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 kdim subspaces in Fqn, called blocks,
such that each t-dim subspace of Fqn
is contained in exactly  blocks
• Trivial design: all k-dim subspaces
• Question: find nontrivial designs
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
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
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
7
4
n=7,k=3,
t=2,=1
6
5
Regular combinatorial objects
• 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
Regular combinatorial objects
• 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 
1234
2341
3412
4123
n=4,t=1,
=1
Regular combinatorial objects
• Example 4: t-designs over finite fields
• Collection of k-dim subspaces of Fqn,
called blocks, such that each t-dim
subspace of Fqn is contained in exactly 
blocks
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?
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 tdim subspace will be in approximately
the 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!
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 Fq, t, k>12(t+1); and n
large enough (n>>kt suffices)
• But, we don’t know how to find them
efficiently…
Overview
• Regular combinatorial objects
• 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 rows
whose average equals the average of all
the rows
1
|𝑆|
𝑟∈𝑆
1
𝑟𝑜𝑤 𝑟 =
|𝑅|
𝑟𝑜𝑤 𝑟
𝑟∈𝑅
Matrix averaging problem
• A subset S of rows for which
1
|𝑆|
𝑟∈𝑆 𝑟𝑜𝑤 𝑟 =
1
|𝑅|
is exactly a t-design
𝑟∈𝑅 𝑟𝑜𝑤 𝑟
k-dim
subspaces
• For example, if:
R = all k-dim subspaces
C = all t-dim subspaces
M = incidence matrix
t-dim
subspaces
0100101001
1001011000
0000101110
…
0010110100
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|~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,
– coding-theoretic properties, and
– divisibility properties,
then there is a small set of rows S such that
1
|𝑆|
𝑟∈𝑆
1
𝑟𝑜𝑤 𝑟 =
|𝑅|
𝑟𝑜𝑤 𝑟
𝑟∈𝑅
• Small = polynomial in |C|, other parameters
KLP framework (1)
• Condition 1: all the elements in the
matrix are small integers
• Trivially true for incidence matrices
0100101001
1001011000
0000101110
…
0010110100
KLP framework (2)
0100101001
1001011000
0000101110
…
0010110100
• 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.)
KLP framework (3)
0100101001
1001011000
0000101110
…
0010110100
• V = subspace of QR 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 r1,r2 there is a
symmetry of V mapping r1 to r2
KLP framework (3)
• Example: t-designs over finite fields
0100101001
1001011000
0000101110
…
0010110100
• 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)
KLP framework (4)
0100101001
1001011000
0000101110
…
0010110100
• V = subspace of QR spanned by columns
• V = orthogonal subspace (in QR)
• Condition 4: V is spanned by short
integer vectors
• Usually the hardest condition to verify
KLP framework (5)
0100101001
1001011000
0000101110
…
0010110100
• Condition 5: Divisibility. There exist a small
integer c such that
𝑐
|𝑅|
𝑟𝑜𝑤 𝑟
𝑟∈𝑅
expressible as integer combination of rows
• Necessary if we hope to get small S,
1
|𝑆|
𝑟∈𝑆
1
𝑟𝑜𝑤 𝑟 =
|𝑅|
𝑟𝑜𝑤 𝑟
𝑟∈𝑅
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
1
𝑟𝑜𝑤 𝑟 =
|𝑅|
𝑟𝑜𝑤 𝑟
𝑟∈𝑅
• Small = polynomial in |C|, other parameters
Proof idea
• S = random small set of rows
• Analyze the probability that
1
1
𝑟𝑜𝑤 𝑟 =
|𝑆|
|𝑅|
𝑟∈𝑆
𝑟𝑜𝑤 𝑟
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
Overview
• Regular combinatorial objects
• 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 SG such
that S acts uniformly on X,
|{gS: g(x)=y}|=|S|/|X|
x,yX
• Proved for G=Sn, S=all k-sets
• Open: G=GL(n,F); S=k-dim Grasmannian