Contemporary Mathematics

Higher Weights of Affine Grassmann Codes and Their Duals

Mrinmoy Datta and Sudhir R. Ghorpade

Abstract. We consider the question of determining the higher weights or the generalized Hamming weights of affine Grassmann codes and their duals. Several initial as well as terminal higher weights of affine Grassmann codes of an arbitrary level are determined explicitly. In the case of duals of these codes, we give a formula for many initial as well as terminal higher weights. As a special case, we obtain an alternative simpler proof of the formula of Beelen et al for the minimum distance of the dual of an affine Grasmann code.

1. Introduction

A q-ary linear code of length n and dimension k, or in short, a [n, k]_{q}-code, is simply a k-dimensional subspace of the n-dimensional vector space 𝔽_{q}^{n} over the finite field 𝔽_{q} with q elements. A basic example is that of a (generalized) Reed-Muller code RM(ν, δ) of order ν and length n := q^{δ}, given by the image of the evaluation map

Ev : 𝔽_{q}[X_{1}, … X_{δ}]_{≤ν} → 𝔽_{q}^{n} defined by Ev(f) = (f(P_{1}), …, f(P_{n})) ,

where 𝔽_{q}[X_{1}, … X_{δ}]_{≤ν} denotes the space of polynomials in δ variables of (total) degree ≤ ν with coefficients in 𝔽_{q} and P_{1}, …, P_{n} is an ordered listing of the points of the affine space 𝔸^{δ}(𝔽_{q}) = 𝔽_{q}^{δ}. A useful variant of this is the projective Reed-Muller code PRM(ν, δ) of order ν and length n := (q^{δ+1} − 1)/(q − 1), which is obtained by evaluating homogeneous polynomials in δ + 1 variables of degree ν with coefficients in 𝔽_{q} at points of the projective space ℙ^{δ} = ℙ^{δ}(𝔽_{q}) or rather at suitably normalized representatives in 𝔽_{q}^{δ+1} of an ordered listing of the points of ℙ^{δ}.

From a geometric viewpoint, projective Reed-Muller codes PRM(ν, δ) correspond (at least when ν < q) to the Veronese variety given by the image of ℙ^{δ} in ℙ^{k−1} under the Veronese map of degree ν, where k := \binom{ν+δ}{ν}. In this set-up, RM(ν, δ) corresponds to the image of this Veronese map when restricted to an 𝔸^{δ} inside ℙ^{δ} (for instance, the set of points (x_{0} : x_{1} : ⋯: x_{δ}) of ℙ^{δ} with x_{0} = 1).

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2010 Mathematics Subject Classification. Primary 15A03, 11T06 05E99 Secondary 11T71.

The first named author is partially supported by a doctoral fellowship from the National Board for Higher Mathematics, a division of the Department of Atomic Energy, Govt. of India.

The second named author is partially supported by Indo-Russian project INT/RFBR/P-114 from the Department of Science & Technology, Govt. of India and IRCC Award grant 12IRAWD009 from IIT Bombay.

Reed-Muller codes are classical objects and in the generalized setting above, their study goes back at least to Kasami, Lin, and Peterson [10] as well as Delsarte, Goethals, and MacWilliams [4]. One may refer to [2, Prop. 4] for a summary of several of the basic properties of RM(ν, δ). Projective Reed-Muller codes appeared explicitly in the work of Lachaud [11, 12] and Sørensen [17]. Around the same time, a new class of codes called Grassmann codes were studied by Ryan [14, 15], and later by Nogin [13] and several others (see, e.g., [6, 7, 8, 5]). These correspond geometrically to the Grassmann variety G_{ℓ,m} formed by the ℓ-dimensional subspaces of 𝔽_{q}^{m} together with the Plücker embedding G_{ℓ,m} ↪ ℙ^{k−1}, where k = \binom{m}{ℓ}. In effect, the Grassmann code C(ℓ, m) is a linear code whose generator matrix has as its columns certain fixed representatives in 𝔽_{q}^{k} of the Plücker coordinates of all 𝔽_{q}-rational points of G_{ℓ,m}. Affine Grassmann codes were introduced in [1] and further studied in [2] and [5]. Given positive integers ℓ, ℓ′ with ℓ ≤ ℓ′, upon letting m = ℓ + ℓ′ and δ = ℓℓ′, the affine Grassmann code C^{𝔸}(l, m) is defined, like a Reed-Muller code, as the q-ary linear code of length n = q^{δ} given by the image of the evaluation map

(1)  Ev : ℱ(ℓ, m) → 𝔽_{q}^{n} defined by Ev(f) = (f(P_{1}), …, f(P_{n})) ,

where ℱ(ℓ, m) is the space of linear polynomials in the minors of a generic ℓ × ℓ′ matrix X and P_{1}, …, P_{n} is an ordered listing of the δ-dimensional affine space of all ℓ × ℓ′ matrices with entries in 𝔽_{q}. The relationship between affine Grassmann codes C^{𝔸}(l, m) and Grassmann codes C(l, m) is akin to that between Reed-Muller codes RM(ν, δ) and projective Reed-Muller codes PRM(ν, δ).

The notion of higher weight, also known as generalized Hamming weight, of a linear code is a natural and useful generalization of the basic notion of minimum distance. If C is a [n, k]_{q}-code, then for r = 0, 1, …, k, the r^{th} higher weight of C is defined by

d_{r} = d_{r}(C) = min{w_{H}(D) : D is a subspace of C with dim D = r},

where w_{H}(D) denotes the support weight of D [see Section 2 below for a definition]. Clearly, d_{1}(C) is the minimum distance d(C) of C. It is well-known and easy to see that 0 = d_{0} < d_{1} < ⋯ < d_{k} and moreover d_{k} = n provided C is nondegenerate. It is, in general, an interesting and difficult question to determine the weight hierarchy, i.e., all the higher weights, of a given class of codes. For example, in a significant piece of work, Heijnen and Pelikaan [9] completely determined the higher weights of Reed-Muller codes RM(ν, δ). In the case of projective Reed-Muller codes, the minimum distance was determined by Lachaud [12] and independently by Sørrensen [17]. In fact, Lachaud derives it as a consequence of an affirmative answer given by Serre [16] to a question of Tsfasman concerning the maximum number of 𝔽_{q}-rational points on a projective hypersurface of a given degree. The second higher weight was determined by Boguslavsky [3], while the determination of d_{r} (PRM(ν, δ)) is still open for r > 2. In the case of Grassmann codes, the r^{th} higher weight is known for the first few and the last few values of r, thanks to Nogin [13] (see also [6]) and Hansen, Johnsen and Ranestad [8] (see also [7]). More precisely, for r = 0, 1, … μ, where μ := 1 + max{ℓ, m − ℓ}, 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}), where n denotes the length of C(ℓ, m) or in other words, the number of 𝔽_{q}-rational points of G_{ℓ,m}, and it is given by the Gaussian binomial coefficient \left[{m \atop ℓ}\right]_{q}. In case ℓ = 2, we know a little more (cf. [7]), but the general case is still open.

We consider in this paper the problem of determining the higher weights of affine Grassmann codes and their duals. Our main result is an explicit formula for d_{r} (C^{𝔸}(l, m)) for the first few and the last few values of r, or more precisely, for 0 ≤ r ≤ μ′ and for k − μ ≤ r ≤ k, where

(2)  μ′ = 1 + max{ℓ, ℓ′ − ℓ} and μ := 1 + max{ℓ, ℓ′} = ℓ′ + 1.

In the case of the result for the first μ′ higher weights, we have to make an additional mild assumption that ℓ < ℓ′. The result for the last μ higher weights can be deduced from the corresponding results for Grassmann codes using a geometric approach. However, we give here self-contained proofs in the spirit of [1, 2] and this has the advantage that analogous results are also obtained for affine Grassmann codes of arbitrary level introduced in [2]. As for the duals, we can in fact go much farther, and determine many more higher weights of the duals of affine Grassmann codes except that the result we give here is best described recursively. As a corollary, we obtain a new and simpler proof of [2, Theorem 17], which states that if ℓ′ > 1, then the minimum distance of C^{𝔸}(l, m; h)^{⊥} is 3 or 4 according as q > 2 or q = 2. The geometric approach and an alternative proof of the result about the last μ higher weights is also outlined in an appendix for the convenience of the reader.

2. Initial Higher Weights

For any q-ary linear code C of length n, and any D ⊆ C, we let

Supp(D) := {i ∈ {1, …, n} : c_{i} ≠ 0 for some c ∈ D} and w_{H}(D) = |Supp(D)|

denote, respectively, the support and the support weight of D. For a codeword c = (c_{1}, …, c_{n}) ∈ C, we write w_{H}(c) = w_{H}({c}) and note that this is simply the Hamming weight of c.

Fix, throughout this paper, positive integers h, ℓ, ℓ′ with h ≤ ℓ ≤ ℓ′ and an l × l′ matrix X = (X_{ij}) whose entries are algebraically independent indeterminates over 𝔽_{q}. Let 𝔽_{q}[X] denote the ring of polynomials in the ℓℓ′ variables X_{ij}’s with coefficients in 𝔽_{q}. As in [2], we let Δ(ℓ, m; h) denote the set of all minors of X of degree ≤ h. Note that Δ(ℓ, m; h) is a subset of 𝔽_{q}[X] that contains the constant polynomial 1, which corresponds to the 0 × 0 minor of X. Further let

ℱ(ℓ, m; h) := the 𝔽_{q}-linear subspace of 𝔽_{q}[X] generated by Δ(ℓ, m; h).

Note that the space ℱ(ℓ, m) defined in the Introduction contains ℱ(ℓ, m; h) and the equality holds when h = ℓ. The affine Grassmann code of level h, denoted C^{𝔸}(ℓ, m; h), is defined to be the image of ℱ(ℓ, m; h) under the evaluation map Ev given by (1). Evidently, C^{𝔸}(ℓ, m; ℓ) = C^{𝔸}(ℓ, m) and C^{𝔸}(ℓ, m; h) is a subcode of RM(h, δ). Now here is a slightly refined version of a basic result proved in [2].

Proposition 2.1. The minimum distance d(ℓ, m; h) of C^{𝔸}(ℓ, m; h) is

(3)  d(ℓ, m; h) = q^{δ} ∏_{i=1}^{h} (1 − \frac{1}{q^{i}} ) = q^{δ−h^{2}} |GL_{h}(𝔽_{q})| .

Moreover, if M is any h × h minor of X, then w_{H} (Ev (M)) = d(ℓ, m; h).

Proof. The first equality (3) is proved in [2, Theorem 5], while the second is easily deduced. Also it is shown that in [2, Theorem 5] if L_{h} = det (X_{ij})_{1≤i,j≤h} is the h^{th} leading principal minor of X, then w_{H} (Ev (L_{h})) = d(ℓ, m; h). Now if M is any h × h minor of X, then there are positive integers p_{1}, …, p_{h}, q_{1}, …, q_{h} with p_{1} < ⋯ < p_{h} ≤ ℓ and q_{1} < ⋯ < q_{h} ≤ ℓ′ such that M = det(X_{p_{i}q_{j}})_{1≤i,j≤h}. Let σ ∈ S_{ℓ} be a permutation such that σ(i) = p_{i} for 1 ≤ i ≤ ℓ and P ∈ GL_{ℓ}(𝔽_{q}) be the permutation matrix corresponding to σ so that for 1 ≤ i, j ≤ ℓ, the (i, j)^{th} entry of P is 1 is j = σ(i) and 0 otherwise. Likewise, let τ ∈ S_{ℓ′} be such that τ(i) = q_{i} for 1 ≤ i ≤ ℓ′ and Q ∈ GL_{ℓ′}(𝔽_{q}) be the permutation matrix corresponding to τ. Then it is easily seen that M is the the h^{th} leading principal minor of P XQ^{−1}. Moreover, we know from [2, §IV] that X ↦ P XQ^{−1} induces a permutation automorphism of C^{𝔸}(ℓ, m; h). It follows that w_{H} (Ev (M)) = w_{H} (Ev (L_{h})) = d(ℓ, m; h). □

The following general observation about the support weights of linear codes will be useful in the sequel.

Lemma 2.2. Let C be an [n, k]_{q}-code and for i = 1, …, n, let π_{i} : C → 𝔽_{q} denote the i^{th} projection map defined by π_{i}(c_{1}, …, c_{n}) = c_{i}. Also let D be a subcode of C and {y_{1}, …, y_{r}} be a generating set of D. Then

Supp(D) = ⋃_{j=1}^{n} A_{j} where for 1 ≤ j ≤ n, A_{j} := {i ∈ {1, …, n} : π_{i}(y_{j}) ≠ 0} .

Proof. Clearly, ∪_{j=1}^{n}A_{j} ⊆ Supp(D). On the other hand, suppose i ∈ {1, …, n} is such that i ∉ ∪_{j=1}^{n}A_{j}. Then π_{i}(y_{j}) = 0 for all j = 1, …, r. Now for any x ∈ D, we can write x = ∑_{j=1}^{r} c_{j}y_{j} for some c_{1}, …, c_{r} ∈ 𝔽_{q}; hence π_{i}(x) = ∑_{j=1}^{r} c_{j}π_{i}(y_{j}) = 0. Thus i ∉ Supp(D). This shows that Supp(D) ⊆ ∪_{j=1}^{n}A_{j}. □

The next two lemmas extend Proposition 2.1 and show that for a judicious choice of a family {M_{1}, …, M_{r}} of minors of X, the support weight of the product of any nonempty subfamily is given by a formula analogous to (3).

Lemma 2.3. Let r be a positive integer such that r ≤ ℓ′ − h + 1 and let Y be any h × (h + r − 1) submatrix of X. Also for j = 1, …, r, let M_{j} denote the h × h minor of Y (and hence of X) corresponding to the first h − 1 columns of Y together with the (h + j − 1)^{th} column of Y, and let A_{j} = {P ∈ 𝔸^{δ} : M_{j}(P) ≠ 0}. Then for any positive integer s with s ≤ r and any j_{1}, …, j_{s} ∈ {1, …, r} with j_{1} < ⋯ < j_{s},

(4)  |A_{j_{1}} ∩ ⋯ ∩ A_{j_{s}}| = d(ℓ, m; h) (1 − \frac{1}{q})^{s−1} .

Proof. Given any ℓ × ℓ′ matrix P ∈ 𝔸^{δ} with entries in 𝔽_{q}, let Q denote the h × (h + r − 1) submatrix of P formed in exactly the same way as Y, and let Q_{1}, … Q_{h+r−1} denote the column vectors of Q. For any positive integer s with s ≤ r and any j_{1}, …, j_{s} ∈ {1, …, r} with j_{1} < ⋯ < j_{s}, the condition P ∈ A_{j_{1}} ∩⋯∩A_{j_{s}} is equivalent to the condition that the column vectors Q_{1}, …, Q_{h−1}, Q_{h+j−1} in 𝔽_{q}^{h} are linearly independent for each j ∈ {j_{1}, …, j_{s}}. This will hold when the submatrix of Q formed by its first h−1 columns is chosen in exactly (q^{h}−1)(q^{h}−q) ⋯ (q^{h}−q^{h−2}) ways, while each of Q_{h+j_{1}−1}, …, Q_{h+j_{s}−1} are chosen in (q^{h} − q^{h−1}) ways. The remaining r − s columns of Q may be chosen arbitrarily in q^{h(r−s)} ways. Since P has ℓℓ′ − h(h + r − 1), i.e., δ − h^{2} − h(r − 1), entries outside Q, it follows that

|A_{j_{1}} ∩ ⋯ ∩ A_{j_{s}}| = (q^{h} − 1)(q^{h} − q) ⋯ (q^{h} − q^{h−2})(q^{h} − q^{h−1})^{s}q^{δ−h^{2}−h(s−1)}
= d(ℓ, m; h) (1 − \frac{1}{q})^{s−1} ,

where the last equality follows from (3). □

Lemma 2.4. Assume that h < ℓ′. Let r be a positive integer such that r ≤ h+1 and let Y be any h × (h + 1) submatrix of X. Also for j = 1, …, r, let M_{j} denote the determinant of the h×h submatrix of Y formed by all except the (h−r+j +1)^{th} column of Y, and let A_{j} = {P ∈ 𝔸^{δ} : M_{j}(P) ≠ 0}. Then (4) holds for any positive integer s with s ≤ r and any j_{1}, …, j_{s} ∈ {1, …, r} with j_{1} < ⋯ < j_{s}.

Proof. Given P ∈ 𝔸^{δ}, let Q be the h × (h + 1) submatrix of P corresponding to Y, and let Q_{1}, …, Q_{h+1} denote its column vectors. Fix any positive integer s with s ≤ r and j_{1}, …, j_{s} ∈ {1, …, r} with j_{1} < ⋯ < j_{s}. Now M_{j_{1}}(P) ≠ 0 implies that Q has rank h and in particular, Q_{h−r+j_{1}+1} is a 𝔽_{q}-linear combination of the remaining h column vectors of Q. Moreover, for 2 ≤ t ≤ s, if M_{j_{t}}(P) ≠ 0, then the coefficients of Q_{h−r+j_{t}+1} in this 𝔽_{q}-linear combination must be nonzero. Conversely, if all except the (h − r + j_{1} + 1)^{th} column of Q are linearly independent (and these columns can thus be chosen in |GL_{h}(𝔽_{q})| ways), while Q_{h−r+j_{1}+1} is a 𝔽_{q}-linear combination of the remaining h column vectors of Q with a nonzero coefficient for the s − 1 columns Q_{h−r+j_{2}+1}, …, Q_{h−r+j_{s}+1}, then M_{j_{t}}(P) ≠ 0 for each t = 1, …, s. The h coefficients in this 𝔽_{q}-linear combination can thus be chosen in q^{h−s+1}(q − 1)^{s−1} ways. Since P has δ − h(h + 1) entries outside of Q, it follows that

|A_{j_{1}} ∩ ⋯ ∩ A_{j_{s}}| = |GL_{h}(𝔽_{q})| q^{h−s+1}(q − 1)^{s−1}q^{δ−h(h+1)} = d(ℓ, m; h) (1 − \frac{1}{q})^{s−1} ,

where the last equality follows once again from (3). □

Theorem 2.5. Let r be a positive integer such that r ≤ max{ℓ′ − h, h} + 1. Assume that h < ℓ′ in case max{ℓ′ − h, h} = h, i.e., ℓ′ ≤ 2h. Then the r^{th} higher weight d_{r}(ℓ, m; h) of C^{𝔸}(ℓ, m; h) is

(5) d_{r}(ℓ, m; h) = q^{δ−r+1} \frac{(q^{r} − 1)}{(q − 1)} ∏_{i=1}^{h} (1 − \frac{1}{q^{i}} ) = q^{δ−h^{2}−r+1}(q^{r} − 1)\frac{|GL_{h}(𝔽_{q})|}{q − 1}.

Moreover, the r^{th} higher weight of C^{𝔸}(ℓ, m; h) attains the Griesmer-Wei bound.

Proof. The hypotheses on r and h together with Lemmas 2.3 and 2.4 ensure that there exist minors M_{1}, …, M_{r} ∈ Δ(ℓ, m; h) with supports A_{1}, …, A_{r} respectively, such that (4) holds for any positive integer s with s ≤ r and any j_{1}, …, j_{s} ∈ {1, …, r} with j_{1} < ⋯ < j_{s}. Consequently,

| ⋃_{j=1}^{r} A_{j}| = ∑_{s=1}^{r}(−1)^{s−1} ∑_{1≤j_{1}<···<j_{s}≤r} |A_{j_{1}} ∩ ⋯ ∩ A_{j_{s}}|

= ∑_{s=1}^{r}(−1)^{s−1}\binom{r}{s}d(ℓ, m; h) (1 − \frac{1}{q})^{s−1}

= \frac{d(ℓ, m; h)}{1 − q^{−1}} (1 − ∑_{s=0}^{r}(−1)^{s}\binom{r}{s} (1 − q^{−1})^{s})

= \frac{d(ℓ, m; h)}{1 − q^{−1}} (1 − [1 − (1 − q^{−1})]^{r}) .

Hence, in view of Lemma 2.2 and Proposition 2.1, we see that if D_{r} is the subspace of ℱ(ℓ, m; h) spanned by M_{1}, …, M_{r}, then

(6)  w_{H}(D_{r}) = d(ℓ, m; h)\frac{(q^{r} − 1)}{q^{r−1}(q − 1)} = q^{δ−r+1} \frac{(q^{r} − 1)}{(q − 1)} ∏_{i=1}^{h} (1 − \frac{1}{q^{i}} ) .

Moreover, by [1, Lemma 3], we see that D_{r} is of dimension r. Thus,

(7) d_{r}(ℓ, m; h) := d_{r} (C^{𝔸}(ℓ, m; h)) ≤ d(ℓ, m; h)\frac{(q^{r} − 1)}{q^{r−1}(q − 1)} = ∑_{i=0}^{r−1} \frac{d(ℓ, m; h)}{q^{i}}.

On the other hand, the Griesmer-Wei bound (cf. [20]) and Proposition 2.1 yields

(8)  d_{r} (C^{𝔸}(ℓ, m; h)) ≥ ∑_{i=0}^{r−1} ⌈\frac{d(ℓ, m; h)}{q^{i}}⌉ ≥ ∑_{i=0}^{r−1} \frac{d(ℓ, m; h)}{q^{i}}.

Using (6), (7) and (8), we obtain the desired result. □

Remark 2.6. The only case in which the above theorem does not give any higher weights of C^{𝔸}(ℓ, m; h) beyond the minimum distance is when h = ℓ = ℓ′. We believe that in this case even the second higher weight does not meet the Griesmer-Wei bound. In fact, it seems plausible that for C^{𝔸}(ℓ, 2ℓ) = C^{𝔸}(ℓ, 2ℓ; ℓ),

d_{2} (C^{𝔸}(ℓ, 2ℓ)) = q^{ℓ} (1 + \frac{1}{q} − \frac{1}{q^{ℓ}} ) ∏_{i=0}^{ℓ−1} (q^{ℓ} − q^{i}) = d(ℓ, 2ℓ; ℓ) (1 + \frac{q^{ℓ−1}}{q^{ℓ} − 1}) .

Note that the expression on the right is strictly greater than d(ℓ, 2ℓ; ℓ) (1 + q^{−1}). We remark also that the expression on the right is an upper bound for d_{2} (C^{𝔸}(ℓ, 2ℓ)). This can be seen, for example, by considering the 2-dimensional subspace D of ℱ(ℓ, 2ℓ) spanned by the ℓ^{th} leading principal minor M_{1} = det(X) and the (ℓ − 1)^{th} leading principal minor M_{2} of X, and using Lemma 2.2 to show that

|Supp(D)| = |A_{1}| + |A_{2}| − |A_{1} ∩ A_{2}|
= |GL_{ℓ}(𝔽_{q})| + |GL_{ℓ−1}(𝔽_{q})|q^{2ℓ−1} − GL_{ℓ−1}(𝔽_{q})|q^{ℓ−1}(q^{ℓ} − q^{ℓ−1})
= d(ℓ, 2ℓ; ℓ) (1 + \frac{q^{ℓ−1}}{q^{ℓ} − 1}) ,

where, as before, A_{j} = {P ∈ 𝔸^{δ} : M_{j}(P) ≠ 0} for j = 1, 2.

3. Terminal Higher Weights

As in the case of Grassmann codes, determining some of the terminal higher weights is simpler than determining some of the initial higher weights. In fact, we go a little farther than what we could do with the initial higher weights. Thus, as opposed to finding explicitly the first μ′ higher weights of say C^{𝔸}(ℓ, m), where μ′ = max{ℓ, ℓ′ − ℓ} + 1, we are able find explicitly the last ℓ′ + 1 higher weights of not just C^{𝔸}(ℓ, m), but any C^{𝔸}(ℓ, m; h).

The first step is a simple observation that holds, in fact, for any functional code defined by means of an evaluation map on a space of (polynomial) functions. However, we will just restrict to the case of affine Grassmann codes of a given level.

Lemma 3.1. Let D be a subcode of C^{𝔸}(ℓ, m; h) and t a positive integer such that there exist t linearly independent polynomials g_{1}, …, g_{t} ∈ 𝔽_{q}[X] with the property that deg g_{i} ≤ 1 and Ev(g_{i}) ∈ D for all i = 1, …, t. Then w_{H}(D) ≥ q^{δ} − q^{δ−t}.

Proof. As noted in [1, §II], the evaluation map Ev given by (1) is injective. Hence D is in bijection with W := Ev^{−1}(D) = {f ∈ ℱ(ℓ, m; h) : Ev(f) ∈ D}. Moreover, if we let Z(W) := {P ∈ 𝔸^{δ} : f(P) = 0 for all P ∈ W } denote the corresponding affine variety. then it is clear that

w_{H}(D) = |Supp(D)| = |𝔸^{δ} \ Z(W)| = q^{δ} − |Z(W)|.

Now g_{1}, …, g_{t} ∈ W and the number of common zeros in 𝔸^{δ} of g_{1}, …, g_{t} corresponds to the number, say N, of solutions of a system of t linearly independent nonhomogeneous linear equations in δ variables with coefficients in 𝔽_{q}. Hence N = 0 or N = q^{δ−t} according as the system is inconsistent or consistent. Consequently, |Z(W)| ≤ N ≤ q^{δ−t} and so w_{H}(D) ≥ q^{δ} − q^{δ−t}. □

Let k_{h} denote the dimension of C^{𝔸}(ℓ, m; h). We know from [2, Prop. 2] that

(9)  k_{h} = ∑_{i=0}^{h} \binom{ℓ}{i}\binom{ℓ′}{i} and k_{ℓ} = \binom{m}{ℓ}.

In the remainder of this paper, we fix an ordering N_{1}, …, N_{k_{h}} on Δ(ℓ, m; h) such that the 0 × 0 minor appears at the end, preceded by the 1 × 1 minors arranged lexicographically. More precisely, we require N_{k_{h}} = 1 and N_{k_{h}−(i−1)ℓ′−j} = X_{ij} for i = 1, …, ℓ and j = 1, …, ℓ′. For instance, N_{k_{h}−1} = X_{11} and N_{k_{h}−δ} = X_{ℓℓ′}. Note that N_{1}, …, N_{k_{h}} gives an ordered 𝔽_{q}-basis of ℱ(ℓ, m; h), thanks to [2, Prop. 2].

Lemma 3.2. d_{k_{h}−r}(C^{𝔸}(ℓ, m; h)) ≥ q^{δ} − q^{r−1} for all r = 1, …, δ + 1.

Proof. Fix a positive integer r with r ≤ δ+1 and let s := k_{h}−r. Let D be any s-dimensional subcode of C^{𝔸}(ℓ, m; h). Then W := Ev^{−1}(D) is a linear subspace of ℱ(ℓ, m; h) with dim W = s. Suppose f_{1}, …, f_{s} give an ordered 𝔽_{q}-basis of W. Then there is a unique s × k_{h} matrix Λ = (λ_{ij}) of rank s with entries in 𝔽_{q} such that \underline{f} = Λ\underline{N}, where \underline{f} := [f_{1}, …, f_{s}]^{T} and \underline{N} := [N_{1}, …, N_{k_{h}}]^{T} denote the column vectors (over 𝔽_{q}[X]) corresponding to the abovementioned ordered 𝔽_{q}-bases of W and ℱ(ℓ, m; h) respectively. Now let Λ^{∗} = (λ_{ij}^{∗}) be the reduced row-echelon form of Λ and let p_{1}, …, p_{s} with 1 ≤ p_{1} < ⋯ < p_{s} ≤ k_{h} be the column indices in which the pivots occur, so that λ_{ip_{i}}^{∗} = 1 and λ_{ij}^{∗} = 0 if j < p_{i} and also λ_{kp_{i}}^{∗} = 0 if k ≠ i. It is clear that p_{s} ≥ s ≥ k_{h} − δ − 1. Hence if we let t := δ + 1 − r, then in each of the last t rows of Λ^{∗}, the first k_{h} − δ − 1 entries are zero. Consequently, the last t rows of the product Λ^{∗}\underline{N} are 𝔽_{q}-linear combinations of the last δ + 1 minors among N_{1}, …, N_{k_{h}}, and these give rise to linearly independent polynomials g_{1}, …, g_{t} ∈ 𝔽_{q}[X] of degree ≤ 1. Moreover, Λ^{∗} is obtained from Λ by a finite sequence of elementary row operations, and hence there is a nonsingular s×s matrix P with entries in 𝔽_{q} such that Λ^{∗} = P Λ. In particular, g_{1}, …, g_{t} correspond to the last t rows of the product P \underline{f}, and hence they are in W. Now Lemma 3.1 implies that w_{H}(D) ≥ q^{δ} − q^{δ−t} = q^{δ} − q^{r−1}. Since D was an arbitrary s-dimensional subscode of C^{𝔸}(ℓ, m; h), we obtain the desired result. □

It can be shown that the lower bound in Lemma 3.2 is attained when r ≤ ℓ′ +1, and this leads to the following result about the terminal higher weights.

Theorem 3.3. d_{k_{h}}(C^{𝔸}(ℓ, m; h)) = q^{δ} and d_{k_{h}−r}(C^{𝔸}(ℓ, m; h)) = q^{δ} − q^{r−1} for any positive integer r with r ≤ ℓ′ + 1.

Proof. Since 1 ∈ ℱ(ℓ, m; h), we see that d_{k_{h}}(C^{𝔸}(ℓ, m; h)) = q^{δ} or in other words, the code C^{𝔸}(ℓ, m; h) is nondegenerate. Fix a positive integer r ≤ δ+1 and let s := k_{h} −r. Consider the linear subspace W of ℱ(ℓ, m; h) spanned by {N_{1}, …, N_{s}}, i.e., by all the minors in Δ(ℓ, m; h), except 1, X_{11}, …, X_{1r−1}. Observe that the corresponding affine variety Z(W) consists precisely of the ℓ × ℓ′ matrices P = (P_{ij}) ∈ 𝔸^{δ} satisfying P_{ij} = 0 for all i = 1, …, ℓ and j = 1, …, ℓ′, except when (i, j) = (1, 1), …, (1, r − 1). Consequently, |Z(W)| = q^{r−1} and thus if we let D := Ev(W) be the s-dimensional subcode of C^{𝔸}(ℓ, m; h) corresponding to W, then w_{H}(D) = q^{δ} − q^{r−1}. This together with Lemma 3.2 yields the desired result. □

4. Higher Weights of Duals of Affine Grassmann Codes

For determining the higher weights of duals of affine Grassmann code, we use a simple, but powerful, method based on the following key result of Wei [20].

Proposition 4.1. Let C be a [n, k]_{q}-code. Then

(i) (Monotonicity) If k > 0, then 1 ≤ d_{1}(C) < d_{2}(C) < ⋯ < d_{k}(C) ≤ n. (ii) (Duality) The higher weights of C and its dual are related by

{d_{s}(C^{⊥}) : s = 1, …, n − k} = {1, …, n} \ {n + 1 − d_{r}(C) : r = 1, …, k} .

It is convenient and computationally effective to rephrase the above result for nondegenerate linear codes of positive dimension as follows. We will in fact give two equivalent formulations, the first of which is better suited for the terminal weights while the second is better suited for the initial weights.

Corollary 4.2. Let C be a nondegenerate [n, k]_{q}-code with k > 0. Let d_{0} := 0 and for 1 ≤ r ≤ k, let d_{r} denote the r^{th} higher weight of C. Also let

(10)  e_{j} := d_{j} − j and f_{j} := n − j − d_{k−j} for 0 ≤ j ≤ k.

Then the e-sequence and the f-sequence partition {0, 1, …, n − k}; more precisely,

(11) 0 = e_{0} ≤ e_{1} ≤ ⋯ ≤ e_{k} = n − k and 0 = f_{0} ≤ f_{1} ≤ ⋯ ≤ f_{k} = n − k.

Moreover, for 0 ≤ s < n − k, the last s^{th} higher weight of the dual of C is given by

(12) d_{n−k−s} (C^{⊥}) = n−s−j if j is the unique integer < k with e_{j} ≤ s < e_{j+1}.

Equivalently, for 0 < s ≤ n − k, the s^{th} higher weight of the dual of C is given by

(13) d_{s} (C^{⊥}) = s + j + 1 if j is the unique integer < k with f_{j} < s ≤ f_{j+1}.

Proof. Note that d_{k} = n, since C is nondegenerate. With this in view, part (i) of Proposition 4.1 implies (11). Next, parts (i) and (ii) of Proposition 4.1 together with (11) readily imply (12) and (13) . □

Remark 4.3. The above Corollary shows that the higher weights of the dual of a nondegenerate linear code C of positive dimension k take consecutive values in strings of length d_{r+1} −d_{r} −1 for 0 ≤ r < k, where d_{r} denotes the r^{th} higher weight of C. Evidently, this phenomenon is prevalent if there are large gaps among the consecutive higher weights of C. In fact, a duality of sorts seems to prevail here: more the number of consecutive strings among the higher weights of a code, the less there are among the higher weights of its dual, and vice-versa. In this connection, it may useful to note the following result of Tsfasman and Vlăduţ [19, Cor. 3.5], which states that for 1 ≤ r ≤ s ≤ k,

d_{s} ≥ d_{r} + ∑_{i=1}^{s−r} ⌈\frac{(q − 1)d_{r}}{(q^{r} − 1)q} ⌉ and in particular, d_{r+1} − d_{r} ≥ ⌈\frac{(q − 1)d_{r}}{(q^{r} − 1)q} ⌉ .

Another special case of Corollary 4.2 worth noting is that C^{⊥} is nondegenerate if and only if d_{1}(C) > 1.

We now turn to duals of affine Grassmann codes. Recall that we have fixed positive integers h, ℓ, ℓ′ with h ≤ ℓ ≤ ℓ′ and that the length of the corresponding affine Grassmann code C^{𝔸}(ℓ, m; h) of level h is given by n := q^{δ} and the dimension k_{h} is given by (9). To avoid trivialities we will further assume that ℓ′ > 1. Indeed, it is easy to describe what the affine Grassmann code C^{𝔸}(ℓ, m; h) and its dual is in the trivial case ℓ′ = 1 (or another trivial case h = 0 that we have ignored from the beginning) and in fact, this has been done in the paragraph before Theorem 17 in [2]. Using the results of Sections 2 and 3, we obtain a more concrete version of Corollary 4.2, which determines several initial and terminal higher weights of the C^{𝔸}(ℓ, m; h)^{⊥}. As a very special case, we also obtain an alternative and simpler proof of [2, Theorem 17].

Theorem 4.4. Assume that ℓ′ > 1.

(i) The minimum distance of C^{𝔸}(ℓ, m; h)^{⊥} is given by

d_{1} (C^{𝔸}(ℓ, m; h)^{⊥}) = \begin{cases}3 & if q > 2, \\ 4 & if q = 2. \end{cases}

More generally, upon letting Q_{j} = q^{j} − j for j ≥ 0, the s^{th} higher weight of C^{𝔸}(ℓ, m; h)^{⊥} for 1 ≤ s < q^{ℓ′} − ℓ′ is given by

d_{s} (C^{𝔸}(ℓ, m; h)^{⊥}) = s + j + 1

where j is the unique positive integer ≤ ℓ′ such that Q_{j−1} ≤ s < Q_{j}. (ii) With d(ℓ, m; h) as in (3), we have d(ℓ, m; h) ≥ 2 and

d_{n−k_{h}−s} (C^{𝔸}(ℓ, m; h)^{⊥}) = q^{δ} − s for 0 ≤ s ≤ d(ℓ, m; h) − 2.

In particular, C^{𝔸}(ℓ, m; h)^{⊥} is nondegenerate. Further if we assume that h < ℓ′ or ℓ′ > 2h, and we let G_{0} = H_{0} = 0 and G_{j} := ∑_{i=0}^{j−1} q^{−i} and H_{j} := d(ℓ, m; h)G_{j} − j for any positive integer j, then the last s^{th} higher weight of C^{𝔸}(ℓ, m; h)^{⊥} for H_{1} ≤ s ≤ max{H_{ℓ}, H_{ℓ′−ℓ}} is given by

d_{n−k_{h}−s} (C^{𝔸}(ℓ, m; h)^{⊥}) = n − s − j,

where j is the unique positive integer ≤ max{ℓ, ℓ′−ℓ} with H_{j−1} ≤ s < H_{j}.

Proof. Let C = C^{𝔸}(ℓ, m; h), n = q^{δ} and k = k_{h}, and let d_{j}, e_{j}, f_{j} be as in Corollary 4.2. By Theorem 3.3, the code C is nondegenerate and d_{k−j} = n − q^{j−1} for 1 ≤ j ≤ ℓ′ + 1. Consequently, the condition f_{j} < s ≤ f_{j+1} translates to Q_{j−1} ≤ s < Q_{j}, provided 1 ≤ j ≤ ℓ′. Thus (13) implies the desired formula in (i) for d_{s} (C^{𝔸}(ℓ, m; h)^{⊥}). In the particular case when s = 1, we have Q_{1} = 1 < 2 = Q_{2} or 0 = Q_{0} < 1 < Q_{1} according as q = 2 or q > 2, and this yields the formula for the minimum distance of C^{𝔸}(ℓ, m; h)^{⊥}.

Next, d_{1} = d(ℓ, m; h) = q^{δ−ℓ^{2}}|GLℓ(𝔽_{q})| and since ℓ′ > 1 we see that q^{δ−ℓ^{2}} ≥ 2 when ℓ < ℓ′, whereas |GL_{ℓ}(𝔽_{q})| ≥ (q^{2} − 1)(q^{2} − q) ≥ 6 when ℓ = ℓ′. Thus in any case d(ℓ, m; h) ≥ 2 and so (12) implies the first assertion in (ii). Further, Theorem 2.5 shows that e_{j} = H_{j} for 1 ≤ j ≤ 1 + max{ℓ, ℓ′ − ℓ}. Thus (12) implies the remaining assertion in (ii) as well. □

Remark 4.5. With the first of Theorem 4.4 above, we can also describe many of the initial higher weights d_{s}^{⊥} := d_{s} (C^{𝔸}(ℓ, m; h)^{⊥}) by the recursive formula

d_{s}^{⊥} = \begin{cases}d_{s−1}^{⊥} + 2 & if d_{s−1} is a power of q, \\ d_{s−1}^{⊥} + 1 & otherwise, \end{cases}

provided 2 ≤ s ≤ q^{ℓ′} −ℓ′. Likewise, the last higher weight of C^{𝔸}(ℓ, m; h)^{⊥} is n = q^{δ}, and many terminal higher weights of C^{𝔸}(ℓ, m; h)^{⊥} are given by the recursive formula

d_{n−k−s}^{⊥} = \begin{cases}d_{n−k−s+1}^{⊥} − 2 & if d_{n−k−s+1} = n + 1 − d(ℓ, m; h)G_{j} for some j, \\ d_{n−k−s+1}^{⊥} − 1 & otherwise, \end{cases}

provided 1 ≤ s ≤ max{d(ℓ, m; h)G_{ℓ} −ℓ, d(ℓ, m; h)G_{ℓ′−ℓ} −(ℓ′ −ℓ)} and it is assumed that h < ℓ′ or ℓ′ > 2h.

Using the direct formula in Theorem 4.4 or the recursive formula in Remark 4.5, we can easily write down several of the initial and terminal higher weights of C^{𝔸}(ℓ, m; h)^{⊥}. Table 1 illustrates the first few higher weights d_{s}^{⊥} of the dual of C^{𝔸}(ℓ, m; h), where h, ℓ, ℓ′ are sufficiently large, say ℓ′ > ℓ ≥ h ≥ 27.

Appendix A. A Geometric Approach to Higher Weights

Let n, k be positive integers with k ≤ n. A nondegenerate [n, k]_{q}-projective system is simply a (multi)set X of n points in the projective space ℙ^{k−1} over the finite field 𝔽_{q}. If we write ℙ^{k−1} = ℙ(V), where V is a k-dimensional vector space over 𝔽_{q} and fix some lifts, say v_{1}, …, v_{n}, of these n points to V, then the associated nondegenerate linear code C_{X} is the image of the evaluation map

Ev : V^{∗} → 𝔽_{q}^{n} defined by Ev(ϕ) = (ϕ(v_{1}), …, ϕ(v_{n})) ,

where V^{∗} denotes the dual of V, i.e., the space of all linear maps from V to 𝔽_{q}. It is shown in [18, 19] that the association X ⇝ C_{X} is a one-to-one correspondence, modulo natural notions of equivalence, from the class of nondegenerate [n, k]_{q}- projective systems onto the class of nondegenerate [n, k]_{q}-codes. For 1 ≤ r ≤ k, the r^{th} higher weight of C_{X} corresponds to maximal sections of X by (projective) linear subspaces of ℙ^{k−1} of codimension r; more precisely,

d_{r}(C_{X}) = n − max{|X ∩ Π| : Π linear subspace of ℙ^{k−1} with codim Π = r}.

For more on this, we refer to [18, 19].

Table 1. Dual Higher Weights of Affine Grassmann Codes

\begin{matrix}q & 2 & 3 & 4 & 5 & 7 & 8 & 9 & 11 & 13 & 16 & 17 \\ d_{1}^{⊥} & 4 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 \\ d_{2}^{⊥} & 6 & 5 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 \\ d_{3}^{⊥} & 7 & 6 & 6 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 \\ d_{4}^{⊥} & 8 & 7 & 7 & 7 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\ d_{5}^{⊥} & 10 & 8 & 8 & 8 & 7 & 7 & 7 & 7 & 7 & 7 & 7 \\ d_{6}^{⊥} & 11 & 9 & 9 & 9 & 9 & 8 & 8 & 8 & 8 & 8 & 8 \\ d_{7}^{⊥} & 12 & 11 & 10 & 10 & 10 & 10 & 9 & 9 & 9 & 9 & 9 \\ d_{8}^{⊥} & 13 & 12 & 11 & 11 & 11 & 11 & 11 & 10 & 10 & 10 & 10 \\ d_{9}^{⊥} & 14 & 13 & 12 & 12 & 12 & 12 & 12 & 11 & 11 & 11 & 11 \\ d_{10}^{⊥} & 15 & 14 & 13 & 13 & 13 & 13 & 13 & 13 & 12 & 12 & 12 \\ d_{11}^{⊥} & 16 & 15 & 14 & 14 & 14 & 14 & 14 & 14 & 13 & 13 & 13 \\ d_{12}^{⊥} & 18 & 16 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 14 & 14 \\ d_{13}^{⊥} & 19 & 17 & 16 & 16 & 16 & 16 & 16 & 16 & 16 & 15 & 15 \\ d_{14}^{⊥} & 20 & 18 & 18 & 17 & 17 & 17 & 17 & 17 & 17 & 16 & 16 \\ d_{15}^{⊥} & 21 & 19 & 19 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 17 \\ d_{16}^{⊥} & 22 & 20 & 20 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 \\ d_{17}^{⊥} & 23 & 21 & 21 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 \\ d_{18}^{⊥} & 24 & 22 & 22 & 21 & 21 & 21 & 21 & 21 & 21 & 21 & 21 \\ d_{19}^{⊥} & 25 & 23 & 23 & 22 & 22 & 22 & 22 & 22 & 22 & 22 & 22 \\ d_{20}^{⊥} & 26 & 24 & 24 & 23 & 23 & 23 & 23 & 23 & 23 & 23 & 23 \\ d_{21}^{⊥} & 27 & 25 & 25 & 24 & 24 & 24 & 24 & 24 & 24 & 24 & 24 \\ d_{22}^{⊥} & 28 & 26 & 26 & 25 & 25 & 25 & 25 & 25 & 25 & 25 & 25 \\ d_{23}^{⊥} & 29 & 27 & 27 & 27 & 26 & 26 & 26 & 26 & 26 & 26 & 26 \\ d_{24}^{⊥} & 30 & 29 & 28 & 28 & 27 & 27 & 27 & 27 & 27 & 27 & 27 \\ d_{25}^{⊥} & 31 & 30 & 29 & 29 & 28 & 28 & 28 & 28 & 28 & 28 & 28 \\ d_{26}^{⊥} & 32 & 31 & 30 & 30 & 29 & 29 & 29 & 29 & 29 & 29 & 29 \\ d_{27}^{⊥} & 34 & 32 & 31 & 31 & 30 & 30 & 30 & 30 & 30 & 30 & 30 \end{matrix}

Now let ℓ, m be positive integers with ℓ ≤ m and as before let k = \binom{m}{ℓ}, ℓ′ := m − ℓ and δ = ℓℓ′. Assume that 1 < ℓ ≤ ℓ′. Consider the Grassmannian G_{ℓ,m} = G_{ℓ,m}(𝔽_{q}) of ℓ-dimensional subspaces of 𝔽_{q}^{m}. The Plücker embedding

G_{ℓ,m} ↪ ℙ^{\binom{m}{ℓ}−1} = ℙ(∧^{ℓ}𝔽_{q}^{m}) given by W = ⟨w_{1}, …, w_{ℓ}⟩ ↦ [w_{1} ∧ ⋯ ∧ w_{ℓ}]

is known to be nondegenerate and the corresponding nondegenerate linear code is the Grassmann code C(ℓ, m). The formula stated in the Introduction for the last few higher weights of C(ℓ, m) follows readily from the structure of linear subvarieties of G_{ℓ,m} or, in algebraic parlance, the structure of decomposable subspaces of exterior powers. Indeed, G_{ℓ,m} contains a linear subspace Π of dimension r − 1, provided r ≤ μ, where μ := 1 + max{ℓ, m − ℓ}; see, for example, [7, Cor. 7] or [5, Lemma 3.5]. This subspace Π has codimension (k −1)−(r −1) = k −r in ℙ^{k−1}, and clearly, |Π ∩ G_{ℓ,m}| = |Π| = |ℙ^{r−1}| = (1 + q + ⋯ + q^{r−1}). Consequently,

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

Suppose we fix an ordered basis {e_{1}, …, e_{m}} of 𝔽_{q}^{m} and the corresponding basis {e_{α} : α ∈ I(ℓ, m)} of ∧^{ℓ}𝔽_{q}^{m}, where

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

and e_{α} := e_{α_{1}} ∧ ⋯ ∧ e_{α_{ℓ}} for α = (α_{1}, …, α_{ℓ}) ∈ I(ℓ, m). The Plücker coordinates of an ℓ-dimensional subspace W ∈ G_{ℓ,m} spanned by {w_{1}, …, w_{ℓ}} are precisely p = (p_{α})_{α∈I(ℓ,m)}, where p_{α} ∈ 𝔽_{q} are determined by the relation

w_{1} ∧ ⋯ ∧ w_{ℓ} = ∑_{α∈I(ℓ,m)} p_{α}e_{α}.

For α ∈ I(ℓ, m), let H_{α} denote the hyperplane {p ∈ ℙ^{k−1} : p_{α} = 0} in ℙ^{k−1}, and let U_{α} := {p ∈ ℙ^{k−1} : p_{α} ≠ 0} be the corresponding basic open set. It is a classical fact that U_{α} ∩ G_{ℓ,m} is isomorphic to the affine space 𝔸^{δ} of ℓ × ℓ′ matrices over 𝔽_{q}. This correspondence is given explicitly by the Basic Cell Lemma of [6]. For the sake of definitiveness, consider θ := (ℓ′ + 1, ℓ′ + 2, …, m) ∈ I(ℓ, m). Then the Plücker embedding restricted to U_{θ} ∩ G_{ℓ,m} gives a nondegenerate embedding of 𝔸^{δ} into ℙ^{k−1}, and the linear code corresponding to this projective system is, in fact, the affine Grassmann code C^{𝔸}(ℓ, m) of length q^{δ}. If Π is a linear subspace of ℙ^{k−1} of dimension r − 1, then Π ∩ H_{θ} would be a linear subspace of dimension r − 2 or r − 1 according as Π ⊈ H_{θ} or Π ⊆ H_{θ}. Consequently,

|Π ∩ U_{θ}| = |Π| − |Π ∩ H_{θ}| ≤ (1 + q + ⋯ + q^{r−1}) − (1 + q + ⋯ + q^{r−2}) = q^{r−1}.

Consequently, |Π ∩ U_{θ} ∩ G_{ℓ,m}| ≤ q^{r−1} and so d_{k−r}(C^{𝔸}(ℓ, m)) ≥ q^{δ} − q^{r−1} for all r = 1, …, k. This proves a stronger version of Lemma 3.2 in the case h = ℓ. Further, if r ≤ μ = ℓ′ + 1 and if Π is a linear subspace of ℙ^{k−1} of codimension k − r chosen in such a way that Π ⊆ G_{ℓ,m} and Π ⊆ H_{θ}, then |Π ∩ U_{θ} ∩ G_{ℓ,m}| = |Π ∩ H_{θ}| = q^{r−1} and so d_{k−r}(C^{𝔸}(ℓ, m)) = q^{δ} − q^{r−1} for all r = 1, …, μ. Since 1 < ℓ ≤ ℓ′, choosing such a subspace Π is possible for 1 ≤ r ≤ μ; for example, we can take Π = {p ∈ ℙ^{k−1} : p_{(1,2,...,ℓ−1,j)} = 0 for j = ℓ, ℓ + 1, …, ℓ + r − 1} to be the intersection of Plücker coordinate hyperplanes that are “close” to each other. Thus we obtain an alternative proof of Theorem 3.3 when h = ℓ.

On the other hand, deriving the formulas that we have for initial higher weights of C^{𝔸}(ℓ, m) from the corresponding results for the Grassmann code C(ℓ, m) is not so straightforward. To be sure, the optimal linear subspace in G_{ℓ,m} of large dimension (or small codimension) are obtained in [6] by considering close families in I(ℓ, m) and the corresponding linear subsbaces of ℙ^{k−1} given by the intersections of Plücker coordinate hyperplanes. Recall that Λ ⊆ I(ℓ, m) is said to be close if any two distinct elements of Λ have ℓ − 1 coordinates in common. However, determining the maximum possible cardinality of the intersection of the corresponding linear subspace Π with U_{θ} ∩ G_{ℓ,m} is not easy. It may be tempting to consider Λ ⊆ I(ℓ, m) not containing θ such that Λ ∪ {θ} is close. But this doesn’t work even when Λ is singleton (which would correspond to looking at the minimum distance). In fact, it is better to keep the elements of Λ as far away from θ as possible. Thus choosing a close family in I(ℓ, ℓ′) rather than I(ℓ, m) is helpful and this has, in fact, motivated the proofs of Lemma 2.3 and 2.4, which paved the way for Theorem 2.5.

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

E-mail address: mrinmoy.dat@gmail.com

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

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