Critical Sets of Full n-Latin Squares

2015 ◽  
Vol 32 (2) ◽  
pp. 543-552 ◽  
Author(s):  
Nicholas J. Cavenagh ◽  
Vaipuna Raass
Keyword(s):  
Latin Squares ◽  
Critical Sets

scholarly journals A Discussion of a Cryptographical Scheme Based in F-Critical Sets of a Latin Square

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 285
Author(s):  
Laura M. Johnson ◽  
Stephanie Perkins
Keyword(s):  
Necessary Conditions ◽  
Latin Square ◽  
Latin Squares ◽  
Critical Sets ◽  
Long Cycles ◽  
The Given

This communication provides a discussion of a scheme originally proposed by Falcón in a paper entitled “Latin squares associated to principal autotopisms of long cycles. Applications in cryptography”. Falcón outlines the protocol for a cryptographical scheme that uses the F-critical sets associated with a particular Latin square to generate access levels for participants of the scheme. Accompanying the scheme is an example, which applies the protocol to a particular Latin square of order six. Exploration of the example itself, revealed some interesting observations about both the structure of the Latin square itself and the autotopisms associated with the Latin square. These observations give rise to necessary conditions for the generation of the F-critical sets associated with certain autotopisms of the given Latin square. The communication culminates with a table which outlines the various access levels for the given Latin square in accordance with the scheme detailed by Falcón.

scholarly journals A census of critical sets based on non-trivial autotopisms of Latin squares of order up to five

2021 ◽  
Vol 6 (1) ◽  
pp. 261-295
Author(s):  
Raúl M. Falcón ◽  
◽  
Laura Johnson ◽  
Stephanie Perkins ◽  
Keyword(s):  
Latin Squares ◽  
Critical Sets

Two remarks about Sudoku squares

2006 ◽  
Vol 90 (519) ◽  
pp. 425-430 ◽  
Author(s):  
A. D. Keedwell
Keyword(s):  
Current Problem ◽  
Latin Square ◽  
Latin Squares ◽  
Critical Set ◽  
Critical Sets ◽  
The Given ◽  
A Current ◽  
Defining Sets

Smallest defining setsA standard Sudoku square is a 9 × 9 latin square in which each of the nine 3 × 3 subsquares into which it can be separated contains each of the integers 1 to 9 exactly once.A current problem is to complete such a square when only some of the cells have been filled. These cells are often called ‘givens’. (Such problems are currently (2005) published daily in British newspapers.) In more mathematical terms, the given filled cells constitute a defining set or uniquely completable set for the square if they lead to a unique completion of the square. If, after deletion of any one of these givens, the square can no longer be completed uniquely, the givens form a critical set. The investigation of critical sets for ‘ordinary’ latin squares is a topic of current mathematical interest. (See [1] for more details.)

Critical sets in back circulant latin squares

1996 ◽  
Vol 52 (1) ◽  
pp. 157-179 ◽  
Author(s):  
Diane Donovan ◽  
Joan Cooper
Keyword(s):  
Latin Squares ◽  
Critical Sets

On the spectrum of critical sets in latin squares of order 2n

2007 ◽  
Vol 16 (1) ◽  
pp. 25-43 ◽  
Author(s):  
Diane Donovan ◽  
James LeFevre ◽  
G. H. John van Rees
Keyword(s):  
Latin Squares ◽  
Critical Sets

The minimum size of critical sets in latin squares

1997 ◽  
Vol 62 (2) ◽  
pp. 333-337 ◽  
Author(s):  
Chin-Mei Fu ◽  
Hung-Lin Fu ◽  
C.A. Rodger
Keyword(s):  
Latin Squares ◽  
Minimum Size ◽  
Critical Sets

Critical Sets in Latin Squares: An Intriguing Problem

2001 ◽  
Vol 85 (503) ◽  
pp. 239 ◽  
Author(s):  
A. Donald Keedwell
Keyword(s):  
Latin Squares ◽  
Critical Sets

scholarly journals Some results on defining sets of t-designs

1999 ◽  
Vol 59 (2) ◽  
pp. 203-215 ◽  
Author(s):  
Brenton D. Gray ◽  
Colin Ramsay
Keyword(s):  
Infinite Family ◽  
Latin Squares ◽  
Critical Sets ◽  
Defining Sets

We investigate how varying the parameters of t-(ν, κ, λ) designs affects the sizes of smallest defining sets. In particular, we consider the effect of varying each of the parameters t, ν and λ. We establish a number of new bounds for the sizes of smallest defining sets and find the size of smallest defining sets for an infinite family of designs. We also show how one of our results can be applied to the problem of finding critical sets of Latin squares.

scholarly journals Critical sets in latin squares and associated structures

2002 ◽  
Vol 66 (1) ◽  
pp. 175-176
Author(s):  
Richard Winston Bean
Keyword(s):  
Latin Squares ◽  
Critical Sets
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