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On the Distances between Latin Squares and the Smallest Defining Set Size
Author(s) -
Cavenagh Nicholas,
Ramadurai Reshma
Publication year - 2017
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21529
Subject(s) - latin square , mathematics , combinatorics , square (algebra) , set (abstract data type) , order (exchange) , discrete mathematics , statistics , geometry , computer science , chemistry , rumen , food science , finance , fermentation , economics , programming language
In this note, we show that for each Latin square L of order n ≥ 2 , there exists a Latin squareL ′ ≠ L of order n such that L and L ′ differ in at most 8 ncells. Equivalently, each Latin square of order n contains a Latin trade of size at most 8 n . We also show that the size of the smallest defining set in a Latin square is Ω ( n 3 / 2 ) .