The maximum size of a partial spread in a finite vector space · I Apartial t-spreadof V is a collection of subspaces fW 1;:::;W kg V s.t. I dim(W 1) = :::dim(W k) = t I W i \W j

$Page 1: The maximum size of a partial spread in a finite vector space · I Apartial t-spreadof V is a collection of subspaces fW 1;:::;W kg V s.t. I dim(W 1) = :::dim(W k) = t I W i \W j$
IntroductionKnown Results

Auxiliary ResultsMain ResultsFuture Work

The maximum size of a partial spread in a finitevector space

Esmeralda NastaseXavier University

Joint work with Papa Sissokho

June 12, 2017

E. Nastase The maximum size of a partial spread in a finite vector space



DefinitionsMotivation

I V = V (n, q) the vector space of dimension n over GF(q).

I A partial t-spread of V is a collection of subspaces{W1, . . . ,Wk} ⊆ V s.t.

I dim(W1) = . . . dim(Wk) = t

I Wi ∩Wj = {0} for i 6= j .

I If W1 ∪ · · · ∪Wk contains all the vectors of V , then it is called at-spread.

I Andre (1954), Segre (1964)- A t-spread of V exists if and only ift | n.







I dim(W1) = . . . dim(Wk) = t

I Wi ∩Wj = {0} for i 6= j .









I dim(W1) = . . . dim(Wk) = t

I Wi ∩Wj = {0} for i 6= j .







Question

I What is the maximum size of a partial t-spread of V ?





Applications

I error-correcting codes

I orthogonal arrays

I (s, k , λ)-nets

I subspace codes




ConjectureLower BoundUpper BoundExact values

I Let µq(n, t)=maximum size of a partial t-spread of V , and let

`q(n, t) =qn − qt+r

qt − 1.

Conjecture (Hong and Patel, 1972)

Let n = kt + r , and 0 ≤ r < t. Then

µq(n, t) =qn − qt+r

qt − 1+ 1 = `q(n, t) + 1.





Theorem (Hong and Patel, 1972; Beutelspacher, 1975)


µq(n, t) ≥ qn − qt+r

qt − 1+ 1 = `q(n, t) + 1.





Theorem (Drake and Freeman, 1979)


µq(n, t) ≤ qn − qr

qt − 1− bωc − 1 = `q(n, t) + qr − bωc − 1,

where 2ω =√

4qt(qt − qr ) + 1− (2qt − 2qr + 1).





Theorem (Andre, 1954, Segre, 1964 (r = 0); Hong and Patel,1972 (r = 1, q = 2); Beutelspacher, 1975 (r = 1, q > 2))

Let n = kt + r , and r ∈ {0, 1}. Then


qt − 1+ 1 = `q(n, t) + 1.





Theorem (El-Zanati, Jordon, Seelinger, Sissokho, and Spence,2012)

If r = 2, t = 3, and q = 2, then

µ2(n, 3) =2n − 25

7+ 2 = `2(n, 3) + 2.

Theorem (Kurz, 2016)

If t > 3 and q = r = 2, then

µ2(n, t) =2n − 2t+2

2t − 1+ 1 = `2(n, t) + 1.





Theorem (El-Zanati, Jordon, Seelinger, Sissokho, and Spence,2012)

If r = 2, t = 3, and q = 2, then

µ2(n, 3) =2n − 25

7+ 2 = `2(n, 3) + 2.

Theorem (Kurz, 2016)

If t > 3 and q = r = 2, then

µ2(n, t) =2n − 2t+2

2t − 1+ 1 = `2(n, t) + 1.





Theorem (N., Sissokho, 2017+)

Let n = kt + r , and 0 ≤ r < t. If t >qr − 1

q − 1, then


qt − 1+ 1 = `q(n, t) + 1.




DefinitionsLemmasMain Lemma

I A subspace partition or partition P of V , is a collection ofsubspaces {W1, . . . ,Wk} s.t.

I V = W1 ∪ · · · ∪Wk

I Wi ∩Wj = {0} for i 6= j .

I Let P be any partition of V . We say P hast type [tnt , . . . , 1n1 ],if there are ni > 0 subspaces of dim i in P, and 1 < · · · < t.





Lemma 1 - (Heden and Lehmann, 2013)

Let P be a partition of V of type [tnt , . . . , 1n1 ] and let bH,i be thenumber of i-subspaces in P that are in a hyperplane H of V . Then

|P| = 1 +t∑

i=1

bH,iqi





Lemma 2 - (Heden and Lehmann, 2013)

Let P be a partition of V of type [tnt , . . . , 1n1 ] and let bH,i be thenumber of i-subspaces in P that are in a hyperplane H of V . Then

1.∑H∈H

H =qn − 1

q − 1

2.∑H∈H

bH,iH = niqn−i − 1

q − 1





Main Lemma - (N. and Sissokho, 2017+)


q − 1, then

µq(n, t) ≤ qn − qt+r

qt − 1+ 1 = `q(n, t) + 1.





For any integer i ≥ 1, let Θi =qi − 1

q − 1and δi =

qi − 2qi−1 + 1

q − 1.

Then

I 0 < δi < qi−1 andδiq< δi−1.

Proof Sketch:

I Assume that µq(n, t) > `q(n, t) + 1 . Then V has a partialt-spread of size `q(n, t) + 2.

I ∃ P0 of V (n, q) of type [tnt , 1n1 ],where

nt = `q(n, t) + 2 and

n1 =

(qr − 1

q − 1− 1

)qt +

qt+1 − 2qt + 1

q − 1= (Θr − 1)qt + δt+1.






q − 1and δi =

qi − 2qi−1 + 1

q − 1.

Then


Proof Sketch:



nt = `q(n, t) + 2 and

n1 =

(qr − 1

q − 1− 1

)qt +

qt+1 − 2qt + 1

q − 1= (Θr − 1)qt + δt+1.






q − 1and δi =

qi − 2qi−1 + 1

q − 1.

Then


Proof Sketch:



nt = `q(n, t) + 2 and

n1 =

(qr − 1

q − 1− 1

)qt +

qt+1 − 2qt + 1

q − 1= (Θr − 1)qt + δt+1.





We use induction on j , for 0 ≤ j ≤ Θr − 1:

I Base Case: H0 = V (n, q) and P0 = {H0}

I Inductive Step: there exists Pj of Hj∼= V (n − j , q) of type

[tmj,t , (t − 1)mj,t−1 , . . . , (t − j)mj,t−j , 1mj,1 ],

where mj ,t , . . . ,mj ,t−j , mj ,1, and cj are nonnegative integers s.t.

t∑i=t−j

mj ,i = nt = `q(n, t) + 2,

and

mj ,1 = cjqt−j + δt+1−j , and 0 ≤ cj ≤ Θr − 1− j .





We use induction on j , for 0 ≤ j ≤ Θr − 1:

I Base Case: H0 = V (n, q) and P0 = {H0}I Inductive Step: there exists Pj of Hj

∼= V (n − j , q) of type

[tmj,t , (t − 1)mj,t−1 , . . . , (t − j)mj,t−j , 1mj,1 ],

where mj ,t , . . . ,mj ,t−j , mj ,1, and cj are nonnegative integers s.t.

t∑i=t−j

mj ,i = nt = `q(n, t) + 2,

and

mj ,1 = cjqt−j + δt+1−j , and 0 ≤ cj ≤ Θr − 1− j .





I Using Lemma 2,

bavg ,1 =

∑H∈H bH,1 H∑

H∈H H<

cjqt−j + δt+1−j

q< cjq

t−j−1 + δt−j .

=⇒ ∃ a hyperplane Hj+1 of Hj with

bHj+1,1 ≤ bavg ,1 < cjqt−j−1 + δt−j ,

and applying Lemma 1,

mj+1,1 = bHj+1,1 = cj+1qt−j−1 +δt−j , and 0 ≤ cj+1 ≤ Θr−2− j .





I Using Lemma 2,

bavg ,1 =

∑H∈H bH,1 H∑

H∈H H<


q< cjq

t−j−1 + δt−j .









I Using Lemma 2,

bavg ,1 =

∑H∈H bH,1 H∑

H∈H H<


q< cjq

t−j−1 + δt−j .









I Thus, we let Pj+1 be the subspace partition of Hj+1 defined by:

Pj+1 = {W ∩ Hj+1 : W ∈ Pj}.

=⇒ Pj+1 is a subspace partition of Hj+1 of type

[tmj+1,t , (t − 1)mj+1,t−1 , . . . , (t − j − 1)mj+1,t−j−1 , 1mj+1,1 ], (1)

where mj+1,t ,mj+1,t−1, . . . ,mj+1,t−j−1 satisfy

t∑i=t−j−1

mj+1,i =t∑

i=t−jmj ,i = nt = `q(n, t) + 2, and (2)






I Thus, we let Pj+1 be the subspace partition of Hj+1 defined by:

Pj+1 = {W ∩ Hj+1 : W ∈ Pj}.

=⇒ Pj+1 is a subspace partition of Hj+1 of type

[tmj+1,t , (t − 1)mj+1,t−1 , . . . , (t − j − 1)mj+1,t−j−1 , 1mj+1,1 ], (1)

where mj+1,t ,mj+1,t−1, . . . ,mj+1,t−j−1 satisfy

t∑i=t−j−1

mj+1,i =t∑

i=t−jmj ,i = nt = `q(n, t) + 2, and (2)






Final Step, j = Θr − 1:

I cΘr−1 = 0 and thus, mΘr−1,1 = δt+2−Θr in PΘr−1 of HΘr−1.

I Using the averaging argument, there exists a hyperplane H∗ ofHΘr−1 with

bH∗,1 ≤ bavg ,1 < δt+1−Θr .

I Using Lemma 1 on the partition PΘr−1 and the hyperplane H∗

of HΘr−1 we obtain that

bH∗,1 ≥ δt+1−Θr ,

which is a contradiction.

=⇒ µq(n, t) ≤ `q(n, t) + 1 .








bH∗,1 ≤ bavg ,1 < δt+1−Θr .



bH∗,1 ≥ δt+1−Θr ,


=⇒ µq(n, t) ≤ `q(n, t) + 1 .








bH∗,1 ≤ bavg ,1 < δt+1−Θr .



bH∗,1 ≥ δt+1−Θr ,


=⇒ µq(n, t) ≤ `q(n, t) + 1 .








bH∗,1 ≤ bavg ,1 < δt+1−Θr .



bH∗,1 ≥ δt+1−Θr ,


=⇒ µq(n, t) ≤ `q(n, t) + 1 .




Main ResultsApplications



q − 1, then


qt − 1+ 1 = `q(n, t) + 1.

Proof.

I r = 0: Andre’s, Segre’s Theorem.

I r = 1: Hong and Patel’s, and Beutelspacher’s Theorem.

I r ≥ 2: LB for µq(n, t), by Hong and Patel’s and Beutelspacher’sTheorem and UB for µq(n, t), by Main Lemma, are equal.







q − 1, then


qt − 1+ 1 = `q(n, t) + 1.

Proof.










q − 1, then


qt − 1+ 1 = `q(n, t) + 1.

Proof.










q − 1, then


qt − 1+ 1 = `q(n, t) + 1.

Proof.









Let n = kt + r , and 0 ≤ r < t. If r ≥ 2 and t = Θr , then

µq(n, t) ≤ `q(n, t) + q.





Hypergraphs

I Let Hq(n, t) be the hypergraph whose vertices are the1-subspaces of V (n, q) and whose edges are its t-subspaces.

I Then Hq(n, t) is a (qt − 1)/(q − 1)-uniform hypergraph.

I By our Theorem, if n = kt + r , 0 ≤ r < t, and t >qr − 1

q − 1, then

the maximum size of a matching in Hq(n, t) is

qn − qt+r

qt − 1+ 1 = `q(n, t) + 1.





Hypergraphs




q − 1, then


qn − qt+r

qt − 1+ 1 = `q(n, t) + 1.





Hypergraphs




q − 1, then


qn − qt+r

qt − 1+ 1 = `q(n, t) + 1.




Question

I What is the maximum size of a partial t-spread of V if r ≥ 2

and t ≤ qr − 1

q − 1?




Thank you!


Documents

The maximum size of a partial spread in a finite vector space · I Apartial t-spreadof V is a collection of subspaces fW 1;:::;W kg V s.t. I dim(W 1) = :::dim(W k) = t I W i \W j