New Constructions of Difference Matrices
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1 New Constructions of Difference Matrices Department of Mathematics, Simon Fraser University Joint work with Koen van Greevenbroek
2 Outline Difference matrices Basic results, motivation, history Construction 1: finite field Construction 2: composition Construction 3: abelian noncyclic 2-groups Contracted difference matrices Constructions 1, 2, 3 revisited Computer search results Open questions
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5 Difference Matrices matrix over! (! 3 2, 5, 1) difference matrix (in a nonabelian group, calculate quotients not differences)
6 Difference Matrices (! 3, 9, 3) difference matrix
7 Difference Matrices Given a group G and parameter λ, what is the largest number of rows m for which a (G, m, λ) difference matrix exists? largest m 2 0 G 0 G 0 G 0 G λ copies of each element of G m λ G (Jungnickel 1979) If m 3 and G contains a nontrivial cyclic Sylow 2-subgroup, then λ is even (Hall & Paige 1955)
8 Difference Matrices Connections to orthogonal arrays, transversal designs, whist tournaments, generalised Steiner triple systems, Extremal case m = λ G generalised Hadamard matrix Focus mostly on case λ =1: closely related to mutually orthogonal Latin squares pairwise orthogonal orthomorphisms linking systems of difference sets when G is a 2-group, giving systems of linked symmetric designs and cometric association schemes
9 (1) Finite Field Construction Theorem (Drake 1979). For prime p, the additive form of a multiplication table for GF(p n ) is a (! n p, p n, 1) difference matrix i 0 1 α α 2 α 3 α 4 α 5 α α α α α α α
10 (2) Composition Construction Theorem (Buratti 1998). Let G be a group and H G. If there exist (H, m, λ) and (G / H, m, µ) difference matrices, then their Kronecker product is a (G, m, λµ) difference matrix
11 (2) Composition Construction G =! 4 H = 010,200! 2 (H, 4, 1) difference matrix (G / H, 4, 1) difference matrix H H H H H 001+ H H 101+ H H H 101+ H 001+ H H 101+ H 001+ H H (G, 4, 1) difference matrix
12 (2) Composition Construction Apply composition construction repeatedly 4 (! 2, 4, 1) (! 2, 8, 1) H (! 4! 4, 4, 1) H G / H 4 (! 2, 16, 1) G / H (! 8! 8! 4, 4, 1) G
13 (2) Composition Construction Apply composition construction repeatedly (! 2, 8, 1) H (! 2, 8, 1) G / H (! 4! 4, 8, 1) H (! 2, 8, 1) G / H 8 (! 8! 8! 4, 4, 1) G
14 (2) Composition Construction Apply composition construction repeatedly choose chain of subgroups that maximises number of rows of final difference matrix Theorem (Buratti 1998). Let p be prime and G an abelian group of order and exponent. Then optimal choice of p n subgroup chain gives a (G, p n/e, 1) difference matrix sole ingredient is (! n p, p n, 1) difference matrix could another ingredient give stronger result? p e
15 (3) Abelian Noncyclic 2-Groups Theorem (Pan & Chang 2016). There exists a difference matrix for each integer e 1 (! 2 e, 4,1) apply composition construction repeatedly and Hall-Paige result: for an abelian 2-group G, there exists a (G, 4, 1) difference matrix if and only if G is noncyclic
16 Contracted Difference Matrices (! 4, 4, 1) difference matrix (! 4, 2, 0) contracted difference matrix
17 Contracted Difference Matrices For p prime and G an abelian group of order, a (G, k, s) contracted difference matrix of size k (n + s) expands to a (G, p k, p s ) difference matrix of size p k p n+s p n 1 k n + s case s = 0: a (G, k, 0) contracted difference matrix expands to a (G, p k,1) difference matrix Given a group G and parameter s, what is the largest number of rows k for which a (G, k, s) contracted difference matrix exists?
18 (1) Finite Field Construction i 0 1 α α 2 α 3 α 4 α 5 α α α α α α α Theorem. For prime p and α primitive in GF(p n ), the additive form of the multiplication table for 1, α, α 2,,α n 1 a (! n p, n, 0) contracted difference matrix is a
19 (2) Composition Construction Theorem (Buratti 1998). Let G be a group and H G. If there exist (H, m, λ) and (G / H, m, µ) difference matrices, then their Kronecker product is a (G, m, λµ) difference matrix Theorem. Let G be an abelian p-group and let H < G. If there exist (H, k, s) and (G / H, k, t) contracted difference matrices, then their concatenation is a (G, k, s + t) contracted difference matrix
20 (2) Composition Construction G =! 4 H = 010,200! 2 (H, 2, 0) contracted difference matrix (G / H, 2, 0) contracted difference matrix 101+ H H H 001+ H (G, 2, 0) contracted difference matrix
21 (2) Composition Construction Theorem (Buratti 1998). Let p be prime and G an abelian p n group of order and exponent. Then optimal choice of subgroup chain gives a (G, p n/e, 1) p e difference matrix Theorem. Let p be prime and G an abelian group of order and exponent p e. Then optimal choice of subgroup chain gives a (G, n / e contracted difference matrix, 0) p n
22 (3) Abelian Noncyclic 2-Groups Theorem (Pan & Chang 2016). There exists a difference matrix for each integer e 1 (! 2 e, 4,1) apply composition construction repeatedly and Hall-Paige result: for an abelian 2-group G, there exists a (G, 4, 1) difference matrix if and only if G is noncyclic Theorem. There exists a (! 2 e, 2, 0) contracted difference matrix for each integer e
23 (3) Abelian Noncyclic 2-Groups Theorem (Pan & Chang 2016). There exists a difference matrix for each integer e 1 (! 2 e, 4,1) apply composition construction repeatedly and Hall-Paige result: for an abelian 2-group G, there exists a (G, 4, 1) difference matrix if and only if G is noncyclic Theorem. There exists a (! 2 e, 2, 0) contracted difference matrix for each integer e 1 apply composition construction: for an abelian 2-group G, there exists a (G, 2, 0) contracted difference matrix if and only if G is noncyclic
24 Computer Search Results Easier to search for a contracted difference matrix than for corresponding size of difference matrix search space is exponentially smaller no need to expand potential contracted difference matrix random search as well as exhaustive search is easier Each example of contracted difference matrix with new parameters gives new infinite family, by applying composition construction repeatedly
25 Computer Search Results Examples of (G, 3, 0) contracted difference matrices, and therefore (G, 8, 1) difference matrices, for four new groups G:! 4! 4! 4! 4! 8 each G gives new infinite family; also gives linking systems of difference sets of size 7 in infinitely many 2-groups for which largest previously known size was 3 Existence pattern for (contracted) difference matrices seems to favour groups of smaller exponent and higher rank
26 Open Questions Let p be prime and G an abelian p-group. (G, k, 0) contracted difference matrix? (G, p k,1) difference matrix is the largest number of rows m for which a (G, m, 1) difference matrix exists always a power of p? these questions cannot both have a positive answer when p = 3
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