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

Felipe Ruiz, Esther Seiden
1974 Journal of combinatorial theory. Series A
We also show that for any p > 2n and rz even 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

A Hedayat
1973 Journal of combinatorial theory. Series A
In this note we utilize 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

A Hedayat
1978 Journal of combinatorial theory. Series A
A generalization 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

A. Hedayat, Ester Seiden
1974 Pacific Journal of Mathematics
These characteristics are very important if 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

Zhaoqi ZHANG
2019 DEStech Transactions on Computer Science and Engineering
He demonstrated 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

David Bedford
1993 Discrete Mathematics
We describe a general 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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