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''orthogonal table, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o0702701.png" />''
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$#C+1 = 28 : ~/encyclopedia/old_files/data/O070/O.0700270 Orthogonal array,
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A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o0702702.png" />-dimensional [[Matrix|matrix]] whose entries are the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o0702703.png" />, and possessing the property that in each of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o0702704.png" />-dimensional submatrices any of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o0702705.png" /> possible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o0702706.png" />-dimensional vector-columns with these numbers as coordinates is found in the columns of this submatrix precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o0702707.png" /> times. The definition of an orthogonal array implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o0702708.png" />. One often considers the special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o0702709.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027011.png" />, which is then denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027012.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027013.png" />, an orthogonal array <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027014.png" /> is equivalent to a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027015.png" /> pairwise [[Orthogonal Latin squares|orthogonal Latin squares]]. For given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027016.png" />, the maximum value of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027017.png" /> has been determined only in a number of specific cases, such as, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027018.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027019.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027020.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027021.png" /> is odd and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027022.png" />.
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''orthogonal table,  $  \mathop{\rm OA} ( N, k, n, t, \lambda ) $''
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A $  ( k \times N) $-
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dimensional [[Matrix|matrix]] whose entries are the numbers $  1 \dots n $,  
 +
and possessing the property that in each of its $  ( t \times N) $-
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dimensional submatrices any of the $  n  ^ {t} $
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possible $  t $-
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dimensional vector-columns with these numbers as coordinates is found in the columns of this submatrix precisely $  \lambda $
 +
times. The definition of an orthogonal array implies that $  N = \lambda n  ^ {t} $.  
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One often considers the special case $  \mathop{\rm OA} ( N, k, n, t, \lambda ) $
 +
with $  t = 2 $
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and $  \lambda = 1 $,  
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which is then denoted by $  \mathop{\rm OA} ( n, k) $.  
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When $  k > 3 $,  
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an orthogonal array $  \mathop{\rm OA} ( n, k) $
 +
is equivalent to a set of $  k- 2 $
 +
pairwise [[Orthogonal Latin squares|orthogonal Latin squares]]. For given $  n, t, \lambda $,  
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the maximum value of the parameter $  k $
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has been determined only in a number of specific cases, such as, for example, $  k \leq  ( \lambda n  ^ {2} - 1)/( n- 1) $
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when $  t = 2 $,  
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or $  k _  \max  = t+ 1 $
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when $  \lambda $
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is odd and $  n = 2 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Dénes,  A.D. Keedwell,  "Latin squares and their applications" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Hall,  "Combinatorial theory" , Wiley  (1986)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Dénes,  A.D. Keedwell,  "Latin squares and their applications" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Hall,  "Combinatorial theory" , Wiley  (1986)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Regarding existence, the only general result for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027024.png" /> states the existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027026.png" /> (H. Hanani, cf. [[#References|[a1]]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027027.png" />, see [[Orthogonal Latin squares|Orthogonal Latin squares]]. In geometric terms, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070270/o07027028.png" /> is equivalent to a  "transversal designtransversal design" , respectively a  "netnet" ; cf. [[#References|[a1]]] for some fundamental results and [[#References|[a2]]] for a recent survey.
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Regarding existence, the only general result for $  t= 2 $
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and $  \lambda \neq 1 $
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states the existence of $  \mathop{\rm OA} ( \lambda n  ^ {2} , 7 , n, 2, \lambda ) $
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for all $  n \geq  2 $(
 +
H. Hanani, cf. [[#References|[a1]]]). For $  \lambda = 1 $,  
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see [[Orthogonal Latin squares|Orthogonal Latin squares]]. In geometric terms, an $  \mathop{\rm OA} ( \lambda n  ^ {2} , k, n, 2, \lambda ) $
 +
is equivalent to a  "transversal designtransversal design" , respectively a  "netnet" ; cf. [[#References|[a1]]] for some fundamental results and [[#References|[a2]]] for a recent survey.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Beth,  D. Jungnickel,  H. Lenz,  "Design theory" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Jungnickel,  "Latin squares, their geometries and their groups. A survey" , ''Proc. IMA Workshops on Coding and Design Theory Minneapolis, 1988'' , Springer  (to appear)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Beth,  D. Jungnickel,  H. Lenz,  "Design theory" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Jungnickel,  "Latin squares, their geometries and their groups. A survey" , ''Proc. IMA Workshops on Coding and Design Theory Minneapolis, 1988'' , Springer  (to appear)</TD></TR></table>

Latest revision as of 08:04, 6 June 2020


orthogonal table, $ \mathop{\rm OA} ( N, k, n, t, \lambda ) $

A $ ( k \times N) $- dimensional matrix whose entries are the numbers $ 1 \dots n $, and possessing the property that in each of its $ ( t \times N) $- dimensional submatrices any of the $ n ^ {t} $ possible $ t $- dimensional vector-columns with these numbers as coordinates is found in the columns of this submatrix precisely $ \lambda $ times. The definition of an orthogonal array implies that $ N = \lambda n ^ {t} $. One often considers the special case $ \mathop{\rm OA} ( N, k, n, t, \lambda ) $ with $ t = 2 $ and $ \lambda = 1 $, which is then denoted by $ \mathop{\rm OA} ( n, k) $. When $ k > 3 $, an orthogonal array $ \mathop{\rm OA} ( n, k) $ is equivalent to a set of $ k- 2 $ pairwise orthogonal Latin squares. For given $ n, t, \lambda $, the maximum value of the parameter $ k $ has been determined only in a number of specific cases, such as, for example, $ k \leq ( \lambda n ^ {2} - 1)/( n- 1) $ when $ t = 2 $, or $ k _ \max = t+ 1 $ when $ \lambda $ is odd and $ n = 2 $.

References

[1] J. Dénes, A.D. Keedwell, "Latin squares and their applications" , Acad. Press (1974)
[2] M. Hall, "Combinatorial theory" , Wiley (1986)

Comments

Regarding existence, the only general result for $ t= 2 $ and $ \lambda \neq 1 $ states the existence of $ \mathop{\rm OA} ( \lambda n ^ {2} , 7 , n, 2, \lambda ) $ for all $ n \geq 2 $( H. Hanani, cf. [a1]). For $ \lambda = 1 $, see Orthogonal Latin squares. In geometric terms, an $ \mathop{\rm OA} ( \lambda n ^ {2} , k, n, 2, \lambda ) $ is equivalent to a "transversal designtransversal design" , respectively a "netnet" ; cf. [a1] for some fundamental results and [a2] for a recent survey.

References

[a1] T. Beth, D. Jungnickel, H. Lenz, "Design theory" , Cambridge Univ. Press (1986)
[a2] D. Jungnickel, "Latin squares, their geometries and their groups. A survey" , Proc. IMA Workshops on Coding and Design Theory Minneapolis, 1988 , Springer (to appear)
How to Cite This Entry:
Orthogonal array. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_array&oldid=48073
This article was adapted from an original article by V.M. Mikheev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article