The number of different reduced complete sets of MOLS corresponding to the Desarguesian projective planes

Transcription

1 The number of different reduced complete sets of MOLS corresponding to the Desarguesian projective planes Vrije Universiteit Brussel joint work with K. Hicks, G.L. Mullen and L. Storme July 22, 2013 Fq11: International Conference on Finite Fields and Applications

2 1 Definitions Latin squares Mutually orthogonal latin squares 2 Projective planes 3 On the number of (n 1)MOLS(n) 4 Computational results 5 Bibliography

3 Latin squares Latin square Definition A latin square of order n is an n n matrix in which n distinct symbols from a symbol set S are arranged, such that each symbol occurs exactly once in each row and in each column.

4 Latin squares Reduced latin square Definition We say that a latin square of order n is reduced (or in standard form), if the first row and the first column of this latin square is in the natural order of the symbol set we chose.

5 Latin squares Example Symbol set S = {0, 1, 2,..., n 1}. Example A reduced latin square of order n on the symbol set S n 2 n n n n 3 n 2

6 Mutually orthogonal latin squares Mutually orthogonal latin squares Definition Let us take two latin squares of order n, L and L, on respectively the symbol set S = {a 0, a 1,..., a n 1 } and the symbol set S = {α 0, α 1,..., α n 1 }. If we find all of the n 2 ordered pairs (i, j) S S, when these latin squares are superimposed, then we say that these two latin squares are mutually orthogonal.

7 Mutually orthogonal latin squares Sets of mutually orthogonal latin squares Definition A set of latin squares L 1, L 2,..., L k is mutually orthogonal if L i and L j are orthogonal for all 1 i < j k. Definition We say that a set of k different mutually orthogonal latin squares of order n is reduced (or in standard form), if one of the latin squares is reduced and if the first row in every other latin square in this set is in the natural order of the symbol set we chose.

8 Mutually orthogonal latin squares Sets of MOLS Notation If we have k different mutually orthogonal latin squares of order n, we will also use the notation (k)mols(n). Definition A maximal set MOLS(n) is a set (k)mols(n) such that it is impossible to extend this set to a set (k + 1)MOLS(n). Definition A set of (n 1)MOLS(n) is called a complete set of MOLS of order n.

9 Connection between MOLS and projective planes Theorem There exists a projective plane of order n if and only if there exists a complete set of mutually orthogonal latin squares of order n.

10 Construction of a set (q 1)MOLS(q) arising from the finite field F q. q = p d, p prime, d 1. define a set of (q 1)MOLS(q) arising from the finite field F q.

11 Construction of a set (q 1)MOLS(q) arising from the finite field F q. Primitive polynomial f (x) = x d + f d 1 x d f 0 of degree d over F p. Let α be a root of f (x), meaning that α is a primitive element of F q. F q = {0, 1, α, α 2,..., α q 2 }.

12 Construction of a set (q 1)MOLS(q) arising from the finite field F q. Notation We also use the following notation x 0 x 1 X =. x d 1 Fd p. corresponding to the additive notation x 0 + x 1 α x d 1 α d 1 of an element of F q.

13 Construction of a set (q 1)MOLS(q) arising from the finite field F q. Definition We denote the latin squares by A a, a F q, where for x, y F q. (A a ) x,y = ax + y,

14 Question How many different reduced (q 1)MOLS(q) are there, for q = p d a prime power, describing PG(2, q)?

15 Question How many different reduced (q 1)MOLS(q) are there, for q = p d a prime power, describing PG(2, q)? determine the group of permutations on the (q 1)MOLS(q) stabilizing the set {A 1, A 2,..., A q 1 }, describing PG(2, q).

16 Lemma A permutation on the alphabet commutes with a permutation on the rows and/or columns of the latin squares.

17 Lemma Consider the set of MOLS {A 1, A 2,..., A q 1 } of the previous definition. Suppose we apply σ 1 to A 1, σ 2 to A 2,..., and σ q 1 to A q 1, the same column permutations to all of the latin squares in the set and also the same row permutations to all of the latin squares in the set, whereby row i moves to row 0. Suppose we obtain a new reduced set of latin squares {C 1, C 2,..., C q 1 }, then σ b = σ a τ b, where τ b : y y bi + ai.

18 Lemma There are at least q permutations on the alphabet leaving the set of MOLS {A 1, A 2,..., A q 1 } invariant and these permutations on the alphabet are the same as a permutation on the columns applied on every latin square in the set.

19 Proof. How many distinct permutations are there on the field F q?

20 Proof. There are q! distinct permutations on the field F q.

21 Proof. There are q! distinct permutations on the field F q. Define the permutation γ b ((A a ) x,y ) =...? γ b : F q F q : y y + b, for b F q.

22 Proof. There are q! distinct permutations on the field F q. Define the permutation γ b : F q F q : y y + b, for b F q. γ b ((A a ) x,y ) = ax + y + b = (A a ) x,y+b.

23 Proof. There are q! distinct permutations on the field F q. Define the permutation γ b : F q F q : y y + b, for b F q. γ b ((A a ) x,y ) = ax + y + b = (A a ) x,y+b. Perform the permutation γ b : y y + b on the columns in every latin square of the set of MOLS.

24 Proof. There are q! distinct permutations on the field F q. Define the permutation γ b : F q F q : y y + b, for b F q. γ b ((A a ) x,y ) = ax + y + b = (A a ) x,y+b. Perform the permutation γ b : y y + b on the columns in every latin square of the set of MOLS. We have transformed A a in A a, so we have the same set of MOLS {A 1, A 2,..., A q 1 }.

25 Theorem There are at most (q 2)!/d different reduced complete sets of q 1 MOLS of order q, for q = p d a prime power, describing PG(2, q).

26 q(q 1)d permutations on the alphabet: AΓL 1 (q). This group is the semidirect product AGL 1 (q) Aut(F q ). Subgroup of AGL d (p). Theorem by Thierry Berger[1].

27 Theorem by Thierry Berger Let G be a permutation group on the finite field K = F p d, d 2. If the affine group AGL 1 (p m ) is a subgroup of G, then one of the following assertions holds: 1 G = Sym(p d ). 2 p = 2 and G = Alt(2 d ). 3 There exists a divisor r of d, such that AGL k (p r ) G AΓL k (p r ), where d = rk. 4 p = 3, d = 4, G AΓL 2 (9) and if G 0 is the stabilizer of 0 in G, then G 0 admits a normal subgroup N isomorphic with SL(2, 5), N = G 0 SL 2 (9) and [G 0 : N] = 8.

28 Lemma There are at least three different reduced sets of q 1 MOLS of order q representing PG(2, q) for q = p d > 4.

29 Theorem There are exactly (q 2)!/d different reduced complete sets of q 1 MOLS of order q, for q = p d a prime power, arising from the finite field F q.

30 Proof. d = 1. AGL 1 (p) is maximal in S p. S p would stabilize the set {A 1,..., A p 1 }. if q = p > 3: at least three different reduced sets of (p 1)MOLS(p) describing PG(2, p).

31 Proof. d 2. G is a larger group of permutations on the alphabet, stabilizing the set {A 1, A 2,..., A q 1 }, modulo the same permutations on the rows and the columns of the latin squares. G is described as in the previous theorem by Thierry Berger.

32 Proof. q = 4. AΓL 1 (q) = q(q 1)d = 24 = S q. q > 4. at least three different reduced complete sets of MOLS of order q, describing PG(2, q).

33 Proof. cases 3 and 4 of the previous theorem by Thierry Berger. G always has a normal subgroup of order p d (p r 1), where d = rk. G is represented by matrices, field automorphisms and translations over the subfield GF (p k ) of GF (p d ). no larger group G of permutations on the alphabet stabilizes the set {A 1, A 2,..., A q 1 }, modulo the same permutations on the rows and the columns of the latin squares.

34 Corollary For any prime p, there are (p 2)! different reduced complete sets of MOLS of order p arising from the field F p.

35 O(n, r) is the number of reduced maximal sets of MOLS of order n containing exactly r squares. S(n, r) is the number of reduced maximal sets of MOLS of order n containing exactly r distinct squares.

36 S(3, 2) = 1 = O(3, 2). S(4, 3) = 1 = O(4, 3). S(5, 4) = 6 = O(5, 4). S(7, 6) = 120 = O(7, 6). S(8, 7) = 240 = 8 O(8, 7).

37 q 7: the computational values obtained for O(q, q 1) all agree with the theoretical results for S(q, q 1). q = 8: O(8, 7) = S(8, 7)/8 = 30.

38 References I T. Berger, Classification des groupes de permutations d un corps fini contenant le groupe affine, C. R. Acad. des Sciences de Paris, Ser. I, 1994, R.C. Bose, On the application of the properties of Galois fields to the construction of hyper-graeco-latin squares, Sankhyā 3(1938), C.J. Colbourn and J.H. Dinitz, Handbook of Combinatorial Designs, CRC Press, Taylor and Francis Group, Boca Raton, Fl, 2007.

39 References II J. Deńes and A.D. Keedwell, Latin Squares and their Applications, Academic Press, New York, D.A. Drake and W. Myrvold, The non-existence of maximal sets of four mutually orthogonal latin squares of order 8, Designs, Codes and Cryptography 33(2004), D.A. Drake, G.H.J. van Rees, and W.D. Wallis, Maximal sets of mutually orthogonal latin squares, Discrete Math. 194(1999),

40 References III A. Hedayat and W.T. Federer, On embedding and enumeration of orthogonal Latin squares, Annals of Math. Statist. 42(1971), C.W.H. Lam, G. Kolesova, and L. Thiel, A computer search for finite projective planes of order 9, Discrete Math. 92(1991), C.F. Laywine and G.L. Mullen, Discrete Mathematics Using Latin Squares, Wiley, New York, 1998.

41 References IV M.W. Liebeck, C.E. Praeger, and J. Saxl, A classification of the maximal subgroups of the finite alternating and symmetric groups, J. Algebra 111(1987), B.D. McKay and I.M. Wanless, On the number of latin squares, Ann. Comb. 9(2005),

42 Thanks a lot for your attention!