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Page 1317 of Mathematical Reviews Vol. , Issue 96c [page]

1996 Mathematical Reviews  
Summary: “It is proved in this paper that for any integer n > 53, there exist 3 IMOILS (three incomplete mutually orthogonal idempotent Latin squares) if and only if v > 4n.” 96c:05026 05B15 Roberts, Charles  ...  E., Jr. (1-INS; Terre Haute, IN) Sets of mutually orthogonal Latin squares with “like subsquares”.  ... 

Page 7323 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
42, 46, 54, 58, 62, 66, 70, 94}.” 97m:05049 05B15 Zhang, Xiafu (PRC-KIT; Kunming); Zhang, Hangfu (1-SCA; Los Angeles, CA) Three mutually orthogonal idempotent Latin squares of order 18.  ...  Summary: “Three mutually orthogonal idempotent Latin squares of order 18 are constructed, which can be used to obtain 3 HMOLS of type 5'* and type 23'* and to obtain a (90,5, 1)-PMD.” 97m:05050 05B20 94899  ... 

Page 2086 of Mathematical Reviews Vol. , Issue 98D [page]

1998 Mathematical Reviews  
98d:05034 notice that idempotent orthogonal arrays of strength two are idempotent mutually orthogonal quasigroups. Then, we state some properties of idempotent orthogonal arrays.  ...  Mappings of Latin squares. (English summary) Linear Algebra Appl. 261 (1997), 251-268. Summary: “Let L, denote the set of n by n Latin squares.  ... 

A lower bound on permutation codes of distance n-1 [article]

Sergey Bereg, Peter Dukes
2019 arXiv   pre-print
A classical recursive construction for mutually orthogonal latin squares (MOLS) is shown to hold more generally for a class of permutation codes of length n and minimum distance n-1.  ...  When such codes of length p+1 are included as ingredients, we obtain a general lower bound M(n,n-1) > n^1.079 for large n, gaining a small improvement on the guarantee given from MOLS.  ...  Theorem 2. 3 . 3 If there exist r mutually orthogonal idempotent latin squares of order n, then there exists an r-IPC(n, n − 1). Proof.  ... 
arXiv:1902.04153v2 fatcat:thut5fnrgrfdfiu6hecw4udvji

Latin Squares with Self-Orthogonal Conjugates

Frank E. Bennett, Hantao Zhang
2004 Discrete Mathematics  
In this paper, we investigate the existence of idempotent Latin squares for which each conjugate is orthogonal to precisely its own transpose.  ...  As an application of our results, it is shown that for all integers v ¿ 8, with the possible exception of 10 and 11, there exists an idempotent Latin square of order v that realizes the one-regular graph  ...  Acknowledgements The ÿrst author would like to acknowledge the support of the Natural Sciences and Engineering Research Council of Canada under NSERC Grant OGP0005320 and the second author would like to  ... 
doi:10.1016/j.disc.2003.11.022 fatcat:lxnbrhdsofhl5fgxzjxovalr2y

Difference Covering Arrays and Pseudo-Orthogonal Latin Squares

Fatih Demirkale, Diane Donovan, Joanne Hall, Abdollah Khodkar, Asha Rao
2015 Graphs and Combinatorics  
We will use this connection to settle the spectrum question for sets of 3 mutually pseudo-orthogonal Latin squares of even order, for all but the order 146.  ...  A special class of pseudo-orthogonal Latin squares are the mutually nearly orthogonal Latin squares (MNOLS) first discussed in 2002, with general constructions given in 2007.  ...  A set of t Latin squares, of order n, are said to be mutually pseudo-orthogonal if they are pairwise pseudo-orthogonal.  ... 
doi:10.1007/s00373-015-1649-8 fatcat:lc4uqq2hvzf5fbcwumafdssg6u

Incomplete conjugate orthogonal idempotent latin squares

F.E Bennett, L Zhu
1987 Discrete Mathematics  
Let us denote by COILS(v) a (3, 2, 1)-conjugate orthogonal idempotent Latin square of order v, and by ICOILS(v, n) an incomplete COILS(v) missing a sub-COILS(n).  ...  The construction of an ICOILS(8, 2) has already been instrumental in the construction of a COILS(26), the existence of Which was unknown for some time. A necessary condition for the existence of an.  ...  Acknowledgment The first author acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada under grant A-5320.  ... 
doi:10.1016/0012-365x(87)90207-x fatcat:5fmdqjyzs5eotpm2lm3ev7z4s4

Further results on incomplete (3,2,1)- conjugate orthogonal idempotent Latin squares

F.E. Bennett, Lisheng Wu, L. Zhu
1990 Discrete Mathematics  
Acknowledgements The first author would like to thank the organizing committee of the Third Chinese Combinatorial Conference and the Mathematics Department of Suzhou University for their kind hospitality  ...  A portion of this work was carried out earlier while the second author was a visiting professor at Mount Saint Vincent University, and he gratefully acknowledges the hospitality accorded him.  ...  of T(k + 1, 1; m) or k -1 mutually orthogonal Latin squares (MOLS) of order m.  ... 
doi:10.1016/0012-365x(90)90267-l fatcat:ps7mfunm6vavtnyzn4gvxjjhce

Resolvable bibd and sols

Ronald D. Baker
1983 Discrete Mathematics  
Resolvable grid systems can be constructed from mutually orthogonal self-orthogonal latin squares (SOLS) with symmetric mate.  ...  Together these results prove, as a special case, that: if k -1 is an odd prime power and there exist J(k -2) mutually orthogonal SOLS of order n, with symmetric mate, then there exists a resolvable BIBD  ...  For the remainder of Ithis paper 'a set of SOLS' will mean a set of mutually orthogonal self-orthogonal latin squares, all idempotent, and the phrase 'h SOL!?  ... 
doi:10.1016/0012-365x(83)90003-1 fatcat:xr6bapk2gvchzfybhqrhlcji6i

Complementary partial resolution squares for Steiner triple systems

J.H. Dinitz, E.R. Lamken, A.C.H. Ling
2003 Discrete Mathematics  
array formed by the superposition of three mutually orthogonal Latin squares of order v where v ≡ 1 (mod 6), v ≥ 7, and v /  ...  Our main result is the existence of six complementary partial resolution squares for Steiner triple systems of order v which can be superimposed in a v × v array so that the resulting array is also the  ...  Acknowledgments Research of E.R. Lamken was supported by NSF grant DMS9753244.  ... 
doi:10.1016/s0012-365x(02)00471-5 fatcat:3bo5l7zinvbethed6fofw7auba

Page 6542 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews  
A Vi(m,t) leads to m idempotent pairwise orthogonal Latin squares of order (m7 +1)t+1 with one common hole of order t. For m= 3,4,5,6 and 7 the spectrum for V(m,t) has been de- termined.  ...  The author generalizes the notion of a transversal of a Latin square by defining a k-plex to be a partial Latin square of order n containing kn entries such that each row and column contains exactly k  ... 

Page 18 of Mathematical Reviews Vol. 55, Issue 1 [page]

1978 Mathematical Reviews  
self orthogonal Latin squares (SOLS) of order n having a symmetric and idempotent mate. (2) (kn, k, k—1) of size k and (n(k—1), k, k/2) of size kK—1, when k—1=3 (mod 4) is a prime power and there are  ...  Self-orthogonal Latin squares and BIBD’s.  ... 

Conjugate orthogonal quasigroups

K.T. Phelps
1978 Journal of combinatorial theory. Series A  
INTRODUCTION The construction of orthogonal quasigroups, or equivalently orthogonal latin squares, has long been an area of intense mathematical research, culminating in the celebrated disproof of the  ...  A quasigroup (S, .) of order n is equivalent to an O&n, 3) with (i, j, k) as a row if and only if i *j = k. There are other characterizations of quasigroups, latin squares, and orthogonal arrays.  ... 
doi:10.1016/0097-3165(78)90074-2 fatcat:r2gopbovcjefnoc2j6mzelaajq

The number of the non-full-rank Steiner triple systems [article]

Minjia Shi, Li Xu (Anhui University, Hefei, China), Denis S. Krotov (Sobolev Institute of Mathematics, Novosibirsk, Russia)
2018 arXiv   pre-print
We derive a formula for the number of different Steiner triple systems of order v and given 2-rank r_2<v and the number of Steiner triple systems of order v and given 3-rank r_3<v-1.  ...  The p-rank of a Steiner triple system B is the dimension of the linear span of the set of characteristic vectors of blocks of B, over GF(p).  ...  of idempotent totally symmetric latin squares of order u), Π u is the number of symmetric latin squares of order u, Λ u is the number of latin squares of order u.  ... 
arXiv:1806.00009v1 fatcat:petcjh7mo5cutcgzu7yq6q5n4e

Mutually orthogonal latin squares with large holes [article]

Peter J. Dukes, Christopher M. van Bommel
2014 arXiv   pre-print
More generally, if a set of t incomplete mutually orthogonal latin squares of order n have a common hole of order m, then n > (t+1)m.  ...  Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once.  ...  For prime powers q, there exist q − 2 mutually orthogonal idempotent latin squares of order q.  ... 
arXiv:1410.6743v1 fatcat:khupxleg3fbgfaqrifx7yubfgu
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