Tuscan Squares Stoyan Kapralov ACCT2012 Pomorie 15 21

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

The Idea GOLOMB’S PUZZLE COLUMNTM IEEE Information Theory Society Newsletter September 2010 2

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

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 codes are used to construct binary and nonbinary arrays with a certain window property. Such arrays have practical applications in the coding and transmission of pictures. 4

Definitions - 1 An r x n Tuscan-k rectangle has r rows and n columns such that: (1) each row is a permutation of the n different 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 the right of a. Tuscan square of order n: when r = n. 5

Definitions - 2 A Tuscan square is in standard form when the top row and the leftmost column contain the symbols in the natural order. A Roman square is both Tuscan and 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. 6

Examples - 1 12 21 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 7

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

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

Problems kn 2 3 4 5 1 1 0 2 3 4 5 6 7 8 9 10 11 12 0 1 0 0 6 7 8 736 466144 ≥ 3. 107 1 1 0 0 0 6 0 0 0 9 ≥ 3. 107 ? 0 0 0 10 11 12 13 ≥ 72 ≥ 1 ≥ 964 ≥ 1 ≥ 2 ≥ 1 ≥ 1 ≥ 1 1 1 ? ? ? ? 0 0 ≥ 2 ≥ 1 ≥ 1 ≥ 1 ? ? ? 10

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

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

Realization n Graph preparation n Number of vertices: 56 459 Number of edges: 203 140 075 n We are searching for a clique of size 8. n 13

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