DECOMPOSABLE SUBSPACES, LINEAR SECTIONS OF GRASSMANN VARIETIES, AND HIGHER WEIGHTS OF GRASSMANN CODES

SUDHIR R. GHORPADE, ARUNKUMAR R. PATIL, AND HARISH K. PILLAI

Abstract. Given a homogeneous component of an exterior algebra, we characterize those subspaces in which every nonzero element is decomposable. In geometric terms, this corresponds to characterizing the projective linear subvarieties of the Grassmann variety with its Plücker embedding. When the base field is finite, we consider the more general question of determining the maximum number of points on sections of Grassmannians by linear subvarieties of a fixed (co)dimension. This corresponds to a known open problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties. We recover most of the known results as well as prove some new results. In the process we obtain, and utilize, a simple generalization of the Griesmer-Wei bound for arbitrary linear codes.

1. Introduction

Let V be an m-dimensional vector space over a field F. Given a positive integer ℓ with ℓ ≤ m, consider the ℓ^{th} exterior power ⋀^{ℓ} V of V. A nonzero element ω ∈ ⋀^{ℓ} V is said to be decomposable if ω = v_{1} ∧ ⋯ ∧ v_{ℓ} for some v_{1}, …, v_{ℓ} ∈ V. A subspace of ⋀^{ℓ} V is decomposable if all of its nonzero elements are decomposable. In the first part of this paper, we consider the following question: what are all possible decomposable subspaces of ⋀^{ℓ} V, and, in particular, what is the maximum possible dimension of a decomposable subspace of ⋀^{ℓ} V? We answer this by proving a characterization of decomposable subspaces of ⋀^{ℓ} V. This result can be viewed as an algebraic counterpart of the combinatorial structure theorem for the so called closed families of subsets of a finite set (cf. [5, Thm. 4.2]). As a corollary, we obtain that the maximum possible dimension of a decomposable subspace of ⋀^{ℓ} V is max{ℓ, m − ℓ} + 1. In geometric terms, this corresponds to characterizing the projective linear subvarieties (with respect to the Plücker embedding) of the Grassmann variety G_{ℓ,m} of all ℓ-dimensional subspaces of V, and showing that the maximum possible (projective) dimension of such a linear subvariety is max{ℓ, m − ℓ}. Briefly speaking, the characterization of decomposable subspaces states that they are necessarily one among the two types of subspaces that are described explicitly. Subsequently, using the Hodge star operator, we observe that a nice duality prevails among the two types of decomposable subspaces.

In the second part of this paper, we consider the case when F is the finite field 𝔽_{q} with q elements. For a fixed nonnegative integer s, we consider the linear sections L ∩ G_{ℓ,m} of the Grassmann variety G_{ℓ,m} (with its canonical Plücker embedding) by a linear subvariety L of ℙ(⋀^{ℓ} V) of dimension s, and we ask what is the maximum number of 𝔽_{q}-rational points that such a linear section can have. In light of the abovementioned corollary of the characterization of decomposable subspaces, it is evident that when s is small, or more precisely, when s ≤ max{ℓ, m − ℓ}, the maximum number is 1 + q + q^{2} + ⋯ + q^{s}. But for a general s, the answer does not seem to be known. In fact, enumerative as well as geometric aspects of linear sections of G_{ℓ,m} are not particularly well-understood, in general, except in special cases such as those when the linear sections are Schubert subvarieties of G_{ℓ,m}. (See, for example, Section 6 of [5] and the references therein.) However, the above question admits an equivalent formulation in terms of linear error correcting codes, and as such, it has been considered by various authors. Indeed, if we let C(ℓ, m) denote the linear code associated to G_{ℓ,m}(𝔽_{q}) ↪ ℙ(⋀^{ℓ} V), then its r^{th} higher weight (see Section 4 for definitions) is given by

Date: October 25, 2007.

2000 Mathematics Subject Classification. 15A75, 14M15, 94B05, 94B27.

Key words and phrases. Exterior algebra, decomposable subspace, Grassmann variety, linear code, higher weight, Griesmer-Wei bound, Grassmann code.

d_{r}(C(ℓ, m)) = n − max_{L} |L ∩ G_{ℓ,m}(𝔽_{q})|

where the maximum is taken over projective linear subspaces L of ℙ(⋀^{ℓ} 𝔽_{q}^{m}) of codimension r, and where n denotes the Gaussian binomial coefficient defined by

n = |G_{ℓ,m}(𝔽_{q})| = \left[{m \atop ℓ}\right]_{q} := \frac{(q^{m} − 1)(q^{m} − q) ⋯ (q^{m} − q^{ℓ−1})}{(q^{ℓ} − 1)(q^{ℓ} − q) ⋯ (q^{ℓ} − q^{ℓ−1})}.

With this in view, we shall now consider the equivalent question of determining d_{r} = d_{r}(C(ℓ, m)) for any r ≥ 0, where d_{0} := 0, by convention. This question is open, in general, and the known results can be summarized as follows. From general facts in Coding Theory and the fact that the embedding G_{ℓ,m}(𝔽_{q}) ↪ ℙ(⋀^{ℓ} 𝔽_{q}^{m}) is nondegenerate, one knows that

0 = d_{0} < d_{1} < d_{2} < ⋯ < d_{k} = n where k := \binom{m}{ℓ},

and also that

(1)  d_{r} (C(ℓ, m)) ≥ q^{δ} + q^{δ−1} + ⋯ + q^{δ−r+1} where δ := ℓ(m − ℓ).

The latter is a consequence of the so called Griesmer-Wei bounds for linear codes and a result of Nogin [12] which says that d_{1} = q^{δ}. In fact, Nogin [12] showed that the Griesmer-Wei bound is sometimes attained, that is,

(2)  d_{r} (C(ℓ, m)) = q^{δ} + q^{δ−1} + ⋯ + q^{δ−r+1} for 0 ≤ r ≤ μ,

where

μ := max{ℓ, m − ℓ} + 1.

Alternative proofs of Nogin’s result for higher weights of C(ℓ, m) were given by Ghorpade and Lachuad [4] using the notion of a closed family. Recently, Hansen, Johnsen and Ranestad [7] have observed that a dual result holds as well, namely,

(3)  d_{k−r} (C(ℓ, m)) = n − (1 + q + ⋯ + q^{r−1}) for 0 ≤ r ≤ μ.

In general, the values of d_{r}(C(ℓ, m)) for μ < r < k −μ are not known. For example, if ℓ = 2 and we assume (without loss of generality) that m ≥ 4, then μ = m−1, and d_{r}(C(ℓ, m)) for m ≤ r < \binom{m−1}{2} are not known, except that in the first nontrivial case, Hansen, Johnsen and Ranestad [7] have shown by clever algebraic-geometric arguments that

(4)  d_{5}(C(2, 5)) = q^{6} + q^{5} + 2q^{4} + q^{3} = d_{4} + q^{4}.

Notice that the Griesmer-Wei bound in (1) is not attained in this case. Nonetheless, Hansen, Johnsen and Ranestad [7] conjecture that the difference d_{r} − d_{r−1} of consecutive higher weights of C(ℓ, m) is always a power of q.

Our main results concerning the the determination of d_{r}(C(ℓ, m)) are as follows. First, we recover (2) and (3) as an immediate corollary of our characterization of decomposable subspaces. Next, we further analyze the structure of decomposable vectors in ⋀^{2} V to extend (3) by showing that

(5) d_{k−μ−1} (C(2, m)) = n − (1 + q + ⋯ + q^{μ−1} + q^{2}) = d_{k−μ} − q^{2} for any m ≥ 4.

Finally, we use the abovementioned analysis of decomposable vectors in ⋀^{2} V and also exploit the Hodge star duality to prove the following generalization of (4) for any m ≥ 4.

(6) d_{μ+1} (C(2, m)) = q^{δ} + q^{δ−1} + 2q^{δ−2} + q^{δ−3} + ⋯ + q^{δ−μ+1} = d_{μ} + q^{δ−2}.

In the course of deriving these formulae, we use a mild generalization of the Griesmer-Wei bound, proved here in the general context of arbitrary linear codes, which may be of independent interest.

It is hoped that these results, and more so, the methods used in proving them, will pave the way for the solution of the problem of determination of the complete weight hierarchy of C(ℓ, m) at least in the case ℓ = 2. To this end, we provide, toward the end of this paper, an initial tangible goal by stating conjectural formulae for d_{r} (C(2, m)) when μ + 1 ≤ r ≤ 2μ − 3, and also when k − 2μ + 3 ≤ r ≤ k − μ − 1. It may be noted that these conjectural formulae, and of course both (5) and (6), corroborate the conjecture of Hansen, Johnsen and Ranestad [7] that the differences of consecutive higher weights of Grassmann codes is always a power of q.

2. Decomposable Subspaces

Let us fix, in this as well as the next section, positive integers ℓ, m with ℓ ≤ m, a field F, and a vector space V of dimension m over F. Let

I(ℓ, m) := {α = (α_{1}, …, α_{ℓ}) ∈ ℤ^{ℓ} : 1 ≤ α_{1} < ⋯ < α_{ℓ} ≤ m}.

If {v_{1}, …, v_{m}} is a basis of V, then {v_{α} : α ∈ I(ℓ, m)} is a basis of ⋀^{ℓ} V, where v_{α} := v_{α_{1}} ∧ ⋯ ∧ v_{α_{ℓ}}. Given any ω ∈ ⋀^{ℓ} V, define

V_{ω} := {v ∈ V : v ∧ ω = 0}.

Clearly, V_{ω} is a subspace of V. It is evident that ω = 0 if and only if dim V_{ω} = m. The following elementary characterization will be useful in the sequel. Here, and hereafter, it may be useful to keep in mind that for us, a decomposable vector is necessarily nonzero.

Lemma 1. Assume that ℓ < m and let ω ∈ ⋀^{ℓ} V. Then

ω is decomposable ⟺ dim V_{ω} = ℓ.

Moreover, if dim V_{ω} = ℓ and {v_{1}, …, v_{ℓ}} is a basis of V_{ω}, then ω = c(v_{1} ∧ ⋯ ∧ v_{ℓ}) for some c ∈ F with c ≠ 0.

Proof. If ω is decomposable, then ω = v_{1} ∧ ⋯ ∧ v_{ℓ} for some linearly independent elements v_{1}, …, v_{ℓ} ∈ V. Clearly, {v_{1}, …, v_{ℓ}} ⊆ V_{ω}. Moreover, if v ∈ V_{ω}, then v, v_{1}, …, v_{ℓ} are linearly dependent. It follows that {v_{1}, …, v_{ℓ}} is a basis of V_{ω}. Conversely, let dim V_{ω} = ℓ. Extend a basis {v_{1}, …, v_{ℓ}} of V_{ω} to a basis {v_{1}, …, v_{m}} of V. Write ω = ∑_{α∈I(ℓ,m)} c_{α} v_{α}. Now 0 = v_{i} ∧ ω = ∑_{α∈I(ℓ,m)} c_{α} (v_{i} ∧ v_{α}) for ℓ < i ≤ m. Consequently, c_{α} = 0 if i does not appear in α. It follows that ω = c_{(1,2,...ℓ)} (v_{1} ∧ ⋯ ∧ v_{ℓ}), as desired. □

Corollary 2. If ℓ = 1 or ℓ = m − 1, then the space ⋀^{ℓ} V is decomposable, that is, every nonzero element of ⋀^{ℓ} V is decomposable.

Proof. The result is obvious when ℓ = 1. Suppose ℓ = m − 1. Now ⋀^{m} V is canonically isomorphic to F, and for 0 ≠ ω ∈ ⋀^{ℓ} V, the linear map from V to F given by v ↦ v ∧ ω is nonzero and hence surjective. Clearly, V_{ω} is the kernel of this linear map and so dim V_{ω} = dim V − 1 = ℓ. Thus, Lemma 1 applies. □

Lemma 3. Let ω_{1}, ω_{2} ∈ ⋀^{ℓ} V be decomposable and linearly independent, and let V_{i} = V_{ω_{i}} for i = 1, 2. Then

ω_{1} + ω_{2} is decomposable ⟺ dim V_{1} ∩ V_{2} = ℓ − 1 ⟺ dim V_{1} + V_{2} = ℓ + 1.

Proof. Assume that dim V_{1} ∩ V_{2} = ℓ − 1. Let {f_{1}, …, f_{ℓ−1}} be a basis for V_{1} ∩ V_{2}. Extend it to bases {f_{1}, …, f_{ℓ−1}, g_{1}} and {f_{1}, …, f_{ℓ−1}, g_{2}} of V_{1} and V_{2}, respectively. By Lemma 1, there are c_{1}, c_{2} ∈ F such that ω_{i} = c_{i}(f_{1} ∧ f_{2} ∧ ⋯ ∧ f_{ℓ−1} ∧ g_{i}) for i = 1, 2. Now ω_{1} + ω_{2} ≠ 0 since ω_{1}, ω_{2} are linearly independent, and ω_{1} + ω_{2} = f_{1} ∧ f_{2} ∧ ⋯ ∧ f_{ℓ−1} ∧ (c_{1}g_{1} + c_{2}g_{2}). Thus ω_{1} + ω_{2} is decomposable.

Conversely, suppose ω_{1} + ω_{2} is decomposable. Let W = V_{ω_{1}+ω_{2}}. It is clear that V_{1} ∩ V_{2} ⊆ W. Also, by Lemma 1, dim W = ℓ = dim V_{1} = dim V_{2}. Hence if V_{1} ∩ V_{2} = W, then V_{1} ∩ V_{2} = V_{1} = V_{2}, which contradicts the linear independence of ω_{1} and ω_{2}. Thus, dim V_{1} ∩ V_{2} ≤ ℓ − 1, or equivalently, dim V_{1} + V_{2} ≥ ℓ + 1. Moreover, we can find z ∈ W \ (V_{1} ∩ V_{2}). Note that since z ∧ (ω_{1} + ω_{2}) = 0, we have: z ∧ ω_{1} = 0 ⇔ z ∧ ω_{2} = 0. Hence z ∉ V_{1} ∪ V_{2} and V_{i} + F z has dimension ℓ + 1 for i = 1, 2. Further, since z ∧ ω_{1} = −z ∧ ω_{2}, in view of Lemma 1 we see that V_{1} + F z = V_{z∧ω_{1}} = V_{z∧ω_{2}} = V_{2} + F z. Consequently, V_{1} + V_{2} ⊆ V_{1} + F z = V_{2} + F z and dim V_{1} + V_{2} ≤ ℓ + 1. This proves that dim V_{1} + V_{2} = ℓ + 1 or equivalently, dim V_{1} ∩ V_{2} = ℓ − 1. This proves the desired equivalence. □

Corollary 4. Let v_{1}, v_{2}, v_{3}, v_{4} ∈ V and suppose ω := (v_{1} ∧ v_{2}) + (v_{3} ∧ v_{4}) ∈ ⋀^{2} V is nonzero. Then ω is decomposable if and only if {v_{1}, v_{2}, v_{3}, v_{4}} is linearly dependent.

Proof. When v_{1} ∧ v_{2} and v_{3} ∧ v_{4} are linearly independent, the result follows from Lemma 3. The case when v_{1} ∧ v_{2} and v_{3} ∧ v_{4} are linearly dependent is easy. □

Given a subspace E of ⋀^{ℓ} V, let us define

V_{E} := ⋂_{ω∈E} V_{ω} and V^{E} := ∑_{0≠ω∈E} V_{ω}.

Now, let r = dim E. We say that the subspace E is close of type I if there are ℓ + r − 1 linearly independent elements f_{1}, …, f_{ℓ−1}, g_{1}, …, g_{r} in V such that

E = span{f_{1} ∧ ⋯ ∧ f_{ℓ−1} ∧ g_{i} : i = 1, …, r}.

And we say that E is close of type II if there are ℓ+1 linearly independent elements u_{1}, …, u_{ℓ−r+1}, g_{1}, …, g_{r} in V such that

E = span{u_{1} ∧ ⋯ ∧ u_{ℓ−r+1} ∧ g_{1} ⋯ ∧ ǧ_{i} ∧ ⋯ ∧ g_{r} : i = 1, …, r},

where ǧ_{i} indicates that g_{i} is deleted. We say that E is a close subspace of ⋀^{ℓ} V if E is close of type I or close of type II.

Evidently, every one-dimensional subspace of ⋀^{ℓ} V is close of type I as well as of type II, whereas for two-dimensional subspaces, the notions of close subspaces of type I and type II are identical. A corollary of the following lemma is that in dimensions three or more, the two notions are distinct and mutually disjoint.

Lemma 5. Let E be a close subspace of ⋀^{ℓ} V of dimension r. Then E is decomposable. Moreover, if {ω_{1}, …, ω_{r}} is a basis of E, then V_{E} = V_{ω_{1}} ∩ ⋯ ∩ V_{ω_{r}} and V^{E} = V_{ω_{1}} + ⋯ + V_{ω_{r}}. Further, assuming that r > 1, we have dim V_{E} = ℓ − 1 and dim V^{E} = ℓ + r − 1 if E is close of type I, whereas dim V_{E} = ℓ − r + 1 and dim V^{E} = ℓ + 1 if E is close of type II.

Proof. Since E is close of dimension r, there is a r-dimensional subspace G of V [in fact, G = span{g_{1}, …, g_{r}}, in the above notation] such that E is naturally isomorphic to ⋀^{1} G or to ⋀^{r−1} G according as E is close of type I or of type II. Thus, in view of Corollary 2, we see that E is decomposable. Next, suppose {ω_{1}, …, ω_{r}} is a basis of E. Then obviously, V_{E} = V_{ω_{1}} ∩ ⋯ ∩ V_{ω_{r}}. Moreover, in view of Lemmas 1 and 3, we see that V_{ω+ω′} ⊆ V_{ω} + V_{ω′} for all nonzero ω, ω′ ∈ E such that ω + ω′ ≠ 0. Hence, by induction on r, we obtain V^{E} = V_{ω_{1}} + ⋯ + V_{ω_{r}}. Finally, suppose r > 1. In case E is close of type I, and f_{1}, …, f_{ℓ−1}, g_{1}, …, g_{r} are linearly independent elements of V as in the definition above, then in view of Lemma 1, we see that V_{E} = ∩_{i=1}^{r} span{f_{1}, …, f_{ℓ−1}, g_{i}} = span{f_{1}, …, f_{ℓ−1}} and V^{E} = ∑_{i=1}^{r} span{f_{1}, …, f_{ℓ−1}, g_{i}} = span{f_{1}, …, f_{ℓ−1}, g_{1}, …, g_{r}}. On the other hand, if E is close of type II, and u_{1}, …, u_{ℓ−r+1}, g_{1}, …, g_{r} are linearly independent elements of V as in the definition above, then as before, in view of Lemma 1, we see that V_{E} = span{u_{1}, …, u_{ℓ−r+1}} and V^{E} = span{u_{1}, …, u_{ℓ−r+1}, g_{1}, …, g_{r}}. This proves the desired assertions about dim V_{E} and dim V^{E}. □

The following result may be compared with [5, Thm. 4.2]. Also, the proof is structurally analogous to that of [5, Thm. 4.2], except that the arguments here are a little more subtle.

Theorem 6 (Structure Theorem for Decomposable Subspaces). A subspace of ⋀^{ℓ} V is decomposable if and only if it is close.

Proof. Lemma 5 proves that a close subspace of ⋀^{ℓ} V is decomposable. To prove the converse, let E be a decomposable subspace of ⋀^{ℓ} V. We induct on r := dim E. The case r = 1 is trivial, whereas if r = 2, then the desired result follows from Lemmas 1 and 3. Now, suppose r = 3. Let {ω_{1}, ω_{2}, ω_{3}} be a basis of E, and let V_{i} = V_{ω_{i}} for i = 1, 2, 3. Then dim V_{i} = ℓ and dim V_{i} ∩ V_{j} = ℓ − 1 for 1 ≤ i, j ≤ 3 with i ≠ j, thanks to Lemmas 1 and 3. Thus, if we let W = V_{1} ∩ V_{2} ∩ V_{3}, then

ℓ − 2 = dim V_{1} ∩ V_{2} + dim V_{1} ∩ V_{3} − dim V_{1} ≤ dim W ≤ dim V_{1} ∩ V_{2} = ℓ − 1.

If dim W = ℓ−1, then we can find ℓ+2 elements f_{1}, …, f_{ℓ−1}, g_{1}, g_{2}, g_{3} in V such that {f_{1}, …, f_{ℓ−1}} is a basis of W and {f_{1}, …, f_{ℓ−1}, g_{i}} is a basis of V_{i} for i = 1, 2, 3. We may assume without loss of generality that ω_{i} = f_{1} ∧⋯∧f_{ℓ−1} ∧g_{i} for i = 1, 2, 3, thanks to Lemma 1. Since ω_{1}, ω_{2}, ω_{3} are linearly independent, it follows that g_{i} ∉ ∑_{j≠i} V_{j} for i = 1, 2, 3. Consequently, f_{1}, …, f_{ℓ−1}, g_{1}, g_{2}, g_{3} are linearly independent elements of V and E is close of type I. On the other hand, if dim W = ℓ−2, then we can find ℓ+1 elements u_{1}, …, u_{ℓ−2}, g_{1}, g_{2}, g_{3} in V such that {u_{1}, …, u_{ℓ−2}} is a basis of W, and {u_{1}, …, u_{ℓ−2}, g_{i}} is a basis of ∩_{j≠i}V_{j}, and moreover, g_{i} ∉ V_{i} for i = 1, 2, 3. Consequently, u_{1}, …, u_{ℓ−2}, g_{1}, g_{2}, g_{3} are linearly independent elements of V [indeed, the vanishing of a linear combination of u_{1}, …, u_{ℓ−2}, g_{1}, g_{2}, g_{3} in which the coefficient of g_{i} is nonzero implies that g_{i} is in V_{i}]. Hence in view of Lemma 1, we see that for i = 1, 2, 3, the set {u_{1}, …, u_{ℓ−2}, g_{1}, g_{2}, g_{3}} \ {g_{i}} is a basis of V_{i} and ω_{i} = c_{i} (u_{1} ∧ ⋯ ∧ u_{ℓ−2} ∧ g_{i_{1}} ∧ g_{i_{2}}) for some c_{i} ∈ F \ {0}, where 1 ≤ i_{1} < i_{2} ≤ 3 with i_{1} ≠ i ≠ i_{2}. It follows that E is close of type II.

Finally, we assume that r > 3 and that every decomposable subspace of dimension < r is close of type I or of type II. Let {ω_{1}, …, ω_{r}} be a basis of E, and let V_{i} = V_{ω_{i}} and E_{i} = span{ω_{1}, …, ω_{i−1}, ω_{i+1}, …, ω_{r}} for i = 1, …, r. Each E_{i} is decomposable and by the induction hypothesis, we are in one of the following two cases.

Case 1: E_{i} is close of type I for some i ∈ {1, …, r}.

Fix i ∈ {1, …, r} such that E_{i} is close of type I, and let W_{i} := V_{E_{i}} = ∩_{j≠i}V_{j}. Then dim W_{i} = ℓ − 1 and since V_{E} = W_{i} ∩ V_{i}, by picking any j ∈ {1, …, r} with j ≠ i, and using Lemma 3, we find

ℓ − 2 = dim W_{i} + dim V_{i} − dim(V_{j} + V_{i}) ≤ dim V_{E} ≤ dim W_{i} = ℓ − 1,

If dim V_{E} = ℓ − 1, then it is readily seen that E is close of type I. Suppose, if possible, dim V_{E} = ℓ − 2. Let {f_{1}, …, f_{ℓ−2}} be a basis of V_{E} and f_{ℓ−1} be any element of W_{i} \ V_{E}. Then {f_{1}, …, f_{ℓ−1}} is a basis of W_{i} and f_{ℓ−1} ∉ V_{i}. Since dim V_{j} ∩ V_{i} = ℓ − 1 for j ≠ i, we can find g_{1}, …, g_{i−1}, g_{i+1}, …, g_{r} ∈ V_{i} such that {f_{1}, …, f_{ℓ−2}, g_{j}} is a basis of V_{j} ∩ V_{i} for j ∈ {1, …, r} with j ≠ i. Also, since f_{ℓ−1} ∈ W_{i} \ V_{E}, we see that {f_{1}, …, f_{ℓ−1}, g_{j}} is a basis of V_{j}, and so by Lemma 1, each ω_{j} is a nonzero scalar multiple of f_{1} ∧ ⋯ ∧ f_{ℓ−1} ∧ g_{j} for j ∈ {1, …, r} with j ≠ i. Now ω_{1}, …, ω_{i−1}, ω_{i+1}, …, ω_{r} are linearly independent elements of ⋀^{ℓ} V, and therefore f_{1}, …, f_{ℓ−1}, g_{1}, …, g_{i−1}, g_{i+1}, …, g_{r} are linearly independent of V. In particular, the ℓ-dimensional space V_{i} contains ℓ + r − 3 linearly independent elements f_{1}, …, f_{ℓ−2}, g_{1}, …, g_{i−1}, g_{i+1}, …, g_{r}, which is a contradiction since r > 3. Thus we have shown that E is close of type in I.

Case 2: E_{i} is close of type II for each i ∈ {1, …, r}.

In this case each W_{i} := V_{E_{i}} is of dimension ℓ − r + 2 and as before, picking any j ∈ {1, …, r} with j ≠ i, and using Lemma 3, we find

ℓ − r + 1 = dim W_{i} + dim V_{i} − dim(V_{j} + V_{i}) ≤ dim V_{E} ≤ dim W_{i} = ℓ − r + 2.

First, suppose dim V_{E} = ℓ − r + 1. Fix a basis {u_{1}, …, u_{ℓ−r+1}} of V_{E}. For each i ∈ {1, …, r}, choose g_{i} ∈ W_{i} \ V_{E}. Then g_{i} ∉ V_{i} and {u_{1}, …, u_{ℓ−r+1}, g_{i}} is a basis of W_{i} for i = 1, …, r. Observe that the ℓ + 1 elements u_{1}, …, u_{ℓ−r+1}, g_{1}, …, g_{r} of V are linearly independent [indeed, the vanishing of a linear combination of u_{1}, …, u_{ℓ−r+1}, g_{1}, …, g_{r} in which the coefficient of g_{i} is nonzero implies that g_{i} is in V_{i}]. Hence the subset {u_{1}, …, u_{ℓ−r+1}, g_{1}, …, g_{i−1}, g_{i+1}, …, g_{r}} of V_{i} is a basis of V_{i}, and so in view of Lemma 1, we see that ω_{i} is a nonzero scalar multiple of u_{1} ∧ ⋯ ∧ u_{ℓ−r+1} ∧ g_{1} ⋯ ∧ ǧ_{i} ∧ ⋯ ∧ g_{r} for i = 1, …, r. Thus we have shown that if dim V_{E} = ℓ − r + 1, then E is close of type II. Now suppose, if possible, dim V_{E} = ℓ − r + 2. Then V_{E} = W_{i} for i = 1, …, r. Since r > 3 and E_{r−1} is close of type II, we see that the subspace E^{∗} := span{ω_{1}, ω_{2}, ω_{r}} of E_{r−1} is close of type II. In particular, dim V_{E^{∗}} = dim V_{1} ∩ V_{2} ∩ V_{r} = ℓ − 2. Thus, in view of Lemma 3, we see that dim(V_{1} + V_{2} + V_{r}) is at most

dim V_{1} +dim V_{2} +dim V_{r} −dim V_{1} ∩V_{2} −dim V_{1} ∩V_{r} −dim V_{2} ∩V_{r} +dim V_{1} ∩V_{2} ∩V_{r},

which is 3ℓ−3(ℓ−1)+(ℓ−2) = ℓ+1. Also, by Lemma 3, we have dim(V_{1}+V_{2}+V_{r}) ≥ dim(V_{1} + V_{2}) = ℓ + 1. It follows that V_{1} + V_{2} + V_{r} = V_{1} + V_{2}, or equivalently, V_{r} ⊆ V_{1} + V_{2}. We will now use this to arrive at a contradiction. To this end, consider the space E_{r}. Since E_{r} is close of type II, we can find ℓ + 1 linearly independent elements u_{1}, …, u_{ℓ−r+2}, g_{1}, …, g_{r−1} in V such that u_{1}, …, u_{ℓ−r+2} span W_{r} and ω_{i} = u_{1} ∧ ⋯ ∧ u_{ℓ−r+2} ∧ g_{1} ⋯ ∧ ǧ_{i} ∧ ⋯ ∧ g_{r−1} for i = 1, …, r − 1. It is clear that {u_{1}, …, u_{ℓ−r+2}, g_{1}, …, g_{r−1}} is a basis of V_{1} + V_{2}. Also, since W_{r} = V_{E}, we can add r − 2 elements to the set {u_{1}, …, u_{ℓ−r+2}} to obtain a basis of V_{r}. But, V_{r} ⊆ V_{1} + V_{2} and so the additional r − 2 basis elements of V_{r} are linear combinations of u_{1}, …, u_{ℓ−r+2}, g_{1}, …, g_{r−1}. Consequently, ω_{r} is a linear combination of ω_{1}, …, ω_{r−1}, which is a contradiction. □

Corollary 7. Let μ := max{ℓ, m−ℓ}+1 and r be any positive integer. Then ⋀^{ℓ} V has a decomposable subspace of dimension r if and only if r ≤ μ. Moreover, a close subspace of type I (resp: type II) of dimension r exists if and only if r ≤ m − ℓ + 1 (resp: r ≤ ℓ + 1).

Proof. Let E be a subspace of ⋀^{ℓ} V of dimension r. By Lemma 5, if E is close of type I, then ℓ + r − 1 = dim V^{E} ≤ m, that is, r ≤ m − ℓ + 1, whereas if E is close of type II, then ℓ − r + 1 = dim V_{E} ≥ 0, that is, r ≤ ℓ + 1. Thus, Theorem 6 implies that if ⋀^{ℓ} V has a decomposable subspace of dimension r, then r ≤ μ. The converse is an immediate consequence of the definition of close subspaces and their decomposability. □

Remark 8. Decomposable vectors in ⋀^{ℓ} V are variously known as pure ℓ-vectors (cf. [2, §11.13]), extensors of step ℓ (cf. [1, §3]), or completely decomposable vectors (cf. [12]). Some of the preliminary lemmas proved initially in this section are not really new. For example, Lemma 1 appears essentially as Exercise 17 (a) in Bourbaki [2, p. 650] or as Theorem 1.1 in Marcus [10], Corollary 2 is basically Theorem 1.3 of [10], and Lemma 3 is a consequence of Exercise 17 (c) in [2, p. 651]. We have stated these results in a form convenient for our purpose, and included the proofs for the sake of completeness. At any rate, as far as we know, Theorem 6 is new. On the other hand, characterization of decomposable subspaces has been studied in the setting of symmetric algebras. Although one comes across subspaces of various types, including those similar to the ones considered in this section, the situation for subspaces of symmetric powers is rather different and the characteristic of the underlying field plays a role. We refer to the papers of Cummings [3] and Lim [9] for more on this topic. In the context of tensor algebras, the opposite of decomposable subspaces has been considered, namely, completely entangled subspaces wherein no nonzero element is decomposable. A neat formula for the maximum possible dimension of completely entangled subspaces of the tensor product of finite dimensional complex vector spaces is given by Parthasarathy [13]. As remarked earlier, determining the structure of decomposable subspaces corresponds to determining the linear subvarieties in the Grassmann variety G_{ℓ,m}. A special case of this has been considered, in a similar, but more general, geometric setting by Tanao [14], where subvarieties of G_{2,m} biregular to ℙ^{m} over an algebraically closed field of characteristic zero are studied.

3. Duality and the Hodge Star Operator

We have seen in Section 2 that a decomposable subspace of ⋀^{ℓ} V is close of type I or of type II. It turns out that the two types are dual to each other. This is best described using the so called Hodge star operator 𝔥 : ⋀^{ℓ}V → ⋀^{m−ℓ}V, which may be defined as follows. Fix an ordered basis {e_{1}, …, e_{m}} of V and use it to identify ⋀^{m} V with F so that e_{1} ∧ ⋯ ∧ e_{m} = 1. Let I(ℓ, m) and e_{α} for α ∈ I(ℓ, m) be as in Section 2. Moreover, for α = (α_{1}, …, α_{ℓ}) ∈ I(ℓ, m), let α^{c} = (α_{1}^{c}, …, α_{m−ℓ}^{c}) denote the unique element of I(m−ℓ, m) such that {α_{1}, …, α_{ℓ}}∪{α_{1}^{c}, …, α_{m−ℓ}^{c}} = {1, …, m}. Then 𝔥 : ⋀^{ℓ} V → ⋀^{m−ℓ} V is the unique F -linear map satisfying

𝔥(e_{α}) = (−1)^{α_{1}+···+α_{ℓ}+ℓ(ℓ+1)/2} e_{α^{c}} for α ∈ I(ℓ, m).

Clearly, 𝔥 is a vector space isomorphism. The key property of 𝔥 is that it is essentially independent of the choice of ordered basis of V, and as such, it maps decomposable elements in ⋀^{ℓ} V to decomposable elements in ⋀^{m−ℓ} V. (See, for example, [1, Sec. 6] and [10, Sec. 4.1].) In particular, decomposable subspace of ⋀^{ℓ} V are mapped to decomposable subspaces of ⋀^{m−ℓ} V. Moreover, it is easy to see that via the Hodge star operator, close subspaces of type I are mapped to close subspaces of type II, whereas close subspaces of type II are mapped to close subspaces of type I. Thus, the two types are dual to each other.

In the case ℓ = 2, both ⋀^{ℓ} V and ⋀^{m−ℓ} V are closely related to the space B_{m} of all m×m skew-symmetric matrices with entries in F, and the relation is compatible with the Hodge star operator. To state this a little more formally, we introduce some terminology below and make a few useful observations. In the remainder of this section we tacitly assume that m > 2.

Given any u ∈ V, let u denote the m × 1 column vector whose entries are the coordinates of u with respect to the ordered basis {e_{1}, …, e_{m}}. In particular, e_{i} has 1 as its ith entry and all other entries are 0. Consider the F -linear maps

σ : ⋀^{2}V → B_{m} and π : ⋀^{m−2}V → B_{m}

defined by

σ(e_{r} ∧ e_{s}) = e_{r}e_{s}^{t} − e_{s}e_{r}^{t} for 1 ≤ r < s ≤ m and π(ω) = A_{ω} for ω ∈ ⋀^{m−2}V,

where e^{t} denotes the transpose of e and A_{ω} denotes the m×m matrix whose (i, j)th entry is (the unique scalar corresponding to) e_{i} ∧ e_{j} ∧ ω.

Lemma 9. σ = π ∘ 𝔥.

Proof. We have 𝔥(e_{r} ∧ e_{s}) = (−1)^{r+s+1} (e_{1} ∧ e_{2} ∧ ⋯ ∧ ě_{r} ∧ ⋯ ∧ ě_{s} ∧ ⋯ ∧ e_{m}) for 1 ≤ r < s ≤ m, whereˇ indicates that the corresponding entry is removed. Now,

e_{i}∧e_{j} ∧(e_{1} ∧ e_{2} ∧ ⋯ ∧ ě_{r} ∧ ⋯ ∧ ě_{s} ∧ ⋯ ∧ e_{m}) = \{ \begin{matrix}(−1)^{i+j−3} & if (r, s) = (i, j), \\ (−1)^{i+j−2} & if (r, s) = (j, i), \\ 0 & otherwise \end{matrix}

for 1 ≤ i, j, r, s ≤ m with r < s. It follows that π ∘ 𝔥(e_{r} ∧ e_{s}) = e_{r}e_{s}^{t} − e_{s}e_{r}^{t} = σ(e_{r} ∧ e_{s}) for 1 ≤ r < s ≤ m. Since {e_{r} ∧ e_{s} : 1 ≤ r < s ≤ m} is a basis of ⋀^{2}V and all the maps are linear, the lemma is proved. □

Given any ω′ ∈ ⋀^{2}V and ω ∈ ⋀^{m−2}V, we refer to the rank of σ(ω′) [resp: π(ω)] as the rank of ω′ [resp: ω], and denote it by rank(ω′) [resp: rank(ω)]. Note that if ω = 𝔥(ω′), then rank(ω′) = rank (ω), thanks to Lemma 9.

Corollary 10. Both σ and π are vector space isomorphisms. Moreover,

(7)  ω′ is decomposable ⟺ rank(ω′) = 2  for any ω′ ∈ ⋀^{2}V,

and

(8)  ω is decomposable ⟺ rank(ω) = 2  for any ω ∈ ⋀^{m−2}V.

Proof. It is evident that σ is an isomorphism. Hence by Lemma 9, so is π. Now, given any ω ∈ ⋀^{m−2}V, the kernel of (the linear map from V to V corresponding to) π(ω) = A_{ω} is the space V_{ω}. Hence (8) follows from Lemma 1. Next, if ω′ ∈ ⋀^{2}V is decomposable, then ω′ = u ∧ v for some u, v ∈ V and σ(ω′) = uv^{t} − vu^{t}. It follows that σ(ω′) is of rank 2. This proves the implication ⟹ in (7). The other implication follows from (8) together with Lemma 9 and the fact 𝔥 gives a one-to-one correspondence between decomposable elements. □

Corollary 11. Let v_{1}, v_{2}, v_{3}, v_{4} ∈ V and suppose ω := (v_{1} ∧ v_{2}) + (v_{3} ∧ v_{4}) ∈ ⋀^{2} V is nonzero. Then the rank of σ(ω) is 2 or 4 according as the set {v_{1}, v_{2}, v_{3}, v_{4}} is linearly dependent or linearly independent.

Proof. Follows from (7) above and Corollary 4 in view of the fact that a skewsymmetric matrix is always of even rank. □

4. Griesmer-Wei Bound and its Generalization

Let us begin by reviewing some generalities about (linear, error correcting) codes. Fix integers k, n with 1 ≤ k ≤ n and a prime power q. Let C be a linear [n, k]_{q}-code, i.e., let C be a k-dimensional subspace of the n-dimensional vector space 𝔽_{q}^{n} over the finite field 𝔽_{q} with q elements. Given any x = (x_{1}, …, x_{n}) in 𝔽_{q}^{n}, let

supp(x) := {i : x_{i} ≠ 0} and ∥x∥ := |supp(x)|

denote the support and the (Hamming) norm of x. More generally, for D ⊆ 𝔽_{q}^{n}, let

supp(D) := {i : x_{i} ≠ 0 for some x = (x_{1}, …, x_{n}) ∈ D} and ∥D∥ := |supp(D)|

denote the support and the (Hamming) norm of D. The minimum distance of C is defined by d(C) := min{∥x∥ : x ∈ C with x ≠ 0}. More generally, for any positive integer r, the r^{th} higher weight d_{r} = d_{r}(C) of the code C is defined by

d_{r}(C) := min {∥D∥ : D is a subspace of C with dim D = r} .

Note that d_{1}(C) = d(C). If C is nondegenerate, that is, if C is not contained in a coordinate hyperplane of 𝔽_{q}^{n}, then it is easy to see that

0 < d_{1}(C) < d_{2}(C) < ⋯ < d_{k}(C) = n.

See, for example, [15] for a proof as well as a great deal of basic information about higher weights of codes. The set {d_{r}(C) : 1 ≤ r ≤ k} is often referred to as the weight hierarchy of the code C. It is usually interesting, and difficult, to determine the weight hierarchy of a given code. Again, we refer to [15] for a variety of examples, such as affine and projective Reed-Muller codes, codes associated to Hermitian varieties or Del Pezzo surfaces, hyperelliptic curves, etc., where the weight hierarchy is completely or partially known.

The following elementary result will be useful in the sequel. It appears, for example, in [8, Lemma 2]. We include a proof for the sake of completeness.

Lemma 12. Let D be a r-dimensional code of a [n, k]_{q}-code C. Then

∥D∥ = \frac{1}{q^{r} − q^{r−1}} ∑_{x∈D} ∥x∥.

In particular,

d_{r}(C) = \frac{1}{q^{r} − q^{r−1}} min \{∑_{x∈D} ∥x∥ : D is a subspace of C with dim D = r\} .

Proof. Clearly, (x, i) ↦ (i, x) gives a bijection of {(x, i) : x ∈ D and i ∈ supp(x)} onto {(i, x) : i ∈ supp(D), x ∈ D and x_{i} ≠ 0}. Hence

∑_{x∈D} ∥x∥ = ∑_{x∈D} ∑_{i∈supp(x)} 1 = ∑_{i∈supp(D)} ∑_{x∈D} ; x_{i}≠0 1 = ∑_{i∈supp(D)}(q^{r}−q^{r−1}) = (q^{r}−q^{r−1})∥D∥,

where the penultimate equality follows by noting that if i ∈ supp(D), then x ↦ x_{i} defines a nonzero linear map of D → 𝔽_{q}. □

We remark that the Griesmer bound as well as the Griesmer-Wei bound is an easy consequence of the above lemma. In fact, as we shall see below, it can also be used to derive a useful generalization of the Griesmer-Wei bound. To this end, we need to look at the elements of minimum Hamming weight as well as the second lowest positive exponent in the weight enumerator polynomial of C, provided of course this polynomial has at least two terms with positive exponents.

Let C be a linear [n, k]_{q}-code. Given any subspace D of C, we let

Δ(D) := |{x ∈ D : ∥x∥ = d(C)}| .

Given any r ∈ ℤ with 1 ≤ r ≤ k, we let

Δ_{r}(C) := max {Δ(D) : D is a subspace of C with dim D = r} .

Further, upon letting S_{C} := {∥x∥ : x ∈ C with ∥x∥ > d(C)}, we define

e(C) := \{ \begin{matrix}min S_{C} & if S_{C} is nonempty, \\ d(C) & if S_{C} is the empty set. \end{matrix}

It may be noted that e(C) ≥ d(C) and also that the equality holds if and only if Δ_{k}(C) = q^{k} − 1. We are now ready to prove a simple, but useful generalization of the Griesmer-Wei bound.

Theorem 13. Let C be a linear [n, k]_{q}-code and r be an integer with 1 ≤ r ≤ k. Then

d_{r}(C) ≥ \frac{d(C)Δ_{r}(C) + e(C)(q^{r} − 1 − Δ_{r}(C))}{q^{r} − q^{r−1}}.

Proof. Let D_{r} be a r-dimensional subspace of C such that

∑_{x∈D_{r}} ∥x∥ = min {∑_{x∈D} ∥x∥ : D is a subspace of C with dim D = r} .

Then D_{r} has q^{r} − 1 nonzero elements and so, in view of Lemma 12, we have

(q^{r} − q^{r−1}) d_{r}(C) = ∑_{x∈D_{r}} ; ∥x∥=d(C) d(C) + ∑_{x∈D_{r}} ; ∥x∥>d(C) ∥x∥

≥ d(C)Δ(D_{r}) + e(C) (q^{r} − 1 − Δ(D_{r}))
≥ e(C) (q^{r} − 1) − Δ_{r}(C) (e(C) − d(C)) ,

where the last inequality follows since Δ(D_{r}) ≤ Δ_{r}(C) and d(C) ≤ e(C). This yields the desired formula. □

Corollary 14 (Griesmer-Wei Bound). Given any linear [n, k]_{q}-code C, we have

d_{r}(C) ≥ ∑_{j=0}^{r−1} ⌈\frac{d(C)}{q^{j}}⌉ for 1 ≤ r ≤ k.

Proof. Using Theorem 13 and the fact that e(C) ≥ d(C), we see that

d_{r}(C) ≥ \frac{d(C)(q^{r} − 1)}{q^{r} − q^{r−1}} = ∑_{i=0}^{r−1} \frac{d(C)q^{i}}{q^{r−1}} = ∑_{j=0}^{r−1} \frac{d(C)}{q^{j}} ≥ ∑_{j=0}^{r−1} ⌈\frac{d(C)}{q^{j}}⌉

for any integer r with 1 ≤ r ≤ k. □

5. The Grassmann Code C(ℓ, m)

Let us fix, throughout this section, a prime power q and integers ℓ, m with 1 ≤ ℓ ≤ m, and let

n := \left[{m \atop ℓ}\right]_{q}, k := \binom{m}{ℓ}, and δ := ℓ(m − ℓ),

where \left[{m \atop ℓ}\right]_{q} is the Gaussian binomial coefficient, which was defined in Section 1. It may be remarked that \left[{m \atop ℓ}\right]_{q} is a polynomial in q of degree δ with positive integral coefficients. The Grassmann code C(ℓ, m) is the linear [n, k]_{q}-code associated to the projective system corresponding to the Plücker embedding of the 𝔽_{q}-rational points of the Grassmannian G_{ℓ,m} in ℙ_{𝔽_{q}}^{k−1} = ℙ( ⋀^{ℓ} 𝔽_{q}^{m}); see, for example, [12, 4] for greater details. Alternatively, C(ℓ, m) may be defined as follows.

Let V := 𝔽_{q}^{m}. Fix a basis {e_{1}, …, e_{m}} of V. Then we can, and will, fix a corresponding basis of ⋀^{ℓ} V given, in the notations of Section 5, by {e_{α} : α ∈ I(ℓ, m)}. Let G_{ℓ,m} = G_{ℓ,m}(𝔽_{q}) be the Grassmann variety consisting of all ℓ-dimensional subspaces of V. The Plücker embedding G_{ℓ,m} ↪ ℙ( ⋀^{ℓ} V ) simply maps a ℓ-dimensional subspace of V spanned by v_{1}, …, v_{ℓ} to the point of ℙ( ⋀^{ℓ} V ) corresponding to v_{1} ∧ ⋯ ∧ v_{ℓ}. It is well-known that this embedding is well defined and nondegenerate. Fix representatives ω′_{1}, …, ω′_{n} in ⋀^{ℓ} V corresponding to distinct points of G_{ℓ,m}(𝔽_{q}). We denote the subset {ω′_{1}, …, ω′_{n}} of ⋀^{ℓ} V by T (ℓ, m). Having fixed a basis of V, we can identify each element of ⋀^{m} V with a unique scalar in 𝔽_{q}. With this in view, we obtain a linear map

τ : ⋀^{m−ℓ}V → 𝔽_{q}^{n} given by τ(ω) := (ω′_{1} ∧ ω, ω′_{2} ∧ ω, …, ω′_{n} ∧ ω) .

Since the Plücker embedding is nondegenerate, it follows that τ is injective. The Grassmann code C(ℓ, m) is defined as the image of the map τ. It is clear that C(ℓ, m) is a linear [n, k]_{q}-code. Given any codeword c ∈ C(ℓ, m), there is unique ω ∈ ⋀^{m−ℓ} V such that τ(ω) = c; we denote this ω by ω_{c}.

Given any subspace E of ⋀^{ℓ} V, we let g(E) := |E ∩ T (ℓ, m)|. Note that since T (ℓ, m) consists of nonzero elements, no two of which are proportional to each other, we always have

(9)  |g(E)| ≤ \frac{q^{r} − 1}{q − 1} for any subspace E of ⋀^{ℓ} V with dim E = r.

Given any integer s with 1 ≤ s ≤ k, we let

g_{s}(ℓ, m) := max {g(E) : E a subspace of ⋀^{ℓ} V of codimension s} .

Note that as a consequence of (9), we have

(10)  g_{s}(ℓ, m) ≤ \frac{q^{r} − 1}{q − 1} where r := k − s.

Lemma 15. Let D be a subspace of C(ℓ, m) and s = dim D. If D := τ^{−1}(D), then E := D^{⊥} := {ω′ ∈ ⋀^{ℓ} V : ω′ ∧ ω = 0} is a subspace of ⋀^{ℓ} V of codimension s and

∥D∥ = n − g(E).

Proof. Since τ is an isomorphism of ⋀^{m−ℓ} V and C(ℓ, m), we have dim D = s. Also, since (ω′, ω) ↦ ω′ ∧ ω gives a nondegenerate bilinear map of ⋀^{ℓ} V × ⋀^{m−ℓ} V → 𝔽_{q}, and so E := D^{⊥} is a subspace ⋀^{ℓ} V of codimension s. For 1 ≤ i ≤ n, we have

i ∉ supp(D) ⟺ ω′_{i} ∧ ω = 0 for all ω ∈ D ⟺ ω′_{i} ∈ E.

It follows that ∥D∥ = n − g(E). □

Corollary 16. d_{s} (C(ℓ, m)) = n − g_{s}(ℓ, m) for s = 1, …, k.

Proof. Clearly, E ↦ τ(E^{⊥}) sets up a one-to-one correspondence between subspaces of ⋀^{ℓ} V of codimension s and subspaces of C(ℓ, m) of dimension s. Hence the desired result follows from Lemma 15. □

We now recall some important results of Nogin [12]. Combining Theorem 4.1, Proposition 4.4 and Corollary 4.5 of [12], we have the following.

Proposition 17. The minimum distance of C(ℓ, m) is q^{δ} and the codewords c of C(ℓ, m) such that ω_{c} is decomposable attain the minimum weight q^{δ}. Moreover, the number of minimum weight codewords in C(ℓ, m) is (q − 1)n.

A useful consequence is the following.

Corollary 18. Given any c ∈ C(ℓ, m), we have

∥c∥ = q^{δ} ⟺ ω_{c} is decomposable.

Moreover Δ(C(ℓ, m)) = (q − 1)n.

Proof. The implication ⟸ follows from Proposition 17. The other implication also follows from Proposition 17 by noting that the number of decomposable elements of ⋀^{m−ℓ} V is equal to the number of decomposable elements of ⋀^{ℓ} V, and that the latter is equal to (q − 1)n. □

In [12], Nogin goes on to determine some of the higher weights of C(ℓ, m) using Proposition 17 and some additional work. More precisely, he proves formula (2) in Section 1. As remarked in Section 1, Introduction, alternative proofs of (2) are given in [4] as well as [7]. The latter also proves the dual version (3). We give below yet another proof of (2) and (3) as an application of Theorem 6 and Corollary 18.

Theorem 19. Let μ := max{ℓ, m − ℓ} + 1. Then for 0 ≤ r ≤ μ we have

d_{r} (C(ℓ, m)) = q^{δ}+q^{δ−1}+⋯+q^{δ−r+1} and d_{k−r} (C(ℓ, m)) = n−(1+q+⋯+q^{r−1}).

Proof. The case r = 0 is trivial. Assume that 1 ≤ r ≤ μ. By Corollary 7, there is a decomposable subspace E of ⋀^{ℓ} V of dimension r. Then 𝔥(E) is a decomposable subspace of ⋀^{m−ℓ} V and hence by Corollary 18, D := τ(𝔥(E)) is a r-dimensional subspace of C(ℓ, m) in which every nonzero vector is of minimal weight. Consequently, by Lemma 12, we have

∥D∥ = \frac{1}{q^{r} − q^{r−1}} ∑_{c∈D} ∥c∥ = \frac{d (C(ℓ, m)) (q^{r} − 1)}{q^{r} − q^{r−1}} = ∑_{j=0}^{r−1} \frac{d (C(ℓ, m))}{q^{j}} = ∑_{j=0}^{r−1} q^{δ−j}.

In other words, the Griesmer-Wei bound is attained. This proves the desired formula for d_{r} (C(ℓ, m)). Next, E is a subspace of ⋀^{ℓ} V of codimension k−r, and since E is decomposable, every ω′ ∈ E with ω′ ≠ 0 can be uniquely written as ω′ = λω′_{i} where λ ∈ 𝔽_{q} \ {0} and i ∈ {1, …, n}. It follows that g(E) = (q^{r} − 1)/(q − 1) = 1 + q + ⋯ + q^{r−1}, and so, in view of (10), we find g_{k−r}(ℓ, m) = 1 + q + ⋯ + q^{r−1}. This, together with Corollary 16, yields the desired formula for d_{k−r} (C(ℓ, m)). □

6. Higher Weights of the Grassmann Code C(2, m)

The results on the higher weights of C(2, m) mentioned in the Introduction will be proved in this section. Throughout, let q, ℓ, m, k, n, δ be as in Section 5, except we set ℓ = 2. Also, we let F := 𝔽_{q} and V := 𝔽_{q}^{m}. Note that the complete weight hierarchy of C(2, m) is easily obtained from Theorem 19 if m ≤ 4. With this in view, we shall assume that m > 4. In particular, μ := max{ℓ, m − ℓ} + 1 = m − 1.

We begin by recalling a result of Nogin concerning the spectrum of C(2, m). To this end, given any nonnegative integer t, let N(m, 2t) denote the number of skew-symmetric bilinear forms of rank 2t on 𝔽_{q}^{m}. We know from [11, §15.2] that

(11)  N(m, 2t) = \frac{(q^{m} − 1)(q^{m−1} − 1) ⋯ (q^{m−2t+1} − 1)}{(q^{2t} − 1)((q^{2t−2} − 1) ⋯ (q^{2} − 1)}q^{t(t−1)}.

The said result of Nogin [12, Thm. 5.1] is the following.

Proposition 20. Given any i ≥ 0, let A_{i} := |{c ∈ C(2, m) : ∥c∥ = i}|. Then

(12)  A_{i} = \{ \begin{matrix}N(m, 2t) & if i = q^{2(m−t−1)} \frac{q^{2t} − 1}{q^{2} − 1} for 0 ≤ t ≤ ⌊m/2⌋, \\ 0 & otherwise. \end{matrix}

Moreover, for any c ∈ C(2, m) and 0 ≤ t ≤ ⌊m/2⌋, we have

∥c∥ = q^{2(m−t−1)} \frac{q^{2t} − 1}{q^{2} − 1} ⟺ rank(ω_{c}) = 2t.

Corollary 21. d (C(2, m)) = q^{δ} and e (C(2, m)) = q^{δ} + q^{δ−2}.

Proof. The numbers θ_{t} := q^{2(m−t−1)} \frac{q^{2t}−1}{q^{2}−1} increase with t and the first two positive values of θ_{t} (t ≥ 0) are q^{δ} and q^{δ} + q^{δ−2}. □

We now prove a number of auxiliary results needed to prove the main theorem.

Lemma 22. Let E and E_{1} be subspaces of ⋀^{2} V such that E ⊂ E_{1} and dim E_{1} = dim E + 1. Assume that E is decomposable and E_{1} is not decomposable. Then we have the following.

(i) The set E_{1} \ E contains at most q^{2}(q − 1) decomposable vectors.

(ii) If E_{1} \E contains a decomposable vector ω such that V_{ω} ⊆ V^{E}, then E_{1} \E contains exactly q^{2}(q − 1) decomposable vectors.

Proof. Both (i) and (ii) hold trivially if E_{1} \ E contains no decomposable vector. Now, suppose E_{1} \ E contains a decomposable vector, say ω. Then E_{1} = E + F ω. Write ω = u ∧ v, where u, v ∈ V, and let r := dim E. By Theorem 6, we are in either of the following two cases.

Case 1: E is close of type I.

In this case, there are linearly independent elements f, g_{1}, …, g_{r} ∈ V such that E = span{f ∧ g_{i} : i = 1, …, r}. Let G := span{g_{1}, …, g_{r}}. Elements ξ of E_{1} are of the form ξ = f ∧ g + λ(u ∧ v), where g ∈ G and λ ∈ 𝔽_{q}. Clearly, ξ and (g, λ) determine each other uniquely, and ξ ∈ E_{1} \ E if and only if λ ≠ 0. Observe that {f, u, v} is linearly independent, lest we can write u ∧ v = f ∧ h for some h ∈ V, and consequently, E_{1} becomes decomposable. Hence, by Corollary 4, we see that if λ ≠ 0, then ξ = f ∧ g + λ(u ∧ v) is decomposable if and only if g ∈ span{f, u, v}. Further, in view of Lemmas 1 and 5, we have V_{ω} = span{u, v} and V^{E} = span{f, g_{1}, …, g_{r}}. Thus, g ∈ span{f, u, v} if and only if f ∧ g = f ∧ x for some x ∈ V_{ω} ∩ V^{E}. It follows that decomposable elements of E_{1} \ E are precisely of the form f ∧ x + λ(u ∧ v), where x ∈ V_{ω} ∩ V^{E} and λ ∈ 𝔽_{q} \ {0}. Since |V_{ω} ∩ V^{E}| ≤ |V_{ω}| = q^{2} and |𝔽_{q} \ {0}| = q − 1, both (i) and (ii) are proved.

Case 2: E is close of type II, but not closed of type I.

In this case, by Corollary 7, we must have dim E = 3. Thus, there are linearly independent elements g_{1}, g_{2}, g_{3} ∈ V such that E = span{g_{2}∧g_{3}, g_{1}∧g_{3}, g_{1}∧g_{2}}. Let G := span{g_{1}, g_{2}, g_{3}}. Note that since G = V^{E} and ω = u ∧ v ∉ E, the possibility that V_{ω} ⊆ V^{E} does not arise in this case. Thus dim V_{ω} ∩ V^{E} ≤ 1 and (ii) holds vacuously. The elements of E_{1} are of the form ξ = g ∧ h + λ(u ∧ v), where g, h ∈ G and λ ∈ 𝔽_{q}. Clearly, ξ is a decomposable element of E_{1} \ E if g ∧ h = 0 and λ ≠ 0. If, in addition, ξ = g ∧ h + λ(u ∧ v) is decomposable for some g, h ∈ G with g ∧ h ≠ 0 and λ ∈ 𝔽_{q} \ {0}, then by Corollary 4, {g, h, u, v} is linearly dependent, and hence dim V_{ω} ∩ V^{E} = 1. So we may assume without loss of generality that u = g_{1}. Then it is clear that the elements of E_{1} \ E are precisely the (unique) linear combinations of the form λ(u ∧ v) + λ_{1}(g_{2} ∧ g_{3}) + λ_{2}(g_{1} ∧ g_{3}) + λ_{3}(g_{1} ∧ g_{2}), where λ ∈ 𝔽_{q} \ {0} and λ_{1}, λ_{2}, λ_{3} ∈ 𝔽_{q}; moreover, by Corollary 4, such a linear combination is decomposable if and only if λ_{1} = 0. It follows that E_{1} \ E contains at most q^{2}(q − 1) decomposable elements. □

The bound q^{2}(q − 1) in Lemma 22 can be improved if the dimension of the decomposable subspace E is small.

Lemma 23. Let E and E_{1} be subspaces of ⋀^{2} V such that E ⊂ E_{1} and dim E_{1} = dim E + 1. Assume that E is decomposable of dimension r ≥ 1 and E_{1} is not decomposable. Then E_{1} \ E contains at most q^{r−1}(q − 1) decomposable elements.

Proof. If r ≥ 3, then the result is an immediate consequence of part (i) of Lemma 22. Also, the result holds trivially if E_{1} \ E contains no decomposable element. Thus, let us assume that r ≤ 2 and E_{1} = E + F ω, where ω ∈ E_{1} \ E is decomposable.

First, suppose r = 1. Then E = F ω_{0} for some decomposable ω_{0} ∈ ⋀^{2} V. Since E_{1} is not decomposable and ω ∉ E, in view of Lemmas 1 and 3, we see that dim V_{ω} ∩ V_{ω_{0}} = 0. Hence from Corollary 4, it follows that the only decomposable elements in E_{1} \ E are those of the form λω where λ ∈ 𝔽_{q} \ {0}. Thus E_{1} \ E contains at most (q − 1) decomposable elements, as desired.

Next, suppose r = 2. Then in view of Theorem 6, there are linearly independent elements f, g_{1}, g_{2} ∈ V such that E = span{f ∧ g_{1}, f ∧ g_{2}}. As in the proof of Lemma 22, we can write ω = u∧v, where u, v ∈ V are such that {f, u, v} is linearly independent. Further, if dim V_{ω} ∩ V^{E} = 2, then V_{ω} ⊆ V^{E} and we may assume without loss of generality that g_{1}, g_{2} ∈ V_{ω}; hence E_{1} = span{f ∧ g_{1}, f ∧ g_{2}, g_{1} ∧ g_{2}}, and so E_{1} is close of type II, which is a contradiction. Thus dim V_{ω} ∩V^{E} < 2 and so |V_{ω} ∩ V^{E}| ≤ q. Moreover, as in the proof of Lemma 22, decomposable elements of E_{1} \E are precisely of the form f ∧x+λ(u∧v), where x ∈ V_{ω} ∩V^{E} and λ ∈ 𝔽_{q} \{0}. Thus E_{1} \ E contains at most q(q − 1) decomposable elements, as desired. □

Lemma 24. There exists a (μ+1)-dimensional subspace of ⋀^{2} V containing exactly (q^{μ} − 1)+q^{2} (q − 1) decomposable vectors. Moreover, the remaining (q^{μ} − q^{2}) (q − 1) nonzero elements in this subspace are of rank 4.

Proof. By Corollary 7, there exists a μ-dimensional decomposable subspace of ⋀^{2} V, say E. Since m > 4, we have μ > 3, and so by Theorem 6 and Corollary 7, E is close of type I. Thus there exist μ+1 linearly independent elements f, g_{1}, …, g_{μ} ∈ V such that E = span{f ∧g_{i} : i = 1, …, μ}. Now, consider ω := g_{1} ∧g_{2} and E_{1} := E +F ω. It is clear that ω ∉ E and E_{1} is not decomposable. Moreover, by Theorem 6 and Lemma 5, dim V^{E} = μ+1 = m, and thus V^{E} = V ⊇ V_{ω}. So it follows from part (ii) of Lemma 22 that E_{1} \ E contains exactly q^{2}(q − 1) decomposable elements. Since every nonzero element of E is decomposable, we see that E_{1} is a (μ+1)-dimensional subspace of ⋀^{2} V containing exactly (q^{μ} − 1) + q^{2} (q − 1) decomposable vectors. Since every element of E_{1} is of the form a(f ∧ g) + b(g_{1} ∧ g_{2}) for some a, b ∈ F and g ∈ span{g_{1}, …, g_{μ}}, it follows from Corollary 10 and Corollary 11 that the remaining (q^{μ+1} − 1) − (q^{μ} − 1) − q^{2} (q − 1) elements are of rank 4. □

Lemma 25. Every (μ+1)-dimensional subspace of ⋀^{2} V contains at most (q^{μ} − 1)+ q^{2} (q − 1) decomposable vectors.

Proof. Let E^{∗} be any (μ+1)-dimensional subspace of ⋀^{2} V. Let r be the maximum among the dimensions of all decomposable subspaces of E^{∗}. If r = 0, then E^{∗} contains no decomposable element and the assertion holds trivially. Assume that r ≥ 1. Let E_{r} be a decomposable r-dimensional subspace of E^{∗}. Extending a basis of E_{r} to E^{∗}, we obtain a subspace E′ of E^{∗} such that E_{r} ∩ E′ = {0} and E^{∗} = E_{r} + E′. Clearly,

(13)  E^{∗} = ⋃_{ω∈E′} E_{r} + F ω and E^{∗} \ E_{r} = ⋃_{0≠ω∈E′} (E_{r} + F ω) \ E_{r}.

Given any nonzero ω ∈ E′, the space E_{r} + F ω is not decomposable, thanks to the maximality of r, and so by part (i) of Lemma 22, (E_{r} + F ω) \ E_{r} contains at most q^{2}(q − 1) decomposable elements. Moreover, for any nonzero ω, ω′ ∈ E′, we have E_{r} + F ω = E_{r} + F ω′ if ω and ω′ differ by a nonzero constant, whereas (E_{r} + F ω)∩(E_{r} + F ω′) = E_{r} if ω and ω′ do not differ by a nonzero constant. Thus the second decomposition in (13) is disjoint if we let ω vary over nonzero elements of E′ that are not proportional to each other. It follows that E^{∗} \ E_{r} contains at most \frac{q^{2}(q−1)|E′\\{0\}|}{(q−1)} = q^{2} (q^{μ+1−r} − 1) decomposable elements. In case r ≤ 2, then using Lemma 23 instead of part (i) of Lemma 22, it follows that E^{∗} \E_{r} contains at most q^{r−1} (q^{μ+1−r} − 1) decomposable elements. Thus, if we let s := min{2, r − 1} and N_{r} := (q^{r} − 1) + q^{s} (q^{μ+1−r} − 1), then we see that E^{∗} contains at most N_{r} decomposable elements. To complete the proof it suffices to observe that

(q^{μ} − 1) + q^{2} (q − 1) − N_{r} = \{ \begin{matrix}(q^{r} − q^{3})(q^{μ−r} − 1) & if r ≥ 3, \\ (q^{2} − q^{r−1})(q − 1) & if 1 ≤ r ≤ 2, \end{matrix}

is always nonnegative. □

Corollary 26. We have Δ (C(2, m)) = (q^{μ} − 1) + q^{2} (q − 1) and g_{k−μ−1}(2, m) = 1 + q + q^{2} + ⋯ + q^{μ} + q^{2}.

Proof. The assertion about Δ (C(2, m)) follows from Lemma 24, Lemma 25, and Corollary 18. Further, by Lemma 25, we see that if E is any subspace of ⋀^{2} V of codimension k − μ − 1, that is, of dimension μ + 1, then

g(E) ≤ \frac{(q^{μ} − 1) + q^{2} (q − 1)}{q − 1} = 1 + q + q^{2} + ⋯ + q^{μ−1} + q^{2},

and by Lemma 24, we see that the bound is attained for some subspace of codimension k −μ−1. This proves that g_{k−μ−1}(2, m) = 1+q +q^{2} +⋯+q^{μ−1} +q^{2}. □

Theorem 27. For the Grassmann code C(2, m), we have

d_{k−μ−1} (C(2, m)) = n − (1 + q + ⋯ + q^{μ−1} + q^{2}).

and

d_{μ+1} (C(2, m)) = q^{δ} + q^{δ−1} + 2q^{δ−2} + q^{δ−3} + ⋯ + q^{δ−μ+1}

Proof. The formula for d_{k−μ−1} (C(2, m)) is an immediate consequence of Corollary 26 and Corollary 16. To prove the formula for d_{μ+1} (C(2, m)), we use Corollary 26 and Corollary 21 to observe that for C(2, m), the generalized Griesmer-Wei bound in Theorem 13 can be written as

d_{μ+1} (C(2, m)) ≥ q^{δ} + q^{δ−1} + 2q^{δ−2} + q^{δ−3} + ⋯ + q^{δ−μ+1}.

Moreover, by Lemma 24, there exists a (μ + 1)-dimensional subspace, say E_{1}, of ⋀^{2} V containing (q^{μ} − 1) + q^{2} (q − 1) decomposable elements such that the remaining (q^{μ} − q^{2}) (q − 1) nonzero elements are of rank 4. Thus, in view of Proposition 20, we see that D_{1} := τ(𝔥(E_{1})) is a (μ+1)-dimensional subspace of C(2, m) in which (q^{μ} − 1)+q^{2} (q − 1) elements are of weight q^{δ} while the remaining (q^{μ} − q^{2}) (q − 1) nonzero elements are of weight q^{δ} + q^{δ−2}. Consequently, by Lemma 12, we have

∥D_{1}∥ = \frac{1}{q^{μ+1} − q^{μ}} ∑_{c∈D} ∥c∥

= \frac{q^{δ} [(q^{μ} − 1) + q^{2} (q − 1)]}{q^{μ+1} − q^{μ}} + \frac{(q^{δ} + q^{δ−2}) [(q^{μ} − q^{2}) (q − 1)]}{q^{μ+1} − q^{μ}}
= q^{δ−μ} (q^{μ} + q^{μ−1} + ⋯ + q + 1) + q^{δ−2} − q^{δ−μ}.

This proves that d_{μ+1} (C(2, m)) = q^{δ} + q^{δ−1} + 2q^{δ−2} + q^{δ−3} + ⋯ + q^{δ−μ+1}. □

Remark 28. It appears quite plausible that the new pattern which emerges with d_{μ+1} (C(2, m)) continues for the next several values of d_{r} (C(2, m)). More precisely, we conjecture that for μ + 1 < r ≤ 2μ − 3, one has

d_{r} (C(2, m)) = (q^{δ} + q^{δ−1} + ⋯ + q^{δ−μ+1}) + (q^{δ−2} + q^{δ−3} + ⋯ + q^{δ−r+μ−1}) and

d_{k−r} (C(2, m)) = n − (1 + q + ⋯ + q^{μ−1}) − (q^{2} + q^{3} + ⋯ + q^{r−μ+1})

These conjectural formulae yield the complete weight hierarchy of C(2, 6). In general, we believe that the conjecture of Hansen, Johnsen and Ranestad [7] about d_{r} (C(ℓ, m)) − d_{r−1} (C(ℓ, m)) being a power of q is likely to be true and that determining d_{r} (C(ℓ, m)) from d_{r−1} (C(ℓ, m)) is a matter of deciphering the correct term of the Gaussian binomial coefficient (which can be written as a finite sum of powers of q) that gets added to d_{r−1} (C(ℓ, m)).

References

[1] M. Barnabei, A. Brini, and G.-C. Rota, On the exterior calculus of invariant theory, J. Alg. 96 (1985), 120–160.

[2] N. Bourbaki, Elements of Mathematics: Algebra I, Chapters 1–3, Hermann, Paris, 1974.

[3] L. J. Cummings, Decomposable symmetric tensors, Pacific J. Math. 35 (1970), 65–77.

[4] S. R. Ghorpade and G. Lachaud, Higher weights of Grassmann codes, Coding Theory, Cryptography and Related Areas (Guanajuato, Mexico, 1998), pp. 122–131, Springer-Verlag, Berlin/Heidelberg, 2000.

[5] S. R. Ghorpade and G. Lachaud, Hyperplane sections of Grassmannians and the number of MDS linear codes. Finite Fields Appl. 7 (2001), 468–506.

[6] S. R. Ghorpade and M. A. Tsfasman, Schubert varieties, linear codes and enumerative combinatorics, Finite Fields Appl. 11 (2005), 684–699.

[7] J. P. Hansen, T. Johnsen, and K. Ranestad, Schubert unions in Grassmann varieties, Finite Fields Appl. 13 (2007), 738–750. [Longer version: arXiv.math.AG/0503121 (2005).]

[8] T. Helleseth, T. Kløve, V. I. Levenshtein, and Ø. Ytrehus, Bounds on the minimum support weights, IEEE Trans. Inform. Theory 41 (1995), 432–440.

[9] M. H. Lim, A note on maximal decomposable subspaces of symmetric spaces, Bull. London Math. Soc. 7 (1975), 289–293.

[10] M. Marcus, Finite Dimensional Multilinear Algebra, Part II, Marcel Dekker, New York, 1975.

[11] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North Holland, Amsterdam, 1977.

[12] D. Yu. Nogin, Codes associated to Grassmannians, Arithmetic, Geometry and Coding Theory (Luminy, 1993), pp. 145–154, Walter de Gruyter, Berlin/New York, 1996.

[13] K. R. Parthasarathy, On the maximal dimension of a completely entangled subspace for finite level quantum systems, Proc. Indian Acad. Sci. (Math. Sci.) 114 (2004), 1–10.

[14] H. Tanao, On (n − 1)-dimensional projective spaces contained in the Grassmann variety Gr(n, 1), J. Math. Kyoto Univ. 14 (1974), 415–460.

[15] M. A. Tsfasman and S. G. Vlăduţ, Geometric approach to higher weights, IEEE Trans. Inform. Theory 41 (1995), 1564–1588.

[16] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory 37 (1991), 1412–1418.

Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India.

E-mail address: srg@math.iitb.ac.in

Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India and Shri Guru Gobind Shinghji Institute of Engineering & Technology, Vishnupuri, Nanded 431 606, India

E-mail address: arun.iitb@gmail.com

Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India.

E-mail address: hp@ee.iitb.ac.in