Probabilistic construction of t-designs over finite fields

Transcription

1 Probabilistic construction of t-designs over finite fields Shachar Lovett (UCSD) Based on joint works with Arman Fazeli (UCSD), Greg Kuperberg (UC Davis), Ron Peled (Tel Aviv) and Alex Vardy (UCSD) Gent workshop, 2013

2 t-designs over finite fields Finite field F q t-(n,k, ;q) design is a collection of k- dim subspaces in F qn, called blocks, such that each t-dim subspace of F q n is contained in exactly blocks Trivial design: all k-dim subspaces Question: find nontrivial designs

3 t-designs over finite fields t-designs over finite fields are an extension of the more standard notion of combinatorial t-designs, where subspaces are replaced by subsets Teirlinck 87: First construction of nontrivial combinatorial t-designs, for any t No analog theorem for designs over finite fields (constructions known only for t=1,2,3) This work: existence of nontrivial t-designs over finite fields, for any t Proof by probabilistic argument, non constructive

4 Bigger picture t-designs over finite fields are an instance of regular combinatorial objects [Kuperberg-L-Peled 12]: General framework to prove existence of regular combinatorial objects by probabilistic techniques [Fazeli-L-Vardy 13]: Application to t-designs over finite fields

5 Overview Regular combinatorial objects KLP framework Open problems

6 Overview Regular combinatorial objects KLP framework Open problems

7 Regular combinatorial objects Example 1: Combinatorial t-designs Collection of k-subsets of {1,,n}, called blocks, such that each t-subset of {1,,n} is contained in exactly blocks n=7,k=3, t=2, =

8 Regular combinatorial objects Example 2: Orthogonal arrays Collection of vectors in [q] n, such that on any t coordinates, each one of the possible q t patterns appear exactly times q=2,n=3, t=2, =1

9 Regular combinatorial objects Example 3: t-wise permutations Collection of permutations in S n, such that for any indices i 1,..,i t and j 1, j t, the number of permutations mapping i 1 to j 1,i 2 to j 2,,i t to j t, is exactly n=4,t=1, =1

10 Regular combinatorial objects Example 4: t-designs over finite fields Collection of k-dim subspaces of F qn, called blocks, such that each t-dim subspace of F qn is contained in exactly blocks

11 Regular combinatorial objects highly symmetric objects with many simultaneous conditions of exact counts Constructions known in special cases Existence cannot be exhibited by standard probabilistic techniques. Why?

12 Probabilistic constructions Consider, say, the problem of t-designs over finite fields If we choose randomly a small collection of k-dim subspaces (blocks), than any t- dim subspace will be in approximately the same number of blocks Approximately, but not exactly

13 KLP Framework Theorem [Kuperberg-L-Peled 12]: If the objects satisfy certain symmetric properties, coding-theoretic properties, and divisibility properties, then the probability that a random construction works is positive (but tiny) Hence, the required objects exist!

14 t-designs over finite fields [Fazeli-L-Vardy 13] Application of KLP framework Theorem: t-(n,k, ;q) designs over a finite field F exist for any choice of F q, t, k>12(t+1); and n large enough (n>>kt suffices) But, we don t know how to find them efficiently

15 Overview Regular combinatorial objects KLP framework Open problems

16 Matrix averaging problem Let M be an integer matrix, with rows set R and columns set C row(r) Z C We want to find a small subset S of rows whose average equals the average of all the rows 1 S r S row r = 1 R r R row r

17 k-dim subspaces Matrix averaging problem For example, if: R = all k-dim subspaces C = all t-dim subspaces M = incidence matrix A subset S of rows for which t-dim subspaces S r S row r = 1 R is exactly a t-design r R row r

18 Matrix averaging problem Can we hope that in general, in any 0-1 matrix, there are few rows whose average is the same as the average of all the rows? NO. There are 0-1 matrices with C =n, R ~n n/2 with no such subsets of rows [Alon-Vu] We, on the other hand, would like to have a subset of poly(n) rows

19 KLP theorem Theorem: If matrix M satisfies certain symmetric properties, coding-theoretic properties, and divisibility properties, then there is a small set of rows S such that 1 S r S row r = 1 R r R row r Small = polynomial in C, other parameters

20 KLP framework (1) Condition 1: all the elements in the matrix are small integers Trivially true for incidence matrices

21 KLP framework (2) V = subspace of Q R spanned by columns Condition 2: constant vector in V For t-designs over finite fields, holds because sum of columns is a constant vector (#t-dim subsp. in a k-dim subsp.)

22 KLP framework (3) V = subspace of Q R spanned by columns Symmetry group of V = group of permutations of rows which preserve V Condition 3: Symmetry group of V is transitive e.g. for any pair of rows r 1,r 2 there is a symmetry of V mapping r 1 to r 2

23 KLP framework (3) Example: t-designs over finite fields Rows = k-dim subsp., Cols = t-dim subsp. V = subspace of Q R spanned by columns GL(F q,n) acts on rows and columns, preserve the incidence matrix. Hence, GL(F q,n) < Sym(V) Action of GL(F q,n) on R is transitive (can map any k-dim subspace to any k-dim subspace)

24 KLP framework (4) V = subspace of Q R spanned by columns V = orthogonal subspace (in Q R ) Condition 4: V is spanned by short integer vectors Usually the hardest condition to verify

25 KLP framework (5) Condition 5: Divisibility. There exist a small integer c such that c R r R row r expressible as integer combination of rows Necessary if we hope to get small S, 1 S r S row r = 1 R r R row r

26 KLP theorem Theorem: If matrix M satisfies certain symmetric properties, coding-theoretic properties, and divisibility properties, then there is a small set of rows S such that 1 S r S row r = 1 R r R row r Small = polynomial in C, other parameters

27 Proof idea S = random small set of rows Analyze the probability that 1 row r = 1 row r S R r S r R If the conditions hold, can approximate probability up to 1+o(1) by an appropriate Gaussian process restricted to a lattice Proof utilizes new connections between Fourier analysis, coding theory and local central limit theorems

28 Overview Regular combinatorial objects KLP framework Open problems

29 Summary New probabilistic technique Can prove existence of regular combinatorial structures Application: t-designs over finite fields

30 Open problems (1) Algorithmic: Can prove existence, but we don t know how to find the objects efficiently For other probabilistic techniques for rare events this was accomplished Lovász Local Lemma [Moser, Moser-Tardos, ] Spencer s six standard deviations suffice [Bansal, L-Meka] So, I am hopeful

31 Open problems (2) Other applications Large sets (e.g. partitions) Sparse systems (Steiner systems, Hadamard matrices)

32 Open problems (3) Perfect pseudo-randomness in group theory Conjecture: for any group G acting transitively on a set X, there is a small subset S G such that S acts uniformly on X, {g S: g(x)=y} = S / X x,y X Proved for G=S n, S=all k-sets Open: G=GL(n,F); S=k-dim Grasmannian

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