An Asymptotic Formula in $q$ for the Number of $[n, k]~q$ -Ary MDS Codes

$Page 1: An Asymptotic Formula in $q$ for the Number of $[n, k]~q$ -Ary MDS Codes$
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 11, NOVEMBER 2014 7047

An Asymptotic Formula in q for the Numberof [n, k] q-Ary MDS Codes

Krishna V. Kaipa

Abstract— We obtain an asymptotic formula in q, as well asnew upper and lower bounds, for the number of MDS codes oflength n and dimension k over a finite field with q elements.

Index Terms— MDS codes, Grassmannian.I. INTRODUCTION

MDS CODES of dimension k and length n over Fq

are codes for which the minimum distance equals theSingleton bound n − k + 1. (In this paper all codes consideredare linear codes). Equivalently these are codes for which anyk × n generator matrix has the property that all its k × ksubmatrices are nonsingular. Thus [n, k]q MDS codes can beidentified with the set of row equivalence classes of k × nmatrices over Fq having the property that any k columnsare linearly independent. Let γ (k, n; q) denote the number ofq-ary [n, k] MDS codes. It is well known that the dual of anMDS code is also MDS; therefore γ (k, n; q) = γ (n−k, n; q).Consequently, we henceforth assume without loss of generalitythat k ≤ n/2. Exact values of γ (k, n; q) are hard to determineand are known only for k = 1, 2, all n, and for k = 3 andn ≤ 9. The formula for γ (3, 9; q) was found in the work [8]in 1995. The formula for γ (3, 8; q) and γ (3, 7; q) were foundin [5, Th. 4.4 and 4.2]. In the case of characteristic 2, aformula for γ (3, 7; q) was found independently in [12, Th. 2].The formula for γ (3, 6; q) can be found in [13, p. 220].Formulae for γ (3, 10; q) and γ (4, 8; q) for example, are stillunknown. The known exact formulae can be written in theasymptotic form:

γ (k, n; q) = qδ + (1 − N) qδ−1 + a2(k, n) qδ−2 + O(qδ−3),

(1)

where

δ := k(n − k) and N :=(

n

k

). (2)

We reproduce from the known exact formulae for γ (k, n; q)(see [8, §1]) the corresponding values of a2(k, n).

a2(1, n) = n2 − 3n + 2

2,

a2(2, n) = 3n4 − 10n3 + 9n2 − 26n + 48

24,

Manuscript received November 1, 2013; revised August 3, 2014; acceptedAugust 4, 2014. Date of publication August 20, 2014; date of current versionOctober 16, 2014. K. Kaipa was supported by the Indo-Russian Project underGrant INT/RFBR/P-114 through the Department of Science and Technology,Government of India.

The author is with the Department of Mathematics, Indian Instituteof Science and Education Research, Pune 411008, India (e-mail:[email protected]).

Communicated by N. Kashyap, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2014.2349903

a2(3, 6) = 152,

a2(3, 7) = 506,

a2(3, 8) = 1360,

a2(3, 9) = 3158. (3)

For general (k, n), certain bounds on γ (k, n; q) wereobtained in [3] which imply the following asymptotic formula:

γ (k, n; q) = qδ − (N − 1) qδ−1 + O(qδ−2). (4)

Our main result is:Theorem 1.1: There is a positive constant c(k, n) depend-

ing only on k and n, such that:∣∣γ (k, n; q) − qδ − (1 − N) qδ−1 − a2(k, n) qδ−2∣∣

≤ c(k, n) qδ−3,

where,

a2(k, n) = N2 − 5N + 4

2− Nδ

2

(δ − n − 3

δ + n + 1

). (5)

In particular:

γ (k, n; q) = qδ − (N − 1) qδ−1 + a2(k, n) qδ−2 + O(qδ−3).

(6)An expression for the constant c(k, n) is given in (44). The

formulae in (3) agree with the formula (6) of Theorem 1.1.The quantity

γ (k, n; q) := γ (k, n; q)/(q − 1)n−1

is always an integer, as it is the number of n-arcs inPG(k − 1; q) (see [13, pp. 219–220], [8, Lemma 2]).We rewrite the theorem in terms of γ (k, n; q):

Corollary 1.2: The number γ (k, n; q) of n-arcs inPG(k − 1, q) is of the asymptotic form

qδ−n+1 − b1(k, n) qδ−n+2 + b2(k, n) qδ−n+3 + O(qδ−n+4),

where b1(k, n) = N − n, and

b2(k, n) = a2(k, n) − (n − 1)(N − n) − (n2 − 3n + 2)/2.

In particular:

γ (3, 10; q) = q12 − 110 q11 + 5561 q10 + O(q9),

γ (4, 8; q) = q9 − 62 q8 + 1710 q7 + O(q6). (7)The ideas of the proof of Theorem 1.1 are as follows.

Following [3] and [13], we identify the set of MDS codeswith a subset U(k, n) of the Grassmannian G(k, n) ofk-dimensional subspaces of F

nq . The subset U(k, n) is an

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7048 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 11, NOVEMBER 2014

intersection of N Schubert cells of G(k, n), and hence its car-dinality |U(k, n)| can be expressed using inclusion-exclusionprinciple as a sum

E1 − E2 + E3 − · · · + (−1)N−1 EN .

For each r , the quantity Er is a sum of(N

r

)terms each of

which is of the form |G(k, n)\L| where L is a codimensionr linear subspace of the Plücker space P(∧k

Fnq) (in which

G(k, n) embeds). In [3] asymptotic formulae of the form

qδ + a qδ−1 + O(qδ−2)

were obtained for each |G(k, n)\ L|, and hence for the Er ’sto obtain the result given in equation (4). We study linearsections |G(k, n) ∩ L| more closely in section III, leading toTheorem 3.12 which gives an asymptotic formula of the form

qδ + a qδ−1 + b qδ−2 + O(qδ−3)

for |G(k, n)\L|. These formulae in turn imply the desiredasymptotic formulae for the Er ’s and hence for γ (k, n; q).

II. MDS CODES AS A SUBSET OF THE GRASSMANNIAN

Let V = Fnq with standard basis {e1, . . . , en}. Let

Ik,n := {(i1, i2, . . . , ik) : 1 ≤ i1 < · · · < ik ≤ n}, (8)

and for a multi-index I = (i1, . . . , ik) ∈ Ik,n let

eI := ei1 ∧ · · · ∧ eik ∈ ∧k V .

The multivectors {eI : I ∈ Ik,n} form the standard basisfor ∧k V . Let G(k, n) or Gk(V ) denote the Grassmannianof k dimensional subspaces of F

nq . We recall the the Plücker

embedding of Gk(V ) into P(∧k V ). For any � ∈ Gk(V ) andk ×n matrix M whose rows b1, . . . , bk form a basis for �, let

i(�) := [b1 ∧ · · · ∧ bk] ∈ P(∧k V ).

It is easy to see that i(�) depends only on the row equivalenceclass of the matrix M , and hence only on �. Therefore we havea function i : Gk(V ) → P(∧k V ) which can also be shown tobe injective. The image i(Gk(V )), which we will denote againby Gk(V ) is cut out by the quadratic Plücker relations (19),and hence i : Gk(V ) ↪→ P(∧k V ) realizes Gk(V ) as aprojective variety. Expanding the expression b1∧· · ·∧bk abovein terms of the standard basis of ∧k V we obtain

b1 ∧ · · · ∧ bk =∑

I∈Ik,n

pI (�) eI ,

where pI (�) is the minor of the matrix M on the columnsindexed by I . It follows that pI (�), I ∈ Ik,n regarded ashomogeneous coordinates, depend only on � and are knownas its Plücker coordinates.

Any q-ary code of dimension k and length n, being ak dimensional subspace of F

nq , can be regarded as point

� of Gk(V ). The matrix M in the above discussion isprecisely a generator matrix of the code �. As observed inthe introduction, the code � is [n, k]-MDS, if and only ifall k × k submatrices of M are non-singular. Hence the set of

[n, k]q MDS codes can be identified with the subset of Gk(V )defined as:

U(k, n) := {� ∈ G(k, n) : pI (�) �= 0, ∀ I ∈ Ik,n}. (9)

In particular, we note

γ (k, n; q) = |U(k, n)|.

For each of the N = (nk

)multi-indices I ∈ Ik,n let

CI := {� ∈ G(k, n) : pI (�) �= 0}. (10)

For each � ∈ CI , there is a unique k × n matrix M whoserows forms a basis for �, and which satisfies the property thatthe k × k submatrix of M on the columns indexed by I is thek ×k identity matrix. Conversely the row space of a matrix Mwhich has the k × k identity matrix as the submatrix on thecolumns indexed by I , is a point of CI . Therefore CI can beidentified with the set of all such matrices. Since the columnsof M indexed by I are fixed and the remaining n − k columnsfree, we see that

|CI | = qk(n−k) = qδ. (11)

It follows from the definitions (9) and (10), that

U(k, n) =⋂

I∈Ik,n

CI . (12)

For each r with 1 ≤ r ≤ N , let I rk,n denote the subsets of Ik,n

of cardinality r . Let

Er :=∑

{I1,...,Ir }∈I rk,n

|CI1 ∪ · · · ∪ CIr |. (13)

It follows from the inclusion-exclusion principle appliedto (12) that

|U(k, n)| = E1 − E2 + E3 − · · · + (−1)N−1 EN . (14)

The quantity Er is a sum of(N

r

)terms of the form |CI1 ∪· · ·∪

CIr |. The set

Gk(V )\(CI1 ∪ · · · ∪ CIr )

consists of points of Gk(V ) having Plücker coordinatespI1, . . . , pIr zero. Therefore if L is the codimension r sub-space of ∧k V spanned by

{eI : I /∈ {I1, . . . , Ir }},

and L = PL, then

Gk(V )\(CI1 ∪ · · · ∪ CIr ) = Gk(V ) ∩ L .

The set Gk(V )∩ L is an example of a linear section of Gk(V )which we study in the next section.

KAIPA: ASYMPTOTIC FORMULA IN q FOR THE NUMBER OF [n, k] q-ARY MDS CODES 7049

III. LINEAR SECTIONS OF THE GRASSMANNIAN

We say L ⊂ P(∧k V ) is a codimension r linear subspaceif L = PL with L being a codimension r linear subspaceof ∧k V . Corresponding to the standard basis e1, . . . , en of V ,the dual space V ∗ has the standard dual basis e1, . . . , en

defined by 〈ei , e j 〉 = δi j . Here 〈, 〉 is the natural pairingbetween V ∗ and V . Similarly the space ∧k V ∗ has the standardbasis {eI : I ∈ Ik,n} where eI = ei1 ∧ · · · ∧ eik . Given kelements ω1, . . . , ωk of V ∗ and k elements v1, . . . , vk of V ,the determinant of the k × k matrix with entry in row i andcolumn j being 〈ωi , v j 〉, is multilinear and alternating in both(ω1, . . . , ωk) and (v1, . . . , vk). It therefore defines a bilinearpairing 〈, 〉 between ∧k V ∗ and ∧k V . In terms of the standardbases we have 〈eI , eJ 〉 = δI J , which also shows that thepairing is non-degenerate and hence gives an isomorphismbetween ∧k V ∗ and (∧k V )∗. We refer to elements of ∧k V ∗in short as k-forms.

Given a codimension r linear subspace L = P(L)of P(∧k V ), let

Ann(L) := {ω ∈ ∧k V ∗ : 〈ω, ξ〉 = 0, ∀ ξ ∈ L},and let Ann(L) = P(Ann(L)). We note that Ann(L) is a r −1dimensional linear subspace of P(∧kV ∗). The correspondenceL ↔ Ann(L) allows us to identify codimension r linearsubspaces of P(∧kV ) with r −1 dimensional linear subspacesof P(∧k V ∗). We define

||L|| := |Gk(V )\L|. (15)

We call a codimension 1 subspace H of P(∧k V ), a hyperplane.It follows from the above discussion that each hyperplanecorresponds to a unique k-form (upto a nonzero scalar mul-tiple) ω. We write H = Hω to emphasize this, and we alsodefine

||ω|| := ||Hω|| = |Gk(V )\Hω|. (16)

The main results of this section are Theorems 3.11 and 3.12which appear at the end. In the remaining part of this section,we either recall or develop several results leading up to themain results.

Definition 3.1: A linear subspace of Gk(V ) is a linearsubspace of P(∧k V ) which is entirely contained in Gk(V ).

For each α ∈ Gk−1(V ) and each γ ∈ Gk+1(V ), we define:

πα := {β ∈ Gk(V ) : β ⊃ α},πγ := {β ∈ Gk(V ) : β ⊂ γ }. (17)

Since πα is projectively isomorphic to P(V/α), it is alinear subspace of Gk(V ) of dimension n − k. Similarly πγ

is projectively isomorphic to P(γ ∗), and hence it is a linearsubspace of Gk(V ) of dimension k. It is a classical fact[1, §2.1], [11, Proposition 3.2] that, for k ≥ 2, the πα’sand the πγ ’s are the maximal linear subspaces of Gk(V ).Some facts about these spaces that we will need are asfollows. They easily follow from the definitions of πα and πγ

(also see [2, p. 88]).Fact 3.2: 1) For α �= α′ ∈ Gk−1(V ) the intersection

πα ∩πα′ is empty if dim(α +α′) > k and consists of thesingle point α + α′ if dim(α + α′) = k.

2) For γ �= γ ′ ∈ Gk+1(V ) the intersection πγ ∩ πγ ′is

empty if dim(γ ∩ γ ′) < k and consists of the singlepoint γ ∩ γ ′ if dim(γ ∩ γ ′) = k.

3) The intersection πα ∩ πγ is empty if α �⊂ γ , and it isthe line {β : α ⊂ β ⊂ γ } if α ⊂ γ .

4) For any pair β, β ′ ∈ πα the intersection β ∩β ′ is α, andfor any pair β, β ′ ∈ πγ the vector space sum β + β ′is γ .

We recall the definition of the interior multiplication oper-ator: for ξ ∈ ∧�V , ζ ∈ ∧m V , and ω ∈ ∧�+m V ∗, ιξω ∈ ∧m V ∗is defined by:

〈ιξ ω, ζ 〉 = 〈ω, ξ ∧ ζ 〉.For any nonzero ω ∈ ∧k V ∗, we define subspaces Vω ⊂ V andUω ⊂ V ∗ by

Vω := {v ∈ V : ιvω = 0},Uω := {θ ∈ V ∗ : 〈θ, v〉 = 0 ∀ v ∈ Vω}. (18)

Fact 3.3: We recall [6, p. 210] that ω is decompos-able if and only if dim(Vω) = n − k or equivalentlydim(Uω) = k. We also note that if ω is indecomposable thendim(Uω) ≥ k + 2 because ω ∈ ∧kUω and every element of∧k

Fk+1 is decomposable.

The next lemma will be used in the proof of Theorem 3.11.Lemma 3.4: Let k ≥ 2. A k + 1-form ω ∈ ∧k+1V ∗ is

decomposable if and only if ιvω is decomposable for all v ∈ V .Proof: If ω is decomposable, then so is ιvω. Suppose, by

way of contradiction, that ιvω is decomposable for all v ∈ Vbut ω is indecomposable. In this case dim(Uω) = k + 1 + s,where s ≥ 2 as noted in Fact 3.3. Using the fact thatω ∈ ∧kUω, we note that

Sω := P{ιvω : v ∈ V } ⊂ P(∧kUω)

is a k + s dimensional linear subspace of Gk(Uω). Thedimensions of the maximal linear subspaces of Gk(Uω) are kand dim(Uω) − k = s + 1. Clearly k < k + s, and s + 1 <k + s because k ≥ 2. Therefore Sω is not contained in anymaximal subspace of Gk(Uω). This contradiction shows thatour assumption that ω is indecomposable is false.

The next lemma will be used in the proof of Proposition 3.8.Lemma 3.5: If P is a subset of Gk(V ) such that for

every pair P, Q ∈ P , the line P Q joining them is containedin Gk(V ), then the linear subspace of P(∧k V ) generated byP is completely contained in Gk(V ).

Proof: An element λ ∈ ∧k V is decomposable if andonly if it satisfies the Plücker relations ( [6, pp. 210–211] or[7, p. 312, eq. (4)])

(ιξ λ) ∧ λ = 0 ∀ ξ ∈ ∧k−1V ∗. (19)

Let λ1, λ2, . . . , λn be decomposable elements of ∧k V repre-senting points P1, P2, . . . , Pn of P . For each ξ ∈ ∧k−1V ∗, letai (ξ) := (ιξ λi ) ∧ λi for 1 ≤ i ≤ n, and let

ai j (ξ) := (ιξ λi ) ∧ λ j + (ιξ λ j ) ∧ λi

for 1 ≤ i < j ≤ n. The condition for λ = ∑tiλi to be

decomposable is:∑i

t2i ai (ξ) +

∑i< j

ti t j ai j (ξ) = 0, ∀ ξ ∈ ∧k−1V ∗.


Using (19), ai (ξ) = 0 because λi is decomposable, andai j (ξ) = 0 because Pi Pj ⊂ Gk(V ).

Some of the results we will need about the cardinalities ||L||(as defined in (15)) appear in the literature in the context of lin-ear codes associated to the embedding G(k, n) ↪→ P(∧k

Fnq).

We briefly describe the construction of these Grassmann codesC(k, n), the details of which can be found in [10]. These arelinear codes of length n and dimension k where:

n = |G(k, n)| = (qn − 1)(qn−1 − 1). . .(qn−k+1 − 1)

(qk − 1)(qk−1 − 1). . .(q − 1),

k =(

n

k

). (20)

Let P1, P2, . . . , Pn denote representatives in ∧kF

nq of the n

points of G(k, n) arranged in some order. We also arrange insome order, the k basic vectors {eI : I ∈ Ik,n} of ∧k

Fnq . A k×n

generator matrix for the code C(k, n) has for its (i, j)-th entry,the i -th Plücker coordinate pi(Pj ). The fact that the Plückerembedding is non-degenerate (i.e no hyperplane of P(∧k

Fnq)

contains G(k, n)) is equivalent to the fact that the generatormatrix above has full row rank. A codeword of C(k, n) is ofthe form (∑

I

aI pI (P1), . . . ,∑

I

aI pI (Pn)

).

If ω is the k-form∑

I aI eI , then the above codeword canbe expressed as (ω(P1), . . . , ω(Pn)). In this way we identifycodewords of C(k, n) with k-forms on V = F

nq . Moreover,

the Hamming weight of the above codeword clearly coincideswith the expression ||ω|| as defined in (16). More generally, letL be a codimension r subspace of ∧k

Fnq , and let {ω1, . . . , ωr }

be a basis for Ann(L). Evaluating elements of Ann(L), i.eforms

∑ri=1 aiωi on P1, . . . , Pn gives a r dimensional subcode

of C(k, n). Moreover, the Hamming weight of this subcodeis clearly ||L|| as defined in (15) (where L = P(L)). Theminimum distance d1(C(k, n)) of the code C(k, n) equalsminimum value of ||ω|| over all nonzero k-forms on V , andthe higher weight dr (C(k, n)) equals the minimum value of||L|| over all codimension r subspaces of P(∧k V ).

Using the well known formula (see [14, p. 7])

||D|| = 1

qr − qr−1

∑c∈D

||c||,

relating the Hamming weight of a r dimensional subcode Dto the Hamming weights of its constituent codewords, we getthe corresponding formula

||L|| = 1

qr−1

∑[ω]∈Ann(L)

||ω|| = 1

qr−1

∑H⊃L

||H ||, (21)

where the sum H ⊃ L is over all hyperplanes Hcontaining L.

The next theorem summarizes the information we will needabout the minimum distance, higher weights and the weightspectrum of the code C(k, n).

Theorem 3.6 [Nogin] [10]:1) The minimum distance d1(C(k, n)) equals qδ. The

codewords of minimum weight correspond precisely withdecomposable k-forms.

2) More generally, for 1 ≤ r ≤ n−k +1, the higher weight

dr (C(k, n)) = qδ + qδ−1 + · · · + qδ−r+1.

The r dimensional subcodes with minimum weight corre-spond exactly to codimension r subspaces L of P(∧kV )such that Ann(L) is a linear subspace of Gk(V ∗).

3) For the code C(2, n), up to a code-isomorphism anycodeword corresponds to one of the 2-forms

ωr = e1 ∧ e2 + e3 ∧ e5 + · · · + e2r−1 ∧ e2r,

where 1 ≤ r ≤ �n/2�. Moreover

||ωr || = qδ + qδ−2 + · · · + qδ−2r+2.Proof: We refer to [10] for the proofs of parts 1) and 3).

We give a quick proof of part 2) of the theorem (see also[4, Th. 20]). For a codimension r subspace L ⊂ P(∧k V ),there are

1 + q + q2 + · · · + qr−1

elements in Ann(L), and each [ω] ∈ Ann(L) satisfies||ω|| ≥ qδ (by the first part of the theorem). Theformula (21) gives

||L|| ≥ qδ(1 + q + · · · + qr−1)

qr−1 = qδ + qδ−1 + · · · + qδ−r+1.

Moreover, equality holds above if and only if each [ω] inAnn(L) satisfies ||ω|| = qδ, or equivalently is decomposable.In other words equality holds if and only if Ann(L) is ar − 1 dimensional linear subspace of Gk(V ∗). Since linearsubspaces of Gk(V ∗) exist only in dimensions less than orequal to n − k, taking such subspaces for Ann(L) weobtain the desired formula for dr (C(k, n)) when1 ≤ r ≤ n − k + 1.

The next lemma relates the weight of a C(k, n) codewordto certain codewords of C(k − 1, n − 1) associated with it.For a nonzero ω ∈ ∧k V ∗ and a vector u /∈ Vω, let ωu

denote ιuω regarded as a k − 1 form on the quotient vectorspace V/(Fqu). The weight ||ω|| can be expressed in terms ofthe weights ||ωu || for u /∈ Vω using a refinement of the proof of[10, Proposition 4.4]:

Lemma 3.7: For a vector u /∈ Vω, and ωu as defind above,we have ||ωu || = ||ιu ω||/qk−1. Moreover,

||ω|| = 1

qk − 1

∑u /∈Vω

||ωu||.Proof: Let us denote the cardinality of the general linear

group GL(m, Fq) by

[m]q := qm(m−1)/2 (qm − 1)(qm−1 − 1) · · · (q − 1).

Let [v1, v2, . . . , vm ] denote an ordered set of m vectors of V .By definition of ||ω|| = |Gk(V )\Hω| we get

[k]q · ||ω|| = ∣∣{[v1, v2, . . . , vk ] : 〈ω, v1 ∧ · · · ∧ vk〉 �= 0}∣∣

=∑

u /∈Vω

|{[v2, . . . , vk ] : 〈ιuω, v2 ∧ · · · ∧ vk〉 �= 0}|

= [k − 1]q ·∑

u /∈Vω

||ιuω||

It follows from the definition of ωu that 〈ιuω, v2∧· · ·∧vk〉 �= 0if and only if 〈ωu , v2 ∧ . . . ∧vk〉 �= 0, where vi for 2 ≤ i ≤ k


denotes the class of vi in V/(Fqu). Since the function taking[v2, . . . , vk ] to [v2, . . . , vk ] is qk−1 -to-one, we obtain that

[k − 1]q · ||ωu || = |{[v2, . . . , vk ] : 〈ωu , v2 ∧ · · · ∧ vk〉 �= 0}|= [k − 1]q

qk−1 ||ιuω||.

Therefore, we get ||ωu || = ||ιu ω||/qk−1. Using this formulain the equation above for [k]q · ||ω||, we get:

||ω|| = qk−1 [k − 1]q

[k]q

∑u /∈Vω

||ωu ||.

The lemma now follows by noting that

[k]q = qk−1(qk − 1) [k − 1]q .

Proposition 3.8: For 1 ≤ m ≤ n − k + 1, let L bean m dimensional subspace of P(∧k V ) such that L is notcontained in Gk(V ). Suppose L1 ⊂ L is an m−1 dimensionalsubspace which is contained in Gk(V ). Then we have:

|L ∩ Gk(V )| − |L1| ≤

⎧⎪⎨⎪⎩

1 if m = 1,

q if m = 2,

q2 if m ≥ 3.

(22)

Proof: If there is no point P ∈ L ∩ Gk(V ) which is notin L1, then |L ∩Gk(V )|− |L1| being zero, clearly satisfies thestated bounds. So we assume such a point P exists. Let L2be the subset of L1 defined by

L2 := {Q ∈ L1 : P Q ⊂ Gk(V )},where, for any two points P1 �= P2 ∈ Gk(V ), P1 P2 denotesthe line joining P1 and P2. By extending a basis of P1 ∩ P2to a basis of P1 and P2, it is easy to see that

P1 P2 ∩ Gk(V ) ={

P1 P2 if dim(P1 ∩ P2) = k − 1,

{P1, P2} if dim(P1 ∩ P2) < k − 1.

This together with the fact that L = ∪P ′∈L1 P P ′, gives:

|L ∩ Gk(V )| − |L1| = 1 + (q − 1)|L2|. (23)

Let P = L2 ∪{P}. Given P1, P2 ∈ P , if P1 or P2 equals P ,then by definition of L2 the line P1 P2 is contained in Gk(V ).Otherwise P1, P2 ∈ L2 ⊂ L1, and since L1 ⊂ Gk(V ) isa linear subspace we again get P1 P2 ⊂ Gk(V ). Thereforewe can apply Lemma 3.5 to P to conclude that the linearsubspace L(P) generated by P is contained in Gk(V ). Letπ ′ be a maximal linear subspace of Gk(V ) containing L(P).If Q, Q′ ∈ L2 and Q′′ ∈ QQ′, then P Q′′ ∈ L(P), and henceP Q′′ ⊂ Gk(V ). This shows that Q′′ ∈ L2, and hence that L2is a linear subspace of Gk(V ).

Let π be a maximal linear subspace of Gk(V ) contain-ing L1. We note that P /∈ π , for otherwise all lines joiningP to L1 are contained in Gk(V ), and hence L itself, being theunion of these lines, is contained in Gk(V ), which is not trueby hypothesis. Therefore π �= π ′. By parts 1)-3) of Fact 3.2,π ∩ π ′ is either a line, a point or empty. Since L2 is a linearsubspace of π ∩ π ′, we conclude that

|L2| ∈ {0, 1, 1 + q}.

Using this in (23) we get:

|L ∩ Gk(V )| − |L1| ∈ {1, q, q2}.The fact that L �⊂ Gk(V ) implies |L ∩ Gk(V )| −|L1| < qm . Therefore, in the case m = 1 we get|L ∩ Gk(V )| − |L1| = 1, and in the case m = 2, we get|L ∩ Gk(V )| − |L1| ∈ {1, q}.

Remark : The above proposition generalizes [4, Lem-mas 23 and 24] which obtain the same result in the casewhen k = 2.

Proposition 3.9: Let L be an � dimensional linear subspaceof P(∧k V ) which is not contained in Gk(V ). Also suppose� ≥ 3. Then:

|L ∩ Gk(V )| ≤ 1 + q + 2q2 + q3 + · · · + q�−1.Proof: Let L ′′ be a subspace of L which is maximal with

respect to the property of being contained in Gk(V ). If L ′′ = ∅

then |L ∩ Gk(V )| = 0 satisfies the asserted bound. So weassume dim(L ′′) = μ ≥ 0 and let

F := {L ′ ⊂ L : dim(L ′) = μ + 1, L ′ ⊃ L ′′}.We note that |F | = |P�−μ−1|. Each point of L is containedin some L ′ ∈ F , and any two distinct elements of F intersectin L ′′. Therefore

|L ∩ Gk(V )| = |L ′′| +∑

L ′∈F(|L ′ ∩ Gk(V )| − |L ′′|).

The pair of spaces L ′′ ⊂ L ′ satisfies the hypothesis ofProposition 3.8, therefore |L ′ ∩ Gk(V )| − |L ′′| satisfies thebounds of (22). Consequently,∑

L ′∈F(|L ′ ∩ Gk(V )| − |L ′′|)

≤

⎧⎪⎨⎪⎩

1 + q + · · · + q�−1 if μ = 0,

q + q2 + · · · + q�−1 if μ = 1,

q2 + q3 + · · · + q�−μ+1 if μ ≥ 2.

Adding ||L ′′|| = 1 + q + · · · + qμ to the above equation weget:

|L ∩ Gk(V )|≤

{2 + q + · · · + q�−1 if μ = 0,

1 + 2q + q2 + · · · + q�−1 if μ = 1,(24)

and if μ ≥ 2 then,

|L ∩ Gk(V )| ≤ 1 + q + 2(

q2 + · · · + qmin{μ,�−μ+1})

+(

qmin{μ+1,�−μ+2} + · · · + qmax{μ,�−μ+1}).

(25)

Consider the quantity:

A := 1 + q + 2q2 + q3 + · · · + q�−1,

and let B denote the right hand side of (24) if μ = 0 orμ = 1, and the right hand side of (25) if 2 ≤ μ ≤ � − 1.The assertion in the proposition statement that needs to beestablished is |L ∩ Gk(V )| ≤ A. Hence it suffices to showA ≥ B . If μ = 0, then A − B = q2 − 1 > 0 and if μ = 1


then A − B = q2 − q > 0. If μ = 2 or � − 1, then A = B .In the remaining cases 3 ≤ μ ≤ � − 2, we get:

(q − 1)(A − B)

q3 = (q�−1−μ − 1)(qμ−2 − 1).

Since 3 ≤ μ ≤ � − 2 implies � − 1 − μ ≥ 1 and μ − 2 ≥ 1,we obtain A > B .

Corollary 3.10: Let L be an � dimensional linear subspaceof P(∧k V ) which is not contained in Gk(V ). The quantity|L\Gk(V )| can be bounded as:

|L\Gk(V )| ≤ |L| = 1 + q + q2 + · · · + q�,

|L\Gk(V )| ≥

⎧⎪⎨⎪⎩

q� − q2 if � ≥ 3,

q� − q if � = 2,

q − 1 if � = 1.Proof: It suffices to show

|L ∩ Gk(V )| ≤ (1 + q + 2q2 + q3 + · · · + q�−1),

if � ≥ 3, and

|L ∩ Gk(V )| ≤ (1 + 2q + q2 + q3 + · · · + q�−1),

if � = 2, and |L ∩ Gk(V )| ≤ 2 if � = 1. The case � = 3follows from Proposition 3.9. In the case � = 2, the maximumpossible dimension of a linear subspace of Gk(V ) containedin L is 0 or 1. Therefore by (24) in the proof of Proposition 3.9,we see

|L ∩ Gk(V )| ≤ (1 + 2q + q2 + q3 + · · · + q�−1).

If � = 1, then L is a line in P(∧k V ) which is not containedin Gk(V ), and it is well known such a line has atmost twopoints on Gk(V ). This also follows from the case m = 1 ofProposition 3.8.

Theorem 3.11: Let ω ∈ ∧k V ∗ be a nonzero k-form. If ωis decomposable then ||ω|| = qδ, and if ω is indecomposablethen:

−(k − 2) qδ−3 ≤ ||ω|| − qδ − qδ−2 ≤ k qδ−3, (26)

where δ = δ(k, n) = k(n − k).In particular ||ω|| = qδ + qδ−2 + O(qδ−3).

Proof: If ω is decomposable then ||ω|| = qδ by part 1)of Theorem 3.6. For ω indecomposable we consider the casesk = 2, k = 3 and k ≥ 4 separately.

Case (1) k = 2: If ω is indecomposable and k = 2, then bypart 3) of Theorem 3.6, we have ||ω|| = ∑r−1

i=0 qδ−2i where2 ≤ r ≤ n/2. If r = 2, then ||ω|| = qδ + qδ−2 clearly satis-fies (26). If r ≥ 3, then ||ω|| = qδ + qδ−2 + ∑r−1

i=2 qδ−2i , and

r−1∑i=2

qδ−2i < qδ−4∞∑

i=0

q−2i = qδ−4

1 − q−2 ≤ 4

3qδ−4 ≤ 2

3qδ−3,

where we have used the inequality q ≥ 2, to get theinequalities 1/(1 − q−2) ≤ 4/3 and 4/3 qδ−4 ≤ 2/3 qδ−3.Therefore

0 ≤ ||ω|| − qδ − qδ−2 ≤ 2/3 qδ−3,

which satisfies (26).Setup for the Cases k ≥ 3: For k ≥ 3, let dim(Uω) = k + s,

where s ≥ 2 as noted in Fact 3.3. We now use lemma 3.7 and

the notation therein. Let Wω be a complement of Vω in V ,so that every element of V \Vω can be written uniquely asu + v with u ∈ Wω \{0} and v ∈ Vω. By definition of Vω,we have ιu+v ω = ιu ω, and hence ||ιu+v ω|| equals ||ιu ω||.Using the formula ||ωx || = ||ιx ω||/qk−1 for any x /∈ Vω (seeLemma 3.7), we get ||ωu+v || = ||ωu ||, and hence the righthand side of the formula of Lemma 3.7:

(qk − 1) ||ω|| =∑x /∈Vω

||ωx ||,

becomes qdimVω∑

u∈Wω\{0} ||ωu ||. Since Uω is the annihilatorof Vω (see (18)), we see that dim(Vω) = n − k − s anddim(Wω) = k + s. Hence we get:

||ω|| = qn−k−s

qk − 1

∑u∈Wω\{0}

||ωu ||. (27)

For any u ∈ Wω , the form ωu ∈ ∧k−1(V/Fu)∗ is decompos-able if and only if ιuω is decomposable. This easily followsby working with a basis for V = Wω ⊕ Vω that extends u.Let Lω denote the k + s − 1 dimensional linear subspace ofP(∧k−1Uω) defined by:

Lω = P{ιuω : u ∈ Wω}.Since ω is indecomposable, it follows by Lemma 3.4, thatLω is not contained in Gk−1(Uω). Moreover dim(L) is k +s − 1 ≥ 3 + 2 − 1 ≥ 3, and hence by Corollary 3.10 we get

qk+s−1 − q2 ≤ |Lω\Gk−1(Uω)|≤ 1 + q + q2 + · · · + qk+s−1. (28)

We introduce the notation

δ′ := δ(k − 1, n − 1) = (k − 1)(n − k),

and note that qδ′ = qδ/qn−k . The weight ||ωu || equals qδ′

if ιuω is decomposable, and hence we can expand the for-mula (27) as:

||ω|| = qδ qk+s − 1

qk+s − qs+

∑ιuω indecomposable

qn−k−s

qk − 1(||ωu || − qδ′

).

(29)

The following inequalities will be useful:

1 ≤ qk+s − 1

qk+s − qs= 1 + q−k 1 − q−s

1 − q−k≤ 1 + q−k

1 − 2−k,

(30)(qk+s−1 − q2)(q − 1)

qk+s − qs= (1 − q−1)

(1 + (1 − q3−s)

qk − 1

)

≥ 1 − q−1 for s ≥ 3. (31)

We first consider the case s = 2 which is special, and thenthe case s > 2.

Case (2) k ≥ 3, s = 2: We show that

||ω|| = qδ + qδ−2 + · · · + qδ−2r+2,

for some r ≥ 2. First, we assume n = k +2. There is a naturalisomorphism

∗ : P(∧kV ) → P(∧n−k V ∗)


(inner-product free version of Hodge star operator) definedas follows. Let ξ ∈ ∧k V represent a given element of[ξ ] ∈ P(∧k V ), and let η ∈ ∧n V ∗ represent the unique elementof P(∧n V ∗). We define ∗ξ ∈ ∧n−k V ∗ by the equation

α ∧ ∗ξ = 〈α, ξ〉 η, ∀α ∈ ∧k V ∗.

Although ∗ξ is ambiguous upto a scalar multiple, its class inP(∧n−k V ∗), which we denote by ∗[ξ ] is well defined. Theisomorphism ∗ is defined in each dimension 0 ≤ k ≤ n, andwhen needed, we indicate the dimension using the notation ∗k .If [ξ ] ∈ Gk(V ), then picking a basis f1, . . . , fn for V such thatξ = f1 ∧ . . . ,∧ fk and η = f 1 ∧· · ·∧ f n (where f 1, . . . , f n isthe dual basis), it follows that ∗[ξ ] is represented by f k+1∧. . .∧ f n . Thus ∗ carries Gk(V ) bijectively to Gn−k(V ∗). If Hω

is a hyperplane in P(∧kV ) given by

Hω = P{ξ ∈ ∧k V : 〈ω, ξ〉 = 0},and if [ω] = ∗n−k [ζ ], then it follows that ∗Hω is thehyperplane in P(∧n−k V ∗) given by

Hζ = P{α ∈ ∧n−k V ∗ : 〈α, ζ 〉 = 0}.Since ∗ bijectively carries Hω and Gk(V ) to Hζ and Gn−k V ∗respectively, we see that

||ω|| = |Gk(V )\Hω| = |Gn−k(V ∗)\Hζ |.Since we are assuming n = k + 2, we see that ζ ∈ ∧2V andhence by part (3) of Theorem 3.6,

|Gn−k(V ∗)\Hζ | = qδ + qδ−2 + · · · + qδ−2r+2.

Next, we relax the assumption n = k + 2. Let n = k + 2, sothat n = dim(Uω). By definition (see (18)) of Uω and Vω, wenote that Uω can be identified with (V/Vω)∗, and hence ∧kUω

can be identified with ∧k(V/Vω)∗. Denoting V/Vω as V ,let ω ∈ ∧k(V )∗ denote the image of ω ∈ ∧kUω under thisidentification. The special case n = k + 2 considered aboveapplies to ω, and hence ||ω|| = qd + qd−2 + · · · + qd−2r+2,where d = 2k. In general, when dim(Uω) = k + s andω denotes the element of ∧k(V/Vω)∗ induced by ω, thequantities ||ω|| and ||ω|| are related by

||ω|| = qk(n−k−s) ||ω||. (32)

The proof is elementary, and is given in [9, Proposition II.3],for k = 3, but the same idea works for general k. Using (32),and the fact that d + k(n − k − 2) = k(n − k) = δ, we get||ω|| = qδ + qδ−2 + · · · + qδ−2r+2.

Case (3) k = 3, s ≥ 3: If ιuω is indecomposable, then

0 ≤ ||ωu || − qδ′ − qδ′−2 ≤ 4/3 qδ′−4,

as noted in Case (1) above. Using this inequality together withthe inequality (28) in the formula (29), we get the inequalities:

qδ q3+s − 1

q3+s − qs + qδ−2 (q2+s − q2)(q − 1)

q3+s − qs ≤ ||ω||

≤[

qδ + qδ−2 + 4

3qδ−4

]q3+s − 1

q3+s − qs. (33)

For the upper bound on ||ω||, we simplify the right handinequality of (33) using (30) for k = 3: q3+s−1

q3+s−qs ≤ 1+8/7 q−3

to get:

||ω|| ≤ qδ + qδ−2 + qδ−3[

4

3q+ 8

7+ 8

7q2 + 4 × 8

3 × 7q4

]

≤ qδ + qδ−2 + 3 qδ−3,

where we have used the inequality

4

3q+ 8

7+ 8

7q2 + 4 × 8

3 × 7q4

≤ 4

3 × 2+ 8

7+ 8

7 × 22 + 4 × 8

3 × 7 × 24

= 46/21 < 3.

For the lower bound on ||ω||, we simplify the left handinequality in (33) to get:

||ω|| − qδ − qδ−2 ≥ qδ−5 1 − q3−s

1 − q−3 ≥ 0 ≥ −qδ−3,

where we have used the fact that s ≥ 3 implies 1 − q3−s ≥ 0.Case (4) k ≥ 4, s ≥ 3: If ιuω is decomposable, then

||ωu || = qδ′, and if ιuω is an indecomposable k −1 form, then

by induction we assume that the result (26) holds for ||ωu ||.−(k − 3) qδ′−3 ≤ ||ωu || − qδ′ − qδ′−2 ≤ (k − 1) qδ′−3.

Using this inequality together with the inequality (28) in theformula (29), we get the inequalities:

qδ qk+s − 1

qk+s − qs+ (qδ−2 − (k − 3)qδ−3)

(qk+s−1 − q2)(q − 1)

qk+s − qs

≤ ||ω|| ≤ (qδ + qδ−2 + (k − 1) qδ−3)qk+s − 1

qk+s − qs. (34)

For the upper bound on ||ω||, we simplify the right handinequality of (34) using (30):

qk+s − 1

qk+s − qs≤ 1 + q−k/(1 − 2−k) ≤ 1 + 16q−k/15,

(because k ≥ 4) to get:

||ω|| ≤ qδ + qδ−2 + (k − 1) qδ−3

+ 16qδ−3

15

(1

qk−3 + 1

qk−1 + k − 1

qk

).

The function x �→ (x − 1)/qx is decreasing for x > 3 andhence for k ≥ 4, we have

(k − 1)/qk ≤ (4 − 1)/q4 ≤ 3/16.

Also k ≥ 4 and q ≥ 2 imply 1/qk−3 ≤ 1/2 and 1/qk−1 ≤ 1/8.Therefore, the parenthetical term above satisfies:

1/qk−3 + 1/qk−1 + (k − 1)/qk ≤ 13/16,

and hence:

||ω|| − qδ − qδ−2 ≤ (k − 1 + 13/16) qδ−3 ≤ k qδ−3.


For the lower bound on ||ω||, we simplify the left handinequality in (34). We have:

qδ qk+s − 1

qk+s − qs≥ qδ,

and using the inequality (31), we have

(qk+s−1 − q2)(q − 1)

qk+s − qs≥ 1 − q−1.

Using these inequalities we get:

||ω|| − qδ − qδ−2 ≥ −(k − 2) qδ−3 + (k − 3)qδ−4

≥−(k − 2) qδ−3.

Theorem 3.12: Let L be a codimension r subspaceof P(∧kV ). If Ann(L) ⊂ Gk(V ∗) then

||L|| = qδ + qδ−1 + · · · + qδ−r+1.

If Ann(L) �⊂ Gk(V ∗) and r ≥ 3 then:

| ||L|| − qδ − qδ−1 − 2 qδ−2| ≤ (2k + 4) qδ−3.

If Ann(L) �⊂ Gk(V ∗) and r = 2 then:

| ||L|| − qδ − qδ−1 − qδ−2| ≤ (2k + 4) qδ−3.

In particular:

||L|| = qδ + qδ−1 +{

2 qδ−2 + O(qδ−3) if r ≥ 3,

qδ−2 + O(qδ−3) if r = 2.Proof: When Ann(L) ⊂ Gk(V ∗), the assertion is a

restatement of part 2) of Theorem 3.6. We now assumeAnn(L) �⊂ Gk(V ∗). Using the formula (21) and Theorem 3.11,we get:

qr−1||L|| = qδ |Ann(L) ∩ Gk(V ∗)| +∑

[ω]∈Ann(L)\Gk(V ∗)|ω||.

We rewrite the above equation as:

||L|| = (qδ + qδ−1 + · · · + qδ−r+1)

+ 1

qr−1

∑[ω]∈Ann(L)\Gk(V ∗)

(||ω|| − qδ). (35)

Using Corollary 3.10, we get the lower bound

|Ann(L)\Gk(V ∗)| ≥

⎧⎪⎨⎪⎩

qr−1 − q2 if r ≥ 4,

qr−1 − q if r = 3,

q − 1 if r = 2.

We also have the trivial bound:

|Ann(L)\Gk(V ∗)| ≤ |Ann(L)| = (1 + q + q2 + · · · + qr−1).

For [ω] ∈ Ann(L)\Gk(V ∗), by Theorem 3.11 we have theupper and lower bounds:

−(k − 2) qδ−3 ≤ ||ω|| − qδ − qδ−2 ≤ k qδ−3.

We now use these bounds in (35).Case r ≥ 4: We get the upper bound for ||L||:

||L|| ≤ (qδ + qδ−1 + · · · + qδ−r+1)

+ (qδ−2 + k qδ−3)(1 + q−1 + · · · + q1−r ).

Hence:

||L|| − (qδ + qδ−1 + 2 qδ−2 + (k + 2) qδ−3)

≤ qδ−4 (2 + k)

∞∑i=0

q−1 ≤ qδ−4 (4 + 2k)

≤ (k + 2) qδ−3.

Therefore,

||L|| ≤ qδ + qδ−1 + 2 qδ−2 + (2k + 4) qδ−3.

We get the lower bound for ||L||:||L|| − (qδ + qδ−1 + · · · + qδ−r+1)

≥ (qδ−2 − (k − 2)qδ−3)(1 − q3−r).

If r = 4, this simplifies to:

||L|| − (qδ + qδ−1 + 2 qδ−2)

≥ −(k − 2) qδ−3 + (k − 2) qδ−4 − (2k + 4) qδ−3,

where as if r > 4, the inequality simplifies to:

||L||−(qδ + qδ−1 + 2 qδ−2)

≥ −(k − 3) qδ−3 + (k − 2) qδ−r +r−2∑i=4

qδ−i

≥ −(2k + 4) qδ−3.

Case r = 3: We get the upper bound for ||L||:||L|| − (qδ + qδ−1 + qδ−2)

≤ (qδ−2 + kqδ−3)(1 + q−1 + q−2)

≤ qδ−2 + qδ−3 (k + 1 + k/q + 1/q + k/q2).

Using q−1 ≤ 1/2, we get:

||L|| − (qδ + qδ−1 + 2 qδ−2) ≤ 7k + 6

4qδ−3

≤ (2k + 4) qδ−3.

We get the lower bound for ||L||:||L||−(qδ + qδ−1 + qδ−2)

≥ (qδ−2 − (k − 2)qδ−3)(1 − q−1),

which simplifies to:

||L|| − (qδ + qδ−1 + 2 qδ−2) ≥ −(k − 1) qδ−3 + (k − 2) qδ−4

≥−(2k + 4) qδ−3.

Case r = 2: We get the upper bound for ||L||:||L|| − (qδ + qδ−1) ≤ (qδ−2 + k qδ−3)(1 + q−1)

≤ qδ−2 + qδ−3 (k + 1 + k/q).

Therefore,

||L|| − (qδ + qδ−1 + qδ−2)

≤ qδ−3 (k + 1 + k/2) ≤ (2k + 4) qδ−3.

We get the lower bound:

||L|| − (qδ + qδ−1) ≥ (qδ−2 − (k − 2)qδ−3)(1 + q−1)

≥ qδ−2 − qδ−3 (k − 2 − 1 + (k − 2)/q).


Therefore,

||L||− (qδ + qδ−1 + qδ−2)

≥ −(k − 3 + (k − 2)/2) qδ−3 ≥ −(2k + 4) qδ−3.

IV. PROOF OF THE MAIN RESULT

We recall from §II the formulas (13) and (14)

γ (k, n; q) = E1 − E2 + E3 − · · · + (−1)N−1 EN ,

Er =∑

{I1,...,Ir }∈I rk,n

|CI1 ∪ · · · ∪ CIr |.

From (11), we get

E1 = N qδ. (36)

If L is the codimension r subspace of P(∧k V ) defined by

Ann(L) = P(Span(eI1 , . . . , eIr )), (37)

where {I1, . . . , Ir } ∈ I rk,n , then

|CI1 ∪ · · · ∪ CIr | = ||L||.If r = 2, the line Ann(L) of P(∧kV ∗) joining eI and eJ

is contained in Gk(V ∗) if and only if, the k-dimensionalsubspaces of V ∗ represented by eI and eJ have a k − 1dimensional intersection. In other words there is a multi-index K ∈ Ik−1,n with I and J of the form I = K ∪ {i}and J = K ∪ { j}. Clearly there are(

n

k − 1

) (n − k + 1

2

)= Nδ/2

such elements of I 2k,n . Therefore, by Theorem 3.12 we get:

∣∣E2 − Nδ

2(qδ + qδ−1) −

((N

2

)− Nδ/2

)

×(

qδ + qδ−1 + qδ−2)∣∣ ≤

((N

2

)− Nδ/2

)(2k + 4) qδ−3.

which simplifies to:

∣∣E2 −(

N

2

)(qδ + qδ−1) −

((N

2

)− Nδ/2

)qδ−2

∣∣≤

[(N

2

)− Nδ/2

](2k + 4) qδ−3. (38)

Now, let r ≥ 3. The subspace Ann(L) of (37) is containedin Gk(V ∗) if and only if it is contained in a maximallinear subspace πα or πγ for some α ∈ Gk−1(V ∗) or someγ ∈ Gk+1(V ∗). Moreover, these two cases are disjoint,because Ann(L) is r −1 dimensional and the intersection of aπα and a πγ is at most one dimensional (part 3) of Fact 3.2).Suppose Ann(L) ⊂ πα. By part 4) of Fact 3.2, it follows that αis the intersection of the points of Gk(V ∗) represented by eIμ

and eIν for any pair 1 ≤ μ < ν ≤ r . Now, the dimension of theintersection of the k dimensional subspaces of V ∗ representedby eIμ and eIν (where Iμ, Iν ∈ Ik,n ) is equal to the numberof indices common to the multi-indices Iμ and Iν . Since α isa k − 1 dimensional subspace of V ∗, it follows that there is aJ ∈ Ik−1,n (with eJ representing α), common to eIμ and eIν .

Since 1 ≤ μ < ν ≤ r were arbitrary, we see that J ∈ Ik−1,n

is common to all the multi-indices I1, . . . , Ir .

c1(r) :={( n

k−1

) (n−k+1r

) = kNn−k+1

(n−k+1r

)if r ≤ n − k + 1,

0 if r > n − k + 1,

(39)

such elements of I rk,n .

Suppose Ann(L) ⊂ πγ . By part 4) of Fact 3.2, it followsthat for any pair 1 ≤ μ < ν ≤ r , γ is the vector space sumof the points of Gk(V ∗) represented by eIμ and eIν . In otherwords, there is a J ∈ Ik+1,n containing all the multi-indicesI1, . . . , Ir . Clearly there are

c2(r) :={( n

k+1

) (k+1r

) = N(n−k)k+1

(k+1r

)if r ≤ k + 1,

0 if r > k + 1,(40)

such elements of I rk,n . (A derivation of c1(r) and c2(r) can also

be found in [3, Corollary 4.4]). Therefore, by Theorem 3.12we get:

∣∣Er − (c1(r) + c2(r)) (qδ + qδ−1 + · · · + qδ−r+1)

−((

N

r

)− c1(r) − c2(r))(qδ + qδ−1 + 2qδ−2

)∣∣≤

((N

r

)− c1(r) − c2(r)

)(2k + 4) qδ−3.

If r = 3 this simplifies to:

E3 =(

N

3

)(qδ+qδ−1)+

(2

(N

3

)−c1(3)−c2(3)

)qδ−2 + E ′

3,

where:

|E ′3| ≤

((N

3

)− c1(3) − c2(3)

)(2k + 4)qδ−3

≤[(

N

3

)(2k + 4) − (c1(3) + c2(3)) (2k + 2)

]qδ−3.

(41)

If r ≥ 4, we get:

Er =(

N

r

)(qδ + qδ−1)

+(

2

(N

r

)− c1(r) − c2(r)

)qδ−2 + Er ,

where

E ′r ≤

((N

r

)− c1(r) − c2(r)

)(2k + 4) qδ−3

+ (c1(r) + c2(r)) (qδ−3 + qδ−4 + · · · + qδ−r+1),

E ′r ≥ −

((N

r

)− c1(r) − c2(r)

)(2k + 4) qδ−3

+ (c1(r) + c2(r)) (qδ−3 + qδ−4 + · · · + qδ−r+1). (42)

Using the inequality

−2 qδ−3 ≤ qδ−3 + qδ−4 + · · · + qδ−r+1 ≤ 2 qδ−3,


we get:

|E ′r | ≤

[(N

r

)(2k + 4) − (c1(r) + c2(r)) (2k + 2)

]qδ−3

for r ≥ 4. (43)

We define:

c(k, n) =[(

N

2

)− Nδ/2

](2k + 4)

+∑r≥3

[(N

r

)(2k + 4) − (c1(r) + c2(r)) (2k + 2)

].

We note that c(k, n) is positive. We can rewrite c(k, n)using (39), (40) as:

c(k, n)

= (2N − N − 1)(2k + 4) + Nδk − (2k + 2)

×[(

n

k−1

)(2n−k+1−n + k − 2)+

(n

k+1

)(2k+1 − k − 2)

].

(44)

We define:

a2(k, n)

= −[(N

2

)− Nδ/2

]+

∑r≥3

(−1)r−1[2

(N

r

)− c1(r) − c2(r)

],

which simplifies (again using (39),(40)) to:

a2(k, n) = N2 − 5N + 4

2− Nδ

2

(δ − n − 3

δ + n + 1

).

Using equations (38)-(43) in the formula

γ (k, n; q) = E1 − E2 + E3 − · · · + (−1)N−1 EN ,

we get:

γ (k, n; q)

= qδ

(∑Nr=1(−1)r−1

(N

r

))+ qδ−1

(∑Nr=2(−1)r−1

(N

r

))

+ qδ−2 a2(k, n) + γ ′(k, n; q)

= qδ + (1 − N)qδ−1 + a2(k, n) qδ−2 + γ ′(k, n; q),

where:

|γ ′(k, n; q)| ≤ c(k, n) qδ−3.

Therefore

∣∣γ (k, n; q) − qδ − (1 − N) qδ−1 − a2(k, n) qδ−2∣∣

≤ c(k, n) qδ−3. (45)

In particular:

γ (k, n; q) = qδ + (1 − N) qδ−1

+ a2(k, n) qδ−2 + O(qδ−3). �

V. CONCLUSION

Since the problem of determining the number γ (k, n; q)of [n, k] q-ary MDS codes is very difficult and complicated,we have studied the problem of determining asymptotic for-mulae in q for γ (k, n; q). We have improved the knownformula (4):

γ (k, n; q) = qδ − (N − 1)qδ−1 + O(qδ−2)

by the formula (1):

γ (k, n; q) = qδ − (N − 1)qδ−1 + a2(k, n)qδ−2 + O(qδ−3)

where a2(k, n) is given by (5). The main tool is a closerstudy of cardinalities of linear sections of the Grassmannian(Theorems 3.11 and 3.12). We have in fact obtained a strongerresult (45), which can be restated as follows: Let P±(X) bethe cubic polynomials given by:

P±(X) = X3 − (N − 1) X2 + a2(k, n) X ± c(k, n), (46)

where c(k, n) is given by (44), then:

P−(q) ≤ γ (k, n; q)

qδ−3 ≤ P+(q). (47)

Further, we are hopeful of determining constants a3(k, n)and d(k, n) satisfying the improved asymptotic formula andbounds:

γ (k, n; q) = qδ − (N − 1)qδ−1 + a2(k, n)qδ−2

−a3(k, n)qδ−3 + O(qδ−4),∣∣ γ (k, n; q) − qδ + (N − 1) qδ−1

−a2(k, n)qδ−2 + a3(k, n) qδ−3∣∣≤ d(k, n) qδ−4.

We also plan to study the implications of these bounds for themain conjecture on MDS codes (see [14, p. 6]).

ACKNOWLEDGMENT

The author thanks Aranya Lahiri for many usefuldiscussions. Thanks are also due to the referees for valuablecomments.

REFERENCES

[1] W.-L. Chow, “On the geometry of algebraic homogeneous spaces,” Ann.Math., vol. 50, no. 1, pp. 32–67, 1949.

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[3] S. R. Ghorpade and G. Lachaud, “Hyperplane sections of Grassmanniansand the number of MDS linear codes,” Finite Fields Appl., vol. 7, no. 4,pp. 468–506, 2001.

[4] S. R. Ghorpade, A. R. Patil, and H. K. Pillai, “Decomposable subspaces,linear sections of Grassmann varieties, and higher weights of Grassmanncodes,” Finite Fields Appl., vol. 15, no. 1, pp. 54–68, 2009.

[5] D. G. Glynn, “Rings of geometries II,” J. Combinat. Theory, Ser. A,vol. 49, no. 1, pp. 26–66, 1988.

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[8] A. V. Iampolskaia, A. N. Skorobogatov, and E. A. Sorokin, “Formulafor the number of [9, 3] MDS codes,” IEEE Trans. Inf. Theory, vol. 41,no. 6, pp. 1667–1671, Nov. 1995.


[9] K. V. Kaipa and H. K. Pillai, “Weight spectrum of codes associatedwith the Grassmannian G(3, 7),” IEEE Trans. Inf. Theory, vol. 59,no. 2, pp. 986–993, Feb. 2013.

[10] D. Y. Nogin, “Codes associated to Grassmannians,” in Arithmetic,Geometry and Coding Theory. Berlin, Germany: de Gruyter, 1996,pp. 145–154.

[11] M. Pankov, Grassmannians of Classical Building. Hackensack, NJ,USA: World Scientific, 2010.

[12] R. Rolland, “The number of MDS [7, 3] codes on finite fields ofcharacteristic 2,” Appl. Algebra Eng. Commun. Comput., vol. 3, no. 4,pp. 301–310, 1992.

[13] A. N. Skorobogatov, “Linear codes, strata of Grassmannians, and theproblems of segre,” in Coding Theory and Algebraic Geometry (LectureNotes in Mathematics), vol. 1518. Berlin, Germany: Springer-Verlag,1992, pp. 210–223.

[14] M. Tsfasman, S. Vladut, and D. Nogin, Algebraic Geometric Codes:Basic Notions (Mathematical Surveys and Monographs), vol. 139.Providence, RI, USA: AMS, 2007.

Krishna V. Kaipa received his B.Tech degree (1999) in Mechanical engi-neering from the Indian Institute of Technology (IIT) Bombay, and the M.S.(2001) and Ph.D. (2009) degrees in Mechanical engineering and Mathematicsrespectively from the University of Maryland, College Park. Currently he is anAssistant Professor at the Indian Institute of Science Education and Research-Pune, India. His research interests lie in classical algebraic geometry, topologyand geometry.

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An Asymptotic Formula in \(q\) for the Number of \([n, k]~q\) -Ary MDS Codes