On the construction of sets of mutually orthogonal Latin squares and the falsity of a conjecture of Euler

Bose, R. C.; Shrikhande, S. S.

doi:10.1090/s0002-9947-1960-0111695-3

Cited by 89 publications

(38 citation statements)

References 10 publications

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“…We start with a well-known class. [7] first described these designs in the geometrical setting that we will give here: let n be a projective plane of order n = 2k (i.e. a 2-(«2 + « + 1, « + 1, 1 ) design) with an oval cf (i.e.…”

Section: Cf(3) = Cf(3) On the Other Hand When Cf(3) = Cf(3) If E =mentioning

confidence: 99%

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The moment map of a Lie group representation

Wildberger¹

1992

Trans. Amer. Math. Soc.

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Abstract. Given an m x m Hadamard matrix one can extract m2 symmetric designs on m -1 points each of which extends uniquely to a 3-design. Further, when m is a square, certain Hadamard matrices yield symmetric designs on m points. We study these, and other classes of designs associated with Hadamard matrices, using the tools of algebraic coding theory and the customary association of linear codes with designs. This leads naturally to the notion, defined for any prime p , of p-equivalence for Hadamard matrices for which the standard equivalence of Hadamard matrices is, in general, a refinement: for example, the sixty 24 x 24 matrices fall into only six 2-equivalence classes. In the 16x16 case, 2-equivalence is identical to the standard equivalence, but our results illuminate this case also, explaining why only the Sylvester matrix can be obtained from a difference set in an elementary abelian 2-group, why two of the matrices cannot be obtained from a symmetric design on 16 points, and how the various designs may be viewed through the lens of the four-dimensional affine space over the two-element field.

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Section: Cf(3) = Cf(3) On the Other Hand When Cf(3) = Cf(3) If E =mentioning

confidence: 99%

“…Proposition 9. If C is a binary doubly-even [24,12] code with the property that there exist seven coordinate places such that the projection onto these seven coordinates is the [7,4] Hamming code, then C is not the binary code of a Hadamard 3-(24, 12, 5) design.…”

Section: Other Examplesmentioning

confidence: 99%

The moment map of a Lie group representation

Wildberger¹

1992

Trans. Amer. Math. Soc.

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“…Two sudoku solutions of order 4 are shown in (1.1) below. [2] Two latin squares are said to be orthogonal if, upon superimposition, each ordered pair of entries occurs exactly once. For example, the latin squares in (1.1) are orthogonal: there is no repetition of ordered pairs upon superimposition, as is indicated in the array below.…”

Section: Background Motivation and Resultsmentioning

confidence: 99%

Mutually Orthogonal Families of Linear Sudoku Solutions

Lorch¹

2009

J. Aust. Math. Soc.

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“…Now 3 IS a primitive element of GS (7). Hence the design can be developed from the initial blocks {(I , 2), (3, 4)} E9 (1,2,4) 4. Designs Belonging to the Series (A), which leads to th e d esign When 4 t +3 is a Prime Power…”

Section: The Methods Of Symmetrically Repeated Differencesmentioning

confidence: 99%

“…This can be adapted for co ns tructing tournament designs. The method depe nds on the use of pairwise balanced designs of index unity which we re first introduced by Bose and Shrikhande [4] as auxiliary designs useful for the construction of mutually orthogonal Latin squares of nonprime power orders, esp ecially 4t + 2.…”

Section: X21-2) (63)mentioning

confidence: 99%

The bridge tournament problem and calibration designs for comparing pairs of objects

Bose¹,

Cameron²

1965

J. RES. NATL. BUR. STAN. SECT. B. MATH. MATH. PHYS.

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The c lassical tournam e nt problem caiJ s for arranging v individuals int o tea ms o f /) p layers so that a pla yer is teamed th e sa m e number of times with each of th e other pla ye rs and a lso that eac h player is pitted eq ually often aga in st each of th e other players. The play of th e tuurn a m en t resu lt s in th e determination of difference in p e rformance of th e various pairings of th e g roups. In th e s pecial case when p = 2 each team co n s ist s of two players and th e d es igri s are call e d brid ge tourn a me nt designs .In high prec is iun ca libration one ca n m eas ure only the difference be tw ee n t wo nominally eq ual g ro up s S(l that if u objects a re to be int e rco mpare d in groups of fJ objects, th en th e so lution s tu th e to urnam e nt probl e m provid e sc hedul es for th e gl'll uping. Th ese des ig ns are useful in we ig hing a nd any o th e r m eas ul'e m c nt s where th e obj ec ts tu b e meas ure d ca n b e co mbin ed intu gl'll ups without loss of prec is ion or acc uracy ill th e compari sons.Tili s pape r pre se nt s ge ne ral m e thud s fill' co nstru c tin g of brid ge lourn a m e nt d es ign s, i. e ., for Lh t' case wh c n p = 2, fur all /1 ~ 50.Key Wurd s : Ca libration , ca li bration designs. co mbin a tori a l analysis , diffe re nce se ts, e xpe riment d es ig ns' , in c omplete bloc k d es igns, tournam e nt s , weighing de s ign s.

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On the construction of sets of mutually orthogonal Latin squares and the falsity of a conjecture of Euler

Cited by 89 publications

References 10 publications

The moment map of a Lie group representation

The moment map of a Lie group representation

Mutually Orthogonal Families of Linear Sudoku Solutions

The bridge tournament problem and calibration designs for comparing pairs of objects

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