probabilistic construction of t-designs over finite...

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

t-designs over finite fields

• Finite field Fq

• t-(n,k,;q) design is a collection of k-dim subspaces in Fq

n, called blocks, such that each t-dim subspace of Fq

n

is contained in exactly blocks

• Trivial design: all k-dim subspaces• Question: find nontrivial designs


• 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

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

Overview

• Regular combinatorial objects

• KLP framework

• Open problems

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

1

2

3 4 5

67

n=7,k=3,t=2,=1


• Example 2: Orthogonal arrays

• Collection of vectors in [q]n, such that on any t coordinates, each one of the possible qt patterns appear exactly times

0 0 0

0 1 1

1 0 1

1 1 0

q=2,n=3,t=2,=1


• Example 3: t-wise permutations

• Collection of permutations in Sn, such that for any indices i1,..,it and j1,…jt, the number of permutations mapping i1 to j1,i2 to j2,…,it to jt, is exactly

n=4,t=1,=1

1 2 3 42 3 4 13 4 1 24 1 2 3


• Example 4: t-designs over finite fields

• Collection of k-dim subspaces of Fqn,

called blocks, such that each t-dim subspace of Fq

n is contained in exactly blocks


• “highly symmetric” objects with many simultaneous conditions of exact counts

• Constructions known in special cases

• Existence cannot be exhibited by standard probabilistic techniques. Why?

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 approximatelythe same number of blocks

• Approximately, but not exactly

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!


• [Fazeli-L-Vardy’13]

• Application of KLP framework

• Theorem: t-(n,k,;q) designs over a finite field F exist for any choice of Fq, t, k>12(t+1); and n large enough (n>>kt suffices)

• But, we don’t know how to find them efficiently…

Overview


• KLP framework

• Open problems

Matrix averaging problem

• Let M be an integer matrix, with rows set R and columns set C– row(r) ZC

• We want to find a small subset S of rowswhose average equals the average of all the rows

1

|𝑆|

𝑟∈𝑆

𝑟𝑜𝑤 𝑟 =1

|𝑅|

𝑟∈𝑅

𝑟𝑜𝑤 𝑟


• For example, if: R = all k-dim subspacesC = all t-dim subspacesM = incidence matrix

• A subset S of rows for which

1

|𝑆| 𝑟∈𝑆 𝑟𝑜𝑤 𝑟 =

1

|𝑅| 𝑟∈𝑅 𝑟𝑜𝑤 𝑟

is exactly a t-design

010010100110010110000000101110

…

0010110100

k-d

im

subsp

aces

t-dim subspaces


• 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|~nn/2

with no such subsets of rows [Alon-Vu]

• We, on the other hand, would like to have a subset of poly(n) rows

KLP theorem

• Theorem: If matrix M satisfies certain – symmetric properties,



then there is a small set of rows S such that

• Small = polynomial in |C|, other parameters

1

|𝑆| 𝑟∈𝑆


|𝑅| 𝑟∈𝑅

𝑟𝑜𝑤 𝑟

KLP framework (1)

• Condition 1: all the elements in the matrix are small integers

• Trivially true for incidence matrices

010010100110010110000000101110

…

0010110100

010010100110010110000000101110

…

0010110100

KLP framework (2)

• V = subspace of QR 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.)

010010100110010110000000101110

…

0010110100

KLP framework (3)


• 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 r1,r2 there is a

symmetry of V mapping r1 to r2

KLP framework (3)

• Example: t-designs over finite fields

• Rows = k-dim subsp., Cols = t-dim subsp.• V = subspace of QR spanned by columns

• GL(Fq,n) acts on rows and columns, preserve the incidence matrix. Hence, GL(Fq,n) < Sym(V)

• Action of GL(Fq,n) on R is transitive (can map any k-dim subspace to any k-dim subspace)

010010100110010110000000101110

…

0010110100

010010100110010110000000101110

…

0010110100

KLP framework (4)


• V = orthogonal subspace (in QR)

• Condition 4: V is spanned by short integer vectors

• Usually the hardest condition to verify

010010100110010110000000101110

…

0010110100

KLP framework (5)

• Condition 5: Divisibility. There exist a small integer c such that

expressible as integer combination of rows

• Necessary if we hope to get small S,

𝑐

|𝑅| 𝑟∈𝑅

𝑟𝑜𝑤 𝑟

1

|𝑆| 𝑟∈𝑆


|𝑅| 𝑟∈𝑅

𝑟𝑜𝑤 𝑟

KLP theorem

• Theorem: If matrix M satisfies certain – symmetric properties,



then there is a small set of rows S such that

• Small = polynomial in |C|, other parameters

1

|𝑆| 𝑟∈𝑆


|𝑅| 𝑟∈𝑅

𝑟𝑜𝑤 𝑟

Proof idea

• S = random small set of rows

• Analyze the probability that

• 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

1

|𝑆| 𝑟∈𝑆


|𝑅| 𝑟∈𝑅

𝑟𝑜𝑤 𝑟

Overview


• KLP framework

• Open problems

Summary

• New probabilistic technique

• Can prove existence of regular combinatorial structures

• Application: t-designs over finite fields

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…

Open problems (2)

• Other applications

• Large sets (e.g. partitions)

• Sparse systems (Steiner systems, Hadamard matrices)

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 SG such that S acts uniformly on X,

|{gS: g(x)=y}|=|S|/|X| x,yX

• Proved for G=Sn, S=all k-sets• Open: G=GL(n,F); S=k-dim Grasmannian

probabilistic construction of t-designs over finite...

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