1. Tuscan Squares Stoyan Kapralov ACCT-2012 Pomorie, 15-21 June 2012

2. The Idea GOLOMB’S PUZZLE COLUMNTM IEEE Information TheorySociety Newsletter September 2010

3. The Beginning Solomon W. Golomb and Herbert Taylor, “Tuscan Squares – A New Family of Combinatorial Designs” Ars Combinatoria, 20 – B(1985), pp. 115–132.

4. Applications IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 36. NO. 4, JULY 1990 A New Construction of Two-Dimensional Arrays with the Window Property J. DENES AND A. D. KEEDWELL Abstract-Tuscan squares, row-complete latin squares and commafree codesare used to construct binary and nonbinary arrays with a certain window property. Such arrays have practical applicationsin the coding andtransmission of pictures.

5. Definitions - 1 • An rxn Tuscan-k rectangle has r rows and n columns • such that: • each row is apermutation of the ndifferent symbols • (2) for any two distinct symbols a and b,and for each m from 1 to k, there is at most one row in which b is m steps to theright of a. • Tuscan square oforder n: when r = n.

6. Definitions - 2 A Tuscan square is in standard form when the top row and the left-mostcolumn contain the symbols in the natural order. A Roman square is both Tuscanand latin, and was originally called a row complete latin square. A Tuscan-(n-1)rectangle is a Florentine rectangle, and is a Vatican rectangle when it is also latin.

7. Examples - 1 1 2 2 1 1 2 3 4 2 4 1 3 3 1 4 2 4 3 2 1 1 2 3 4 5 6 2 4 6 1 3 5 3 6 2 5 1 4 4 1 5 2 6 3 5 3 1 6 4 2 6 5 4 3 2 1

8. Examples -2 Tuscan but not Latin square 6 1 5 2 4 3 7 2 6 3 5 4 7 1 5 7 2 3 1 4 6 4 2 5 1 6 7 3 3 6 2 1 7 4 5 1 3 2 7 5 6 4 7 6 5 3 4 1 2

9. Examples - 3 A Tuscan-2 square of order 8 that is not Tuscan-3 1 2 3 4 5 6 7 8 2 1 6 8 7 3 5 4 3 2 7 1 8 4 6 5 4 1 7 5 3 8 6 2 5 8 1 4 7 2 6 3 6 1 5 2 4 8 3 7 7 4 2 8 5 1 3 6 8 2 5 7 6 4 3 1

10. Problems

11. Our problem If there exists a Tuscan-2 square of order 9 ?

12. Plan of attack • Reducethe task to the problem for searching cliques in graph. • Using the Cliquer program of Patric Ostergard for searching the cliques.

13. Realization • Graph preparation • Number of vertices: 56459 • Number of edges: 203140075 • We are searching for a clique of size 8.

14. The Result There is no Tuscan-2 square of order 9