The Latin squares are combinatorial constructions that support communication through their use in frequency hopping designs, error correcting codes and encryption algorithms. They are quite simple and provide connective bridges between different mathematical fields. They find equivalent representation as a strongly regular graph (graph theory), an n 2×3 orthogonal matrix (matrix theory) and are generated through the algebraic operations of quasigroups (group theory). An unanswered question in security of Latin squares is how many cells of a Latin square require to be known by an adversary to allow the partial Latin square to generate a completed, unique Latin square. This minimal set of Latin square cells is known as a smallest critical set.
Papers:
J. Nelder, Critical sets in Latin squares, CSIRO division of Maths and Stats, Newsletter 38, p91, 1977.
D. Keedwell and J. Denes, Latin squares and their applications, 2nd ed., Elsevier, 2015, ISBN 978-0-444-63555-6.
J. J. Rotman, An introduction to the theory of groups, fourth edition, Graduate texts in mathematics, Springer, 1995.
I. M. Wanless, Cycle switches in Latin squares, Graphs and Combinatorics, Vol 20, Issue 4, 545-570 November 2004
J. D. H. Smith, Sylow theory for quasigroups, Journal of Combinatorial Designs, Vol 23, Issue 3, 115-133, 2015.
C. J. Colbourn and J. H. Dinitz, Handbook of combinatorial designs, 2nd ed. First Indian reprint, Chapman and Hall, 2014.
N. J. Cavenagh, R. Ramadurai, On the distances between Latin squares and the smallest critical sets size, arxiv preprint 1602.07734v1, February 2016.
N. J. Cavenagh, A superlinear lower bound for the size of a critical set in a Latin square, J. Combin Des., 15(4), 369-382, 2007.
- Badges
- Science topic
- Similar topics
- Mathematical Sciences
- Graphs