Grassmann Varieties and Linear Codes - spms.ntu.edu.sgccrg/documents/Grassmann Varieties and...

Grassmann Varieties and Linear Codes

Sudhir R. Ghorpade

Department of MathematicsIndian Institute of Technology Bombay

Powai, Mumbai 400076, Indiahttp://www.math.iitb.ac.in/∼srg/

CCRG SeminarNanyang Technological University

SingaporeApril 24, 2012

Sudhir R. Ghorpade Grassmann Varieties and Linear Codes

Grassmann VarietiesA Quick Introduction

V : vector space of dimension m over a field FFor 1 ≤ ` ≤ m, we have the Grassmann variety:

G`,m = G`(V ) := `-dimensional subspaces of V .Plucker embedding:

G`,m → Pk−1 where k :=

(m

`

).

Explicitly, Pk−1 = P(∧`V ) and

W = 〈w1, . . . ,w`〉 ←→ [w1 ∧ · · · ∧ w`] ∈ P(∧`V ).

For example, G1,m = Pm−1. In terms of coordinates,

W = 〈w1, . . . ,w`〉 ∈ G`(V )←→ p(W ) = (pα(AW ))α∈I (`,m) ,

where AW = (aij) is a `×m matrix whose rows are (thecoordinates of) a basis of W and pα(AW ) is the αth minor of AW ,viz., det

(aiαj

)1≤i ,jj≤`, and where


Introduction to Grassmann Varieties Contd.

I (`,m) :=α = (α1, . . . , α`) ∈ Z` : 1 ≤ α1 < · · · < α` ≤ m

.

Facts:

G`,m is a projective algebraic variety given by the commonzeros of certain quadratic homogeneous polynomials in kvariables. As a projective algebraic variety G`,m isnondegenerate, irreducible, nonsingular, and rational.

There is a natural transitive action of GLm on G`,m and if P`denotes the stabilizer of a fixed W0 ∈ G`,m, then P` is amaximal parabolic subgroup of GLm and G`,m ' GLm/P`.

If F = R or C, then G`,m is a (real or complex) manifold, andits cohomology spaces and Betti numbers are explicitlyknown. In fact, bν = dimH2ν(G`,m;C) is precisely the numberof partitions of ν into at most ` parts, each part ≤ m − `,


Grassmannian Over Finite Fields

Suppose F = Fq is the finite field with q elements. ThenG`,m = G`,m(Fq) is a finite set and its cardinality is given by theGaussian binomial coefficient:[

m

`

]q

:=(qm − 1)(qm − q) · · · (qm − q`−1)

(q` − 1)(q` − q) · · · (q` − q`−1).

This is a polynomial in q of degree δ := `(m − `) and in fact,

|G`,m(Fq)| =

[m

`

]q

=δ∑

ν=0

bνqν = qδ + qδ−1 + 2qδ−2 + · · ·+ 1.

Note that

limq→1

[m

`

]q

=

(m

`

).


(Linear) Codes

Fq : finite field with q elements.

[n, k]q-code: a k-dimensional subspace C of Fnq.

Hamming weight of c = (c1, . . . , cn) ∈ Fnq:

wH(c) := #i : ci 6= 0.Hamming weight of a subcode D of C :

wH(D) := #i : ∃ c = (c1, . . . , cn) ∈ D with ci 6= 0.Minimum distance of a (linear) code C :

d(C ) := minwH(c) : c ∈ C , c 6= 0.The r th higher weight of C (1 ≤ r ≤ k):

dr (C ) := minwH(D) : D ⊆ C , dimD = r.C is nondegenerate if C 6⊆ coordinate hyperplane of Fn

q, orequivalently, if dk(C ) = n.


A Nice ExampleReed-Muller Codes

Write Aδ(Fq) := Fδq =P1,P2, . . . ,Pqδ

. Consider the evaluation

map of the polynomial ring in δ variables:

Ev : Fq[X1, . . . ,Xδ]→ Fqδq

f 7−→(f (P1), . . . , f (Pqδ)

),

The r th order generalized Reed-Muller code of length qδ:

RM(r , δ) := Ev (Fq[X1, . . . ,Xδ]≤r ) for r < q.

This has dimension(δ+rr

)and minimum distance (q − r)qδ−1.

More generally, one can consider RM(r , δ) for r ≤ δ(q − 1).Side Remark: An interesting new variant of R-M codes, calledaffine Grassmann codes, has recently been studied; cf. Beelen,Ghorpade, and Høoholdt, IEEE Trans. Inform. Theory, 56 (2010),3166-3176 and 58 (2012), 3843-3855.


Role of Grassmann Varieties in Coding Theory

Gk,n(Fq) may be viewed as the ‘moduli space’ of all [n, k]q-linearcodes. An optimal class of codes, called MDS codes [these are[n, k]q-linear codes for which the minimum distance d is themaximum possible, viz., d = n− k + 1] correspond to certain openstrata of Gk,n. This viewpoint is useful for the following:

Problem

Given a prime power q and integers k , n with 1 ≤ k ≤ n, determine

γ(q) = γ(q; k , n) = # ( [n, k]q-MDS codes) .

This is equivalent to determining

# (inequivalent representations over Fq of Uk,n)

where Uk,n is the so-called uniform matroid.

The problem is open, in general.


A Quick Recap of Matroids

Matroid M on a finite set S : determined by the rank functionr = rM : P(S)→ Z satisfying(i) 0 ≤ r(I ) ≤ |I |, (ii) I ⊆ J ⇒ r(I ) ≤ r(J), and(iii) r(I ∪ J) + r(I ∩ J) ≤ r(I ) + r(J).

I ⊆ S is independent if r(I ) = |I |.base: maximal independent subset of S .uniform matroid Uk,n: the matroid on n elements s.t. any kelements form a base.representation of M over a field F : map f : S → V , where Vis a vector space over F , such that

dim (spanf (x) : x ∈ I) = rM(I ) ∀I ⊆ S .

Representations f1 : S → V1 and f2 : S → V2 are equivalent if∃ an isomorphism φ : V1 → V2 such that φ f1 = f2.

Reference: S. R. Ghorpade and G. Lachaud, Hyperplane sections ofGrassmannians and the number of MDS linear codes, Finite FieldsAppl., 7, (2001), 468–506.


A Geometric Language for CodesProjective Systems a la Tsfasman-Vladut

A [n, k]q-projective system is a collection P of n not necessarilydistinct points in Pk−1; this is nondegenerate if it is not containedin a hyperplane.

[n, k]q-code C [n, k]q-projective system P

Conversely a nondegenerate [n, k]q-projective system gives rise to anondegenerate [n, k]q-code, and the resulting correspondence is abijection, up to equivalence. Note that

d(C) = n −max#P ∩ H : H hyperplane of Pk−1.

and for r = 1, . . . , k ,

dr (C) = n−max#P∩E : E linear subvariety of codim r in Pk−1.


Grassmann Codes

Thanks to the Plucker embedding,

G`,m(Fq) → Pk−1 [n, k]q-code C (`,m).

Length n is the Gaussian binomial coefficient:

n =

[m

`

]q

:=(qm − 1)(qm − q) · · · (qm − q`−1)

(q` − 1)(q` − q) · · · (q` − q`−1).

and the dimension k is the binomial coefficient:

k =

(m

`

).

Theorem (Ryan (1990, q = 2), Nogin (1996, any q))

d (C (`,m)) = qδ where δ := `(m − `).

It may be noted that δ is the dimension of G`,m as a projectivevariety.


Higher Weights of Grassmann Codes

The first result in this direction is:

Theorem (Nogin (1996), Ghorpade-Lachaud(2000))

More generally, for 1 ≤ r ≤ µ we have

dr (C (`,m)) = qδ + qδ−1 + · · ·+ qδ−r+1,

where µ := max`,m − `+ 1.

The alternative proofs in Ghorpade-Lachaud(2000) used acharacterization of the so-called close families of `-subsets of thefinite set 1, . . . ,m. This can be viewed as an analogue foruniform hypergraphs of the elementary result in graph theory that

A simple graph in which any two edges are incident iseither a star or a triangle.

Note: µ = max`,m− `+ 1 is usually much smaller than k =(m`

)and so the above theorem doesn’t give all the higher weights.


Schubert Codes

Let α be in I (`,m), that is,

α = (α1, . . . , α`) ∈ Z`, 1 ≤ α1 < · · · < α` ≤ m.

Consider the corresponding Schubert variety

Ωα := W ∈ G`,m : dim(W ∩ Aαi ) ≥ i ∀i,

where Aj is the span of the first j vectors in a fixed basis of ourm-space. We have

Ωα(Fq) → Pkα−1 [nα, kα]q-code Cα(`,m)

where

nα = |Ωα(Fq)| and kα = |β ∈ I (`,m) : β ≤ α|,

with ≤ being the componentwise partial order:

β = (β1, . . . , β`) ≤ α = (α1, . . . , α`)⇔ βi ≤ αi ∀i .


Minimum Distance of Schubert Codes

Proposition (Ghorpade-Lachaud(2000))

For any α ∈ I (`,m),

d (Cα(`,m)) ≤ qδα where δα :=∑i=1

(αi − i).

It may be noted that when α is the “maximal element”(m − `+ 1, . . . ,m − 1,m) of I (`,m), then Ωα = G`,m whileδα = δ = `(m − `) and so the above inequality is an equality. Infact, the following conjecture was made in the same paper:

Minimum Distance Conjecture (MDC)

For any α ∈ I (`,m),

d (Cα(`,m)) = qδα .


Length of Schubert Codes

If ` = 2 and α = (m − h − 1,m), then

nα =(qm − 1)(qm−1 − 1)

(q2 − 1)(q − 1)−

h∑j=1

j∑i=1

q2m−j−2−i

and

kα =m(m − 1)

2− h(h + 1)

2.

[Hao Chen (2000)]

In general,

nα =∑ `−1∏

i=0

[αi+1 − αi

ki+1 − ki

]q

q(αi−ki )(ki+1−ki )

where the sum is over (k1, . . . , k`−1) ∈ Z` satisfyingi ≤ ki ≤ αi and ki ≤ ki+1 for 1 ≤ i ≤ `− 1; by convention,α0 = 0 = k0 and k` = `. [Vincenti (2001)]


Length of Schubert Codes (Contd.)

nα =∑β≤α

qδβ , where δβ =∑i=1

(βi − i).

Ehresmann (1934); Ghorpade-Tsfasman (2005)Suppose α has u + 1 consecutive blocks:α = (α1, . . . , αp1 , . . . , αpu+1, . . . , αpu+1). Then

nα =

αp1∑s1=p1

· · ·αpu∑

su=pu

u∏i=0

λ(αpi , αpi+1 ; si , si+1)

where, s0 = p0 = 0; su+1 = pu+1 = `, and

λ(a, b; s, t) :=t∑

r=s

(−1)r−sq(r−s2 )[a− s

r − s

]q

[b − r

t − r

]q

.

[Ghorpade-Tsfasman (2005)]

nα = det

(q(j−i)(j−i−1)/2

[αj − j + 1

i − j + 1

]q

)1≤i ,j≤`

.

[Ghorpade-Krattenthaler]


Dimension of Schubert Codes [G-Tsfasman (2005)]

Let α = (α1, . . . , α`) ∈ I (`,m). The dimension of Cα(`,m) isthe `× ` determinant:

kα =

∣∣∣∣∣∣∣∣∣

(α11

)1 0 . . . 0(

α12

) (α2−11

)1 . . . 0

......(

α1`

) (α2−1`−1) (

α3−2`−2)

. . .(α`−`+1

1

)∣∣∣∣∣∣∣∣∣ .

If α1, . . . , α` are in arithmetic progression, i.e.,αi = c(i − 1) + d ∀i for some c , d ∈ Z, then

kα =α1

`!

`−1∏i=1

(α`+1 − i) =α1

α`+1

(α`+1

`

)where α`+1 = c`+ d = `α2 + (1− `)α1.

kα =

αp1∑s1=p1

αp2∑s2=p2

· · ·αpu∑

su=pu

u∏i=0

(αpi+1 − αpi

si+1 − si

),


What do we know about the MDC?

Recall that the MDC states that d (Cα(`,m)) = qδα , whereδα := (α1 − 1) + · · ·+ (α` − `).

True if α = (m − `+ 1, . . . ,m − 1,m). [Nogin]

True if ` = 2. [Hao Chen (2000)]; and independently,[Guerra-Vincenti (2002)].

Lower bound for d (Cα(`,m)) [G-V (2002)]:

qα1(qα2 − qα1) · · · (qα` − qα`−1)

q1+2+···+` ≥ qδα−`.

MDC is true for C(2,4)(2, 4). [Vincenti (2001)]

MDC is true for all Schubert divisors in G`,m.[Ghorpade-Tsfasman (2005)]

MDC is true, in general! [Xu Xiang (2008)]


Back to Higher Weights of Grassmann Codes

Recall: µ := max`,m − `+ 1 and we had:

Theorem (Nogin (1996), Ghorpade-Lachaud(2000))

For 1 ≤ r ≤ µ, we have

dr (C (`,m)) = qδ + qδ−1 + · · ·+ qδ−r+1.

One has the following counterpart from the other end.

Theorem (Hansen-Johnsen-Ranestad (2007))

On the other hand, for 0 ≤ r ≤ µ,

dk−r (C (`,m)) = n − (1 + q + · · ·+ qr−1).

These results cover several initial and terminal elements of theweight hierarchy of C (`,m). Yet, a considerable gap remains.


Narrowing the gap

Examples:

(`,m) = (2, 5). Here k = 10, µ = 4 and we know:

d1, . . . , d4 as well as d6, . . . , d10.

But d5 seems to be unknown.

(`,m) = (2, 6). Here k = 15, µ = 5 and d6, . . . , d9 are notknown.

For C (2,m) with m ≥ 2, the values of dr for m ≤ r <(m−1

2

)do not seem to be known.

Theorem (Hansen-Johnsen-Ranestad (2007))

d5(C (2, 5)) = q6 + q5 + 2q4 + q3 = d4 + q4.

Conjecture (Hansen-Johnsen-Ranestad (2007))

dr − dr−1 is always a power of q.


One step forward

Theorem (Ghorpade-Patil-Pillai (2009))

Assume that ` = 2 and m ≥ 4 so that

µ = max2,m − 2+ 1 = m − 1 and k =

(m

2

).

Thendµ+1(C (2,m)) = dµ + qδ−2

anddk−µ−1(C (2,m)=n−

(1 + q + · · ·+ qµ + q2

).

Corollary. Complete weight hierarchy of C (2, 6).Remark. The proof of the above theorem uses a characterizationof decomposable subspaces of ∧`V where V is an m-dimensionalvector space, and this auxiliary result can be viewed as an algebraicanalogue of the structure theorem for close families of `-subsets ofan m-set.


Complete weight hierarchy of C (2,m)

Consider the (strict) Young tableau Y = Ym corresponding to thepartition (m − 1,m − 2, . . . , 2, 1) of k =

(m2

)with

Yij = 2i + j − 3 for 1 ≤ i ≤ m − 1 and 1 ≤ j ≤ m − i .

Thenn = Cq(Y ) =

∑ν≥0

cν(Y )qν ,

where cν(Y )= # of times ν appears in Y .

Theorem (Ghorpade-Patil-Pillai)

If T1, . . . ,Th are strict subtableaux of Y of area r = k − s, and

gs(2,m) := maxCq(T1), . . . ,Cq(Th),

thends (C (2,m)) = n − gs(2,m) for 1 ≤ s ≤ k .


Codes associated to flag varieties

Fix an m-dimensional vector space V and a sequence

` = (`1, . . . , `s) ∈ Zs with 0 < `1 < · · · < `s < m.

We can consider the partial flag variety

F`(V ) := (V1, . . . ,Vs) : V1 ⊂ · · · ⊂ Vs , dimVi = ì .

Thanks to Plucker and Segre,

F`(V ) →s∏

i=1

Gì ,m →s∏

i=1

Pki−1 →s∏

i=1

P(k1···ks)−1

where ki =(mì

). As before,

F`(V ) (Fq) [n`, k`]q-code C (`;m).


Special partial flags

The parameters of the code C (`;m) are known in a special case.

Theorem (Rodier (2003))

If s = 2 and ` = (1,m − 1), then

n` =(qm − 1)(qm−1 − 1)

(q − 1)2and k` = m2 − 1.

Moreover,d(C (`;m)) = q2m−3 − qm−2.


The length n` of C (`,m)

n` =

[m

`1, `2 − `1, . . . , `s+1 − `s

]=

s∏i=1

[m − ì−1ì − ì−1

]where, by convention, `0 := 0 and `s+1 := m.Equivalently, the length n` is given by

n` =∑σ∈W`

qinv(σ) =∑τ∈M`

qinv(τ)

where W`: permutations σ ∈ Sm satisfying

σ(ì−1 + 1) < σ(ì−1 + 2) < · · · < σ(ì ),

for i = 1, . . . , s + 1, and M`: permutations of the multiset

1`1 , 2`2−`1 , . . . , s`s−`s−1 , (s + 1)m−`s

and inv denotes the number of inversions.See, for example, [Ghorpade-Lachaud(2002)].


Dimension k` of C (`,m)

Given ` = (`1, . . . , `s), consider the partition

m − `1 > m − `2 > · · · > m − `s ≥ 1

and let λ be the conjugate partition.For example if m = 8 and ` = (1, 3, 6, 7), then the associatedpartition is (7, 5, 2, 1) and the conjugate partition isλ = (4, 3, 2, 2, 2, 1, 1). These partitions can be viewed as folllows.


Description of k` Contd.

For each box (i , j) in the (Young diagram of) λ, let h(i ,j) be thehooklength at (i , j), that is, the number of boxes in the hook at(i , j). For example the hook at (2, 2) in the partitionλ = (4, 3, 2, 2, 2, 1, 1) is shown by shaded boxes and we haveh(2,2) = 5.

Theorem

The dimension k` of C (`,m) is given by

k` =∏

(i ,j)∈λ

m + j − i

h(i ,j).

Idea: Use the connection between flag varieties and representationsof linear groups together with classical results from CombinatorialRepresentation Theory.


Rodier recovered!

Example: If ` = (1,m − 1), then

λ = conjugate of (m − 1, 1) = (2, 1, 1, . . . , 1︸︷︷︸m−2 times

).

Hence by the above formula

k` =m(m + 1)(m − 1)(m − 2) · · · (m − (m − 2))

m(1)(m − 2)(m − 3) · · · 1= (m + 1)(m − 1) = m2 − 1,

as is to be expected.

b

b

b

m− 1


Grassmann codes

Another Example (Grassmann codes): If ` = (`), then

λ = conjugate of (m − `) = (1, 1, . . . , 1︸︷︷︸m−` times

).

Hence by the above formula

k` =m(m − 1)(m − 2) · · · (m − (m − `) + 1)

(m − `)(m − `− 1) · · · 1

=m!

`!(m − `)!=

(m

`

)as is to be expected.

b

b

b

m− ℓ


Thank you!


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