Self orthogonal Latin square design with sub self orthogonal Latin square designs

1978
*
Journal of combinatorial theory. Series A
*

A generalization

doi:10.1016/0097-3165(78)90007-9
fatcat:ah55fokmyrawbebijuay2av4gq
*of**the*theory*of**sum**composition**of**Latin**square*designs is given. ... Additional*results*concerning sets*of**orthogonal**Latin**square*designs are also provided. ... Additional*results*concerning*orthogonal**Latin**square*designs based*on**the*theory*of**sum**composition*are also given throughout*the*paper. ...##
###
On the theory and application of sum composition of Latin squares and orthogonal Latin squares

1974
*
Pacific Journal of Mathematics
*

These characteristics are very important if

doi:10.2140/pjm.1974.54.85
fatcat:7e3nt24tcvblnglexbvnb3b5qa
*one*hopes to*construct*a set consisting*of*more than two*orthogonal**Latin**squares**by**the**sum**composition**method*. We wish to thank Mr. W. ... An application*of**sum**composition*for*the**construction**of*sets*of**orthogonal**Latin**squares*. ... Manuscripts, in duplicate if possible, may be sent to any*one**of**the*five editors. Please classify according to*the*scheme*of*Math. Rev. Index to Vol. 39 . ...##
###
Page 28 of Mathematical Reviews Vol. 48, Issue 1
[page]

1974
*
Mathematical Reviews
*

*The*present author

*constructs*

*the*first self

*orthogonal*

*Latin*

*square*

*of*order 10.

*The*

*method*used is

*the*

*sum*

*composition*

*method*

*of*

*the*author and E. ... Seiden [“

*On*a

*method*

*of*

*sum*

*composition*

*of*

*orthogonal*

*Latin*

*squares*. III”, Dept. Statis. and Probability, Res. Memorandum, No. RM-259, Michigan State Univ., East Lansing, Mich., 1970]. ...

##
###
Page 47 of Mathematical Reviews Vol. 58, Issue 1
[page]

1979
*
Mathematical Reviews
*

*Some*further

*results*are also given. {Reviewer’s remark:

*The*

*method*which is called

*sum*composi- tion

*of*

*Latin*

*squares*seems to be

*the*same as

*the*

*method*which has been introduced

*by*K. ... A generalization

*of*

*the*theory

*of*

*sum*

*composition*

*of*

*Latin*

*squares*is presented. ...

##
###
Page 3166 of Mathematical Reviews Vol. , Issue 90F
[page]

1990
*
Mathematical Reviews
*

Because

*of**the**composition*operation used in*the**construction**of*new designs, these*methods*produce designs mostly with large parameter values. ... Then there does not exist a complete set*of*4t mutually*orthogonal**Latin**squares*, each*orthogonal*to L.*One**of**the*authors (Parker) has pointed out to*the*reviewer that J. Bierbrauer [Geom. ...##
###
Page 5314 of Mathematical Reviews Vol. , Issue 85m
[page]

1985
*
Mathematical Reviews
*

{Reviewer’s remarks: (1)

*The*author’s*method*for*constructing*BCLSs is, he claims, closely related to*sum**composition*, which, he thinks, was introduced*by*A. Hedayat and E. Seiden [Pacific J. ... Using*the*algebraic theory*of**orthogonal**squares*developed*by*T. Evans [Amer. Math. ...##
###
Page 4141 of Mathematical Reviews Vol. , Issue 82j
[page]

1982
*
Mathematical Reviews
*

Chidambaraswamy (Toledo, Ohio)
82):05030
8705034
Seiden, Esther; Wu, Ching Jung 82j:05031

*On**construction**of*three mutually*orthogonal**Latin**squares**by**the**method**of**sum**composition*. ... equivalence classes*of*stationary*Latin**squares*each*of*which contains (n—1)!(n—2)! elements.*Results*found*by*B. Smetaniuk [“A new*construction**on**Latin**squares*. II. ...##
###
Page 1786 of Mathematical Reviews Vol. , Issue 92c
[page]

1992
*
Mathematical Reviews
*

This paper gives

*some*nonexistence re- sults for these generalized difference sets*by*using*methods*mainly borrowed from*the*study*of*difference sets, e.g. arguments involv- ing character*sums*, involutions ... Summary: “In this paper we give*the**construction**of*complete sets*of*mutually*orthogonal*F-*squares*with varying numbers*of*symbols*of**composite*order 2*t, for all positive integers t where an ...##
###
Page 1786 of Mathematical Reviews Vol. , Issue 92d
[page]

1992
*
Mathematical Reviews
*

This paper gives

*some*nonexistence re- sults for these generalized difference sets*by*using*methods*mainly borrowed from*the*study*of*difference sets, e.g. arguments involv- ing character*sums*, involutions ... Summary: “In this paper we give*the**construction**of*complete sets*of*mutually*orthogonal*F-*squares*with varying numbers*of*symbols*of**composite*order 2*t, for all positive integers t where an ...##
###
Page 4385 of Mathematical Reviews Vol. , Issue 81K
[page]

1981
*
Mathematical Reviews
*

A new

*construction**of*mutually*orthogonal**Latin**squares*is shown. ... These theorems gener- alize*the*theorems*of*Ryser and Cruse given earlier for*Latin**squares*. In Section 5, a simple*method**of**construction**by**the*filling-in*of*symbols procedure is provided. M. L. ...##
###
The Map, A New Method to Define Latin Square

2019
*
DEStech Transactions on Computer Science and Engineering
*

He demonstrated

doi:10.12783/dtcse/iccis2019/31931
fatcat:wfr6o2kcfnh55k23lilaoamkbi
*methods*for*constructing**orthogonal**Latin**squares*where 𝑛 is odd or a multiple*of*4. ...*By*exploiting*the*properties*of*bijection,*the*propositions and theorems*of**Latin**square*are rebuilt to be algebraic*results*instead*of**the*usual combinatoric*results*. ...*By**the*Fundamental Theorem*of*Arithmetic, each odd number larger than 1 is*the*product*of**some*odd prime numbers. Each minimal non-trivial*orthogonal**Latin*maps*of*odd prime level exists. ...##
###
Construction of orthogonal latin squares using left neofields

1993
*
Discrete Mathematics
*

We describe a general

doi:10.1016/0012-365x(93)90475-9
fatcat:tpd2ajr3tjdjvjivmbyyodddhm
*method**of**construction*for sets*of*mutually*orthogonal**latin**squares*(MOLS) from left neofields. ... .,*Construction**of**orthogonal**Latin**squares*using left neofields, Discrete Mathematics 115 (1993) 17-38. ... Acknowledgement*The*author is most grateful to A.D. Keedwell for*the*numerous helpful suggestions and comments made in*the*preparation*of*this paper. ...##
###
Page 1439 of Mathematical Reviews Vol. 48, Issue 5
[page]

1974
*
Mathematical Reviews
*

Ruiz, Felipe; Seiden, Esther 8267

*Some**results**on**construction**of**orthogonal**Latin**squares**by**the**method**of**sum**composition*. J. Combinatorial Theory Ser. A 16 (1974), 230-240. ... They also*construct*a pair*of**orthogonal**Latin**squares**of*order p?+ 2n for all primes p>4n, using*the*...
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