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The spectrum for large set of disjoint incomplete Latin squares

Jianguo Lei, Qingde Kang, Yanxun Chang
2001 Discrete Mathematics  
An incomplete Latin square LS(n + a; a) is a Latin square of order n + a with a missing subsquare of order a.  ...  A large set of disjoint LS(n + a; a)s, denoted by LDILS(n + a; a), consists of n disjoint LS(n + a; a)s.  ...  Zhu for his encouragement and the manuscript [8] provided by him.  ... 
doi:10.1016/s0012-365x(00)00410-6 fatcat:prie4edbf5dg3l4z5sq66xutai

Page 8403 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
[Zhu, Lie'] (PRC-SOO; Suzhou) Large set of disjoint incomplete Latin squares. (English summary) Bull. Inst. Combin. Appl. 29 (2000), 49-60.  ...  Summary: “Disjoint incomplete Latin squares (DILS) are useful in constructing maximum constant weight codes and generalized Steiner triple systems.  ... 

Page 2338 of Mathematical Reviews Vol. , Issue 2002D [page]

2002 Mathematical Reviews  
Wallis (1-SIL; Carbondale, IL) 2002d:05032 05B15 Lei, Jianguo (PRC-HNU,; Shijiazhuang) ; Kang, Qingde (PRC-HNU; Shijiazhuang) ; Chang, Yanxun (PRC-NJT; Beijing) The spectrum for large set of disjoint incomplete  ...  Summary: “An incomplete Latin square LS(m+a,a) is a Latin square of order n +a with a missing subsquare of order a.  ... 

Page 8081 of Mathematical Reviews Vol. , Issue 2003k [page]

2003 Mathematical Reviews  
The spectrum of self-orthogonal diagonal Latin squares (SODLS; see below) is completely described in this paper, and the spectrum of incomplete SODLS is determined up to three values of the parameters.  ...  The authors define a set of (G,k,2)-DFs to be a complete set of disjoint difference families (CDDF) if their base blocks partition, altogether, G — {0}.  ... 

Existence of good large sets of Steiner triple systems

Junling Zhou, Yanxun Chang
2009 Discrete Mathematics  
The concept of good large set of Steiner triple systems (or GLS in short) was introduced by Lu in his paper "on large sets of disjoint Steiner triple systems", [J.  ...  Lu, On large sets of disjoint Steiner triple systems, I-III, J. Combin. Theory (A) 34 (1983) 140-182].  ...  Acknowledgement The research was supported by NSFC grants No. 10771013 and 10831002.  ... 
doi:10.1016/j.disc.2008.11.008 fatcat:uxx6bj254rgudk7b5fw67kw3hy

Page 2212 of Mathematical Reviews Vol. , Issue 2001D [page]

2001 Mathematical Reviews  
Improvements are given on the existence results for MQOLS based on groups, and a new construction is given for sets of MQOLS based on groups from sets of mutually orthogonal Latin squares based on groups  ...  However, the more difficult problem of constructing large sets of pure disjoint MTS(v) (briefly LPMTS(v)) remains very much open.  ... 

Author index

2001 Discrete Mathematics  
Chang, The spectrum for large set of disjoint incomplete , S. and S. Li, Some new extremal self-dual codes with lengths 42, 44, 52, and 58 (1-3) 147}150 Zhang, Y., S. Pang and Y.  ...  Zhang, A few more r-orthogonal latin squares (1-3) 183}191 Zhu, L., see D. Wu (1-3) 137}145 Zuanni, F., see M. Buratti (  ... 
doi:10.1016/s0012-365x(01)00245-x fatcat:6ay6ni2bffb53pzzlwieqgjeca

Page 354 of Mathematical Reviews Vol. 55, Issue 2 [page]

1978 Mathematical Reviews  
A row complete Latin square [rectangle] defines a decomposition of the complete directed [undirected] graph into disjoint Hamiltonian paths, and a P-groupoid (quasigroup) defines a decomposition of the  ...  . , The bounds on the number of mutually orthogonal Latin and | F-squares, and the number of constraints in orthogonal arrays were derived using the orthogonal partitions of sets.  ... 

Page 4059 of Mathematical Reviews Vol. , Issue 97G [page]

1997 Mathematical Reviews  
The authors identify a new family of critical sets for back circulant Latin squares.  ...  A critical set is a set of entries from a Latin square of order n which (1) is contained in precisely one Latin square of order n and for which (2) if an entry is removed from the critical set, the remaining  ... 

Page 4210 of Mathematical Reviews Vol. , Issue 80K [page]

1980 Mathematical Reviews  
The spectrum of G, Spec(G), is defined as the set of all integers n such that there is a realization of G by Latin squares of order n.  ...  The main question is: For r and A fixed, how large can v be? For A=0 we have precisely the Latin squares and the answer is r. For A>0 apparently very little is known.  ... 

Page 3199 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
A large set of disjoint Kirkman triple systems of order v, or LKTS(wv), isa partition of the set of all 3-subsets of a v-set into v — 2 Kirkman triple systems of order v.  ...  For u = 1, the existence of a KS; (v; “4, 4) is equivalent to the existence of a doubly resolvable (v,k,2)-BIBD. The spectrum of KS;(v;1,1) or Room squares was completed by Mullin and Wallis in 1975.  ... 

Page 6774 of Mathematical Reviews Vol. , Issue 87m [page]

1987 Mathematical Reviews  
The result can obviously be reinterpreted in terms of bipartite graphs and incomplete Latin squares; in the latter case it follows that, if Z is an incomplete Latin square of side n comprising k completely  ...  In the paper under review the authors present a recursive construction in which, given a pair of doubly diagonal orthogonal Latin squares of order 2k with two other pairwise disjoint but common transversals  ... 

Small embeddings of partial directed cycle systems

C.C. Lindner, C.A. Rodger
1992 Bulletin of the Australian Mathematical Society  
In this paper, a generalisation of the Andersen, Hilton, Rodger Theorem for embedding partial idempotent latin squares is proved.  ...  Lindner proved in 1981 that a MTS of order n can be embedded in a MTS of order t for every admissible t ^ 2n + 1 [3].) available at https://doi.  ...  In particular, for a given m ^ 4, we can ask for the spectrum (that is, the set of all n such that an mDCS of order n exists) of mDCSs as well as for an embedding (as small as possible) of partial mDCSs  ... 
doi:10.1017/s0004972700011850 fatcat:dfcq63fdizhjpcr5v6xurqvcvi

Some recent developments on BIBDs and related designs

Lie Zhu
1993 Discrete Mathematics  
A certain number of related open problems are also discussed.  ...  Teirlinck "A completion of Lu's determination of the spectrum for large sets of disjoint S!  ...  A Latin square is said to be orthogonal idempotent Latin squares of order u, then (S, d) is a B[4, 12; u], where & self-orthogonal if the square and its transpose are orthogonal.  ... 
doi:10.1016/0012-365x(93)90016-m fatcat:7odxmq3bobfapi4dmiiw6v6ace

Diagonally cyclic latin squares

Ian M. Wanless
2004 European journal of combinatorics (Print)  
An explicit construction is given for a latin square of any odd order. The square is conjectured to be N ∞ and this has been confirmed up to order 10 000 by computer.  ...  The deepest of these concern sets of mutually orthogonal Parker squares and their interpretation in terms of orthogonal arrays.  ...  A latin square without proper subsquares is said to be N ∞ . The existence spectrum for N 2 squares is known but the spectrum for N ∞ squares is not completely solved.  ... 
doi:10.1016/j.ejc.2003.09.014 fatcat:mkverzvyhbhaxgxd7q4jp6dn4e
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