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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
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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 − `,
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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
`
).
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
(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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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 .
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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 .
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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)]
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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]
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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
),
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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)]
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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 .
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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 = `i .
Thanks to Plucker and Segre,
F`(V ) →s∏
i=1
G`i ,m →s∏
i=1
Pki−1 →s∏
i=1
P(k1···ks)−1
where ki =(m`i
). As before,
F`(V ) (Fq) [n`, k`]q-code C (`;m).
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
The length n` of C (`,m)
n` =
[m
`1, `2 − `1, . . . , `s+1 − `s
]=
s∏i=1
[m − `i−1`i − `i−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(`i−1 + 1) < σ(`i−1 + 2) < · · · < σ(`i ),
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)].
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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.
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes
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
Sudhir R. Ghorpade Grassmann Varieties and Linear Codes