Grassmannians of codes

Consider the point line-geometry ${\mathcal P}_t(n,k)$ having as points all the $[n,k]$-linear codes having minimum dual distance at least $t+1$ and where two points $X$ and $Y$ are collinear whenever $X\cap Y$ is a $[n,k-1]$-linear code having minimum dual distance at least $t+1$. We are interested in the collinearity graph $\Lambda_t(n,k)$ of ${\mathcal P}_t(n,k).$ The graph $\Lambda_t(n,k)$ is a subgraph of the Grassmann graph and also a subgraph of the graph $\Delta_t(n,k)$ of the linear codes having minimum dual distance at least $t+1$ introduced in~[M. Kwiatkowski, M. Pankov, On the distance between linear codes, Finite Fields Appl. 39 (2016), 251--263, doi:10.1016/j.ffa.2016.02.004, arXiv:1506.00215]. We shall study the structure of $\Lambda_t(n,k)$ in relation to that of $\Delta_t(n,k)$ and we will characterize the set of its isolated vertices. We will then focus on $\Lambda_1(n,k)$ and $\Lambda_2(n,k)$ providing necessary and sufficient conditions for them to be connected.


Introduction
Let V := V (n, q) be a n-dimensional vector space over a finite field F q and 0 < k < n; let Γ(n, k) be the graph whose vertices are the k-subspaces of V and where two vertices X, Y are adjacent if and only if dim(X ∩ Y ) = k − 1. The graph Γ(n, k) is called the k-Grassmann graph of V.
An [n, k]-linear code is just a vertex of Γ(n, k). We say that a [n, k]-linear code C has dual minimum distance at least t + 1 if, given a generator matrix G for C, any set of t columns of G is linearly independent. It is easy to see that this condition does not depend on the particular generator matrix chosen for C and that this is the same as to require that the dual code of C has minimum Hamming distance at least t + 1, whence the name.
Denote by C t (n, k) the set of all [n, k]-codes with dual minimum distance at least t+1 over F q . By construction, the elements of C t (n, k) can be regarded Observe that if Z is a k-vector subspace and it contains a subspace D ∈ C t (n, k ′ ) with k ′ < k, then Z ∈ C t (n, k). In particular, the line ℓ X,Y ∈ L t (n, k) can also be described as ℓ X,Y := {Z : X ⊂ Z ⊂ Y } with X ∈ C t (n, k − 1) and dim(Y ) = k + 1.
The geometry P t (n, k) is a subgeometry of the k-Grassmann geometry G (n, k) (see Section 2) and the collinearity graph of P t (n, k) is precisely the graph Λ t (n, k).
In this paper we shall study the structure of Λ t (n, k) and the interplay between the geometry P t (n, k) and the geometry G (n, k). We point out that from a more applied point of view, the study of the graph Λ t (n, k) is related to some code density problem, but we leave its investigation to further works.
Before stating our main results we need to give the following definitions.
Definition 1.2. A code C ∈ C t (n, k) is isolated if C does not contain any proper subcode D ∈ C t (n, k − 1). We denote the set of all isolated codes in C t (n, k) by the symbol I t (n, k).
If a code C is not isolated, then there exists at least one code C ′ ∈ C t (n, k) with C ′ = C and such that C ∩ C ′ ∈ C t (n, k − 1), that is C ′ ∼ Λ C. So it follows readily that a vertex of Λ t (n, k) is isolated if and only if it corresponds to an isolated code of C t (n, k). On the other hand, the graph induced by Λ t (n, k) on I t (n, k) is totally disconnected, i.e. it contains no edge. We will use the same symbol I t (n, k) to denote both the set of isolated codes of P t (n, k) and the subgraph (I t (n, k), ∅) of isolated vertices of Λ t (n, k).
Given a vector space of dimension k over F q , we denote by PG(k − 1, q) the associated projective space; if X is a set of points of PG(k − 1, q), then X denotes the projective subspace spanned by X. The graph Λ t (n, k) is related to interesting and well studied configurations of points of a projective space such as the t-saturating sets, whose definition we report below. X be the set of all points of PG(k−1, q) on subspaces spanned by t+1 points of Ω. The set Ω is t-saturating if S t (Ω) = PG(k − 1, q) and S t−1 (Ω) = PG(k − 1, q).
We refer to [5,6,7] and the references therein for more information on saturating sets and for the known bounds on the minimal cardinality of a t-saturating set and related constructions.

Main Results
The following are the main results of this paper.
Remark 1.5. When C t (n, k − 1) = ∅, then ∆ t (n, k − 1) is an empty graph and Λ t (n, k) = I t (n, k) consists only of isolated vertices. If we take by convention the empty graph to be connected, then Theorem 1.4 is true also in this case. Also the converse holds, i.e. if Λ t (n, k) = I t (n, k), then C t (n, k − 1) = ∅. By Theorem 1.9, this holds for t = k. Theorem 1.6. A code C ∈ C t (n, k) is isolated if and only if the columns of any generator matrix of C are vector representatives of a (t − 1)-saturating set of PG(k − 1, q).
The following corollaries, holding for t = 1 and t = 2, are consequences of Theorem 1.1, Theorem 1.4 and Theorem 1.6. Corollary 1.7. The following hold.
is isolated if and only if the columns of any generator matrix of C are vector representatives of all the points of a (k − 1)dimensional projective space.
Corollary 1.8. The following hold.
is isolated if and only if the 2-secants of the projective set determined by the columns of any generator matrix of C cover all the points of a (k − 1)-dimensional projective space. Theorem 1.9. If q > n t and t < k then the graph Λ t (n, k) is connected. For t = k, the graph Λ k (n, k) is totally disconnected.
In light of Theorem 1.9 it makes sense to ask for given t, k and q what is the structure of the graph Λ t (n, k), as n grows, under the condition C t (n, k) = ∅. Keeping in mind Definition 1.2 of isolated code, we can consider the following two parameters ν t (k; q) := min{n : Λ t (n, k) is disconnected}, ν + t (k; q) := min{n : I t (n, k) = ∅}. If a graph has an isolated vertex then it is disconnected, so we have Clearly, if ν t (k; q) = ν + t (k; q) then the graph Λ t (n, k) is disconnected if and only if it has isolated vertices. We conjecture the following. Conjecture 1.10. For any q, k and t with k ≥ t, ν t (k; q) = ν + t (k; q).
We observe that Conjecture 1.10 holds true for t = 1 by Corollary 1.7 and for t = 2 by Corollary 1.8.
Moving to the geometry P t (n, k) whose collinearity graph is Λ t (n, k), we shall prove that the inclusion map ι of P t (n, k) in the Grassmann geometry G (n, k) has the property that lines of P t (n, k) are mapped into lines of G (n, k) and the preimage of any line of G (n, k) contained in the image of ι is still a line of P t (n, k). According to [3], this is exactly the definition for an embedding to be transparent (see Section 4.4).
Theorem 1.11. The inclusion map ι : P t (n, k) → G (n, k) is a transparent embedding.
Since G (n, k) is projectively embeddable by means of the Plücker embedding ε k in the projective space PG( k V ), Theorem 1.11 implies that the geometry P t (n, k) is also projectively embeddable in PG( k V ) by means of the restriction of ε k to P t (n, k), and this embedding is transparent.
The resulting point set V t := ε k (C t (n, k)) is a subset of the Grassmann variety in PG( k V ) and, by Theorem 1.11 and the transparency of the Plücker embedding ε k , it has the property that for any two distinct points P, Q ∈ V t , the projective line joining P and Q is fully contained in V t if and only if ε −1 k (P ) and ε −1 k (Q) are adjacent in Λ t (n, k).
In particular, it is possible to reconstruct the graph Λ t (n, k) from just looking at the point set V t . We observe that this property does not hold for the graph ∆ t (n, k), i.e. it can happen that X and Y are adjacent in ∆ t (n, k) but the projective line ℓ joining ε k (X) and ε k (Y ) is not fully contained in V t , i.e. there exists a point Z on ℓ such that ε −1 (Z) ∈ C t (n, k).
Structure of the paper In Section 2 we will set the notation and recall the basic results regarding the objects of our interest. In Section 3 we shall focus on the graph Λ t (n, k) proving some general results for arbitrary values of t,k and n. In Section 4 we will prove our main theorems; in particular we will prove Corollary 1.7 in Subsection 4.1 and Corollary 1.8 in Subsection 4.2. The general case is treated in Subsection 4.3 and Theorem 1.11 is proved in Section 4.4.

Preliminaries
As in the Introduction, let V := V (n, q) be a n-dimensional vector space over a finite field F q and 0 < k < n. The k-Grassmann geometry is defined as the point-line geometry G (n, k) whose points are the k-dimensional subspaces of V and whose lines are the sets This geometry has been widely investigated (see for example [13]) and its collinearity graph is the Grassmann graph Γ(n, k) as defined in the Introduction.
Suppose that B = (e 1 , . . . , e n ) is a given fixed basis of V ; henceforth we shall always write the coordinates of the vectors in V with respect to B.
Given two vectors x = n i=1 x i e i and y = n i=1 y i e i , the Hamming distance (with respect to the basis B) between x and y is d(x, y) := |{i : x i = y i }|. In this setting, a [n, k]-linear code C is just a k-dimensional vector subspace of V together with the restriction to C × C of the Hamming distance induced by B.
If B C is an ordered basis of C, a generator matrix G C for C is the k × n matrix whose rows are the components of the elements of B C with respect to B. Given a [n, k]-linear code C, its dual code is the [n, n − k]-linear code C ⊥ given by where · denotes the standard symmetric bilinear form on V given by Since the bilinear form "·" is non-degenerate, C ⊥⊥ = C. We say that C has dual minimum distance at least t + 1 if and only if the minimum Hamming distance of the dual C ⊥ of C is at least t+1. As mentioned in the Introduction, this condition is equivalent to saying that for any generator matrix G C of C, any set of t columns of G C is linearly independent. For t ∈ N, we already defined the set C t (n, k) of all [n, k]-linear codes with dual minimum distance at least t + 1 and the point-line geometry P t (n, k) = (C t (n, k), L t (n, k)) whose collinearity graph is precisely Λ t (n, k). The geometry P t (n, k), clearly, can be regarded as a subgeometry of G (n, k).
We recall that a graph Γ is connected whenever given any two of its vertices (say P and Q with P = Q) there exists a path in Γ, i.e. a sequence of nonrepeated adjacent vertices, starting at P and ending at Q. We consider the empty graph to be connected. The length of a path between P and Q is defined as the smallest cardinality of a path connecting P and Q diminished by 1; the distance d Γ (P, Q) between P and Q in Γ is the length of a shortest path between P and Q; a connected component of a graph is a maximal nonempty subset of its vertices such that the subgraph induced on it is connected. The diameter of a graph is the maximum distance between any two vertices; see [1] for a general reference about graph theory. In general we say that a subgraph Γ ′ is isometrically embedded in a larger graph Γ if there exists a distance-preserving map Γ ′ → Γ; see [8].
We already mentioned in the Introduction the subgraph ∆ t (n, k) of the Grassmann graph having as vertices the [n, k]-linear codes C t (n, k) with dual minimum distance at least t + 1. The graph ∆ t (n, k) was first introduced in [9] where the authors focused mostly on the case of non-degenerate linear codes, i.e. on ∆ 1 (n, k). In [10] these results, among others, were extended to the graph ∆ 2 (n, k) of projective linear codes. In [4] we considered properties of the graph ∆ t (n, k) for arbitrary t.
Comparing the graph Λ t (n, k) with the graph ∆ t (n, k), we observe that Λ t (n, k) has the same vertices as ∆ t (n, k) but less edges. In particular, there is not isometrically embedded in the Grassmann graph either.

Equivalent codes
Given the basis B, the monomial group M (V ) of V consists of all linear transformations of V which map the set of subspaces { e 1 , . . . , e n } in itself. It is straightforward to see that M (V ) ∼ = F × q ≀ S n where ≀ denotes the wreath product and S n is the symmetric group of order n; see [11,Chapter 8, §5] for more details.
Equivalence between linear codes is an equivalence relation and the equivalence class of a code X corresponds to the orbit of X under the action of M (V ) on the subspaces of V .
A monomial transformation ρ is given by a linear transformation of V which sends the code Y (regarded as a k-dimensional subspace of V ), into the code X = ρ(Y ). In particular, if G Y is a generator matrix for Y , then the matrix ρ(G Y ) obtained from G Y by applying ρ to each of its rows, is a generator matrix for X. We put G ρ(Y ) := ρ(G Y ). Observe that the transformation induced by ρ acts on the columns of G Y . Indeed, if ρ is represented by a n × n matrix R with respect to the basis B, The matrix R factors as a product R = P D where P is a permutation matrix and D is a non-singular diagonal matrix. In the rest of this paper, we shall denote, with a slight abuse of notation, by the same symbol ρ not only the isometry ρ : V → V but also the corresponding map acting on the generator matrices of the codes.
If X is a [n, k]-linear code with generator matrix G X and A ∈ GL(k, F q ) then G ′ X = AG X is also a generator matrix for X. It follows that two [n, k]-linear codes X and Y with generator matrices respectively G X and G Y are equivalent if there exists A ∈ GL(k, F q ), a permutation matrix P ∈ GL(n, F q ) and a diagonal matrix D ∈ GL(n, F q ) such that In particular, two codes are equivalent if and only if any two of their generator matrices belong to the same orbit under the action of the group GL(k, F q ) : where GL(k, F q ) acts on the left of the generator matrix and fixes each code, regarded as a subspace, while F × q ≀ S n acts on the right. The following lemma provides explicit generators for the group M (V ) and describe their action on the set of the linear codes.
In particular, the map τ ij is represented with respect to B by a permutation matrix, while µ is represented by the matrix diag(α, 1, 1, . . . , 1).
Proof. By construction M (V ) is generated by the generators of the constituent groups of the wreath product F × q ≀ S n . As F × q is cyclic generated by α and S n is generated by the swaps, it follows that the above set is enough to generate M (V ).
By Lemma 2.2, the group M (V ) is generated by transpositions τ ij (1 ≤ i < j ≤ n) and the diagonal transformation µ, so it is enough prove the claim above for ρ = τ ij and ρ = µ.
We study the rank of the matrix G C G µ(C) . We have rank In particular, either rank In the former case C = µ(C) (and there is nothing to prove); in the latter dim(C ∩ µ(C)) = k − 1 and C and µ(C) are adjacent in the graph ∆ t (n, k).
We consider now the action of swaps τ ij , 1 ≤ i < j ≤ n, on G C . If P i = P j then τ ij (C) = C and there is nothing to say. If P i = P j , then the code τ ij (C) has generator matrix G τ ij (C) = P 1 . . . P j . . . P i . . . P n . Then, Proceeding as before, we have that either C = τ ij (C) or C ∼ ∆ τ ij (C).
3 General results on C t (n, k) and the graph Λ t (n, k) Let C be a [n, k]-linear code and G C = (P 1 , P 2 , . . . , P n ) a given generator matrix of C, where P 1 , . . . , P n denote column vectors in F k q . Proof. Let D be an [n, k −1]-subcode of C. Then a generator matrix G D for D has, as rows, k −1 linearly independent vectors of F n q , and each of these vectors is a linear combination of the rows of G C . So, there exists a (k − 1) × k matrix F of rank k − 1 such that G D = F G C . If ϕ is the linear application F k q → F k−1 q with matrix F with respect to the canonical bases of F k q and F k−1 q , then ϕ is surjective. Define G ϕ C := (ϕ(P 1 ), . . . , ϕ(P n )) as the matrix having as columns the images under ϕ of the columns P 1 , . . . , P n .
is a surjective linear function, then put G ϕ C := F G C for F a (k −1)×k matrix (of rank k −1) representing ϕ. Clearly rank(G ϕ C ) = k −1 and all rows of G ϕ C are linear combinations of rows of G C ; so G ϕ C is the generator matrix of a subcode of C.
Remark 3.2. Any map ϕ of the previous lemma can be regarded as a function which sends all k-dimensional subspaces C of V into (k − 1)-dimensional subspaces D with D ⊂ C. Its explicit action depends on the choice of a basis for each k-dimensional subspace C of V , i.e. on the choice of a generator matrix G C for C. In particular, ϕ can be regarded as a map F k,n q → F k−1,n q sending a k × n matrix G into a (k − 1) × n matrix G ′ , G → G ′ = F G, where F is the representative matrix of ϕ (with respect to given bases). In any case, by Lemma 3.1, the set of all subcodes of C does not depend on the specific choice of G C for C since ϕ varies among all surjective linear functions F k q → F k−1 q . Lemma 3.3. If C ∈ C t (n, k) \ I t (n, k) and C ′ are equivalent [n, k]-codes then there exist equivalent [n, k − 1]-subcodes D, D ′ ∈ C t (n, k − 1) of C and C ′ respectively. In particular, C ′ ∈ C t (n, k) \ I t (n, k).
Proof. Since C ∈ C t (n, k) \ I t (n, k) and C ′ are equivalent, then C ′ ∈ C t (n, k) and there exists ρ ∈ M (V ) such that ρ(C) = C ′ . Take D ∈ C t (n, k − 1) as a subcode of C (D exists because C is not isolated) and let G C be a generator matrix for C. As seen in Section 2, G ρ(C) (= ρ(G C )) is a generator matrix of C ′ . Denote by ϕ : F k q → F k−1 q the surjective map associated to D (see Lemma 3.1), so G ϕ C is a generator matrix of D. We claim that D is equivalent to the subcode of ρ(C) determined by ϕ, i.e.
Lemma 3.4. Let C, C ′ ∈ C t (n, k) \ I t (n, k) and suppose that D and D ′ are two [n, k − 1]-linear subcodes of C and C ′ respectively. If D, D ′ are in the same connected component of ∆ t (n, k − 1) then C and C ′ are in the same connected component of Λ t (n, k).

Proof. By hypothesis, there exists a path
It follows that C and C ′ are in the same connected component of Λ t (n, k).
Recall that Λ t (n, k) is the graph induced by Λ t (n, k) on the codes in C t (n, k) \ I t (n, k). The following is immediate.
Corollary 3.6. Suppose C, C ′ ∈ C t (n, k)\I t (n, k) are equivalent codes. Then C and C ′ belong to the same connected component of Λ t (n, k).
Proof. By Lemma 3.3, there are equivalent subcodes D, D ′ of C and C ′ respectively with D, D ′ ∈ C t (n, k − 1). By Theorem 2.3 D and D ′ belong to the same connected component of ∆ t (n, k − 1). The thesis now follows directly from Lemma 3.4.
In Lemma 3.8 we shall make use of the following remark.
Remark 3.7. For any D ∈ C t (n, k − 1) and for any [n, k]-linear code C with D ⊂ C we have C ∈ C t (n, k); consequently, for any element D ∈ C t (n, k − 1) with k − 1 < n there exists at least one C ∈ C t (n, k) with D ⊂ C.
Lemma 3.8. Suppose ∆ t (n, k−1) not to be connected. Then the graph Λ t (n, k) is not connected.
Proof. Take two codes D, D ′ ∈ C t (n, k−1) which belong to different connected components of ∆ t (n, k − 1) and let C, C ′ be two codes in C t (n, k) having D, D ′ as subcodes respectively. These codes exist by Remark 3.7. By contradiction, suppose there exists a path In either case, we have a collection of subcodes . This is a contradiction, since we assumed D and D ′ to be in separate connected components of ∆ t (n, k−1).
Proof of Theorem 1.4. Since Λ t (n, k) = Λ t (n, k) ∪ I t (n, k) and the elements of I t (n, k) are isolated vertices of Λ t (n, k), Theorem 1.4 is a direct consequence of Corollary 3.9.
Proof of Theorem 1. 6 We have that C ∈ C t (n, k) is an isolated code if and only if every [n, k − 1]-subcode of C does not belong to C t (n, k − 1). This is the same as to say that for every [n, k − 1]-subcode D of C there exist at least t columns in any of its generator matrices which are linearly dependent. By Lemma 3.1 any subcode D has a generator matrix of the form G ϕ C = (ϕ(P 1 ), . . . , ϕ(P n )) where G C = (P 1 , . . . , P n ) is a given generator matrix for C with columns P 1 , . . . , P n and ϕ is an element of M . For C to be isolated we require that for every ϕ ∈ M , there is at least one set of t columns of G ϕ C which are linearly dependent. This is equivalent to saying that there exist i 1 , . . . i t ∈ {1, . . . , n} such that λ i 1 P i 1 + . . . λ it P it ∈ ker(ϕ), i.e. the projective point given by ker(ϕ)(∈ K ) is in the (t − 1)-dimensional projective space of PG(k − 1, q) spanned by the points P i 1 , . . . , P it . Since this needs to be true for all ϕ ∈ M we have that the columns of the generator matrix G C of C must form a (t − 1)-saturating set of PG(k − 1, q). Theorem 1.6 follows.
The following lemma is a direct consequence of Theorem 1.1 and Theorem 1.4. We also propose an alternative direct proof which holds only for t = 1.
Proof. It is enough to show that if a code is not isolated, then it belongs to the same connected component as any other non-isolated code. First observe that a code which contains the all-1 vector 1, is not isolated. Let now C 1 and C 2 be two non-isolated codes and suppose that D 1 and D 2 are two non-degenerate [n, k − 1]-subcodes of C 1 and C 2 respectively. If 1 ∈ C i then put (Theorem 1.1) we can construct a path in Λ 1 (n, k) from C ′ 1 to C ′ 2 of codes all containing the vector 1. So there is a path from C 1 to C ′ 1 and then from C ′ 1 to C ′ 2 and finally to C ′ 2 to C 2 , i.e. C 1 and C 2 belong to the same connected component. Furthermore observe that Proof. By Lemma 4.1, the graph Λ 1 (n, k) is connected if and only if there exists no isolated vertex in it, i.e. I 1 (n, k) = ∅. This is equivalent to saying that for every C ∈ C 1 (n, k), there exists at least a [n, k − 1]-subcode D which is non-degenerate. By Lemma 3.1, we have that C admits at least a nondegenerate subcode if and only if there is at least one function in M whose kernel does not contain any column of G C . This is always guaranteed if the number of surjective linear functions from F k q onto F k−1 q , and hence the number of their kernels, is strictly greater than the number of columns of a generator matrix of C (i.e. the length of C), hence for |K | = q k −1 q−1 > n. On the other hand, if n = q k −1 q−1 the code whose generator matrix has a column set consisting of all the distinct vector representatives of the 1-dimensional subspaces of F k q is definitely in I 1 (n, k) and so Λ 1 (n, k) is disconnected. Corollary 1.7 follows from Lemma 4.1 and Lemma 4.2 or, alternatively, Theorem 1.6.

Projective codes
We first prove that the Grassmann graph of the projective codes ∆ 2 (n, k) is always connected. Observe that, in any case, there is a natural bound on n, that is n ≤ q k −1 q−1 as the maximum length of a projective code of dimension k is the same as the number of points of PG(k − 1, q). Lemma 4.3. The graph ∆ 2 (n, k) is connected for any q and any 2 ≤ k ≤ n.
Proof. Let C and C ′ be two projective codes with generator matrices respectively G C = (P 1 , P 2 , . . . , P n ) and G C ′ = (P ′ 1 , P ′ 2 , . . . , P ′ n ) where P i , P ′ i ∈ F k q are column vectors. By Theorem 2.3 we can assume, up to code equivalence, that if a column P i in G C is proportional to a column P ′ j in G C ′ , then i = j. Also by Theorem 2.3 we can assume that if P i in G C is proportional to P ′ i in G C ′ , then P i = P ′ i (i.e. the coefficient of proportionality is 1 and the columns are the same). If under these assumptions G C = G C ′ then C and C ′ are equivalent codes and, by Theorem 2.3 they belong to the same connected component of ∆ 2 (n, k). Otherwise can construct a path in ∆ 2 (n, k) joining C and C ′ by considering the codes C 0 := C ∼ ∆ C 1 ∼ ∆ · · · ∼ ∆ C w =: C ′ where the code C i has as generator matrix, the matrix where the first i columns are the first i columns of G C ′ and the remaining columns are the corresponding columns of  Proof. By Corollary 1.8 part 1., the graph Λ 2 (n, k) is connected if and only if there exists no isolated vertex in it. This is equivalent to saying that for every C ∈ C 2 (n, k), there exists at least a [n, k − 1]-subcode D which is projective. By Lemma 3.1, we have that C admits at least a projective subcode if and only if there is at least one function in M whose kernel does not belong to any 2-space spanned by 2 columns of G C . This is always guaranteed if the number of columns of any generator matrix of C (i.e. the length of C) is strictly less than the minimum size of a 1-saturating set of PG(k − 1, q).
On the other hand, if n ≥ µ 1 , where µ 1 is the minimum size of a 1saturating set Ω of PG(k − 1, q), the code whose generator matrix contains as columns vector representatives for all of the 1-dimensional subspaces of Ω is definitely in I 2 (n, k) and so Λ 2 (n, k) is disconnected.

Proof of Theorem 1.9
Lemma 4.5. If t = k then C k (n, k) = I k (n, k). If t < k and q > n t 1/(k−t) , we have C t (n, k) = ∅ and I t (n, k) = ∅.
Proof. If k = t, each set of k columns of any generator matrix of a code C ∈ C k (n, k) is linearly independent, so C is MDS. In particular, the dual minimum distance of each [n, k − 1]-subcode of an MDS code is at most k − 1 < k (as its generator matrix has just k − 1 rows). So, every code C ∈ C k (n, k) is isolated and C k (n, k) = I k (n, k).
Suppose now t ≤ k −1 and q > n. Take C ∈ C t (n, k) with generator matrix G C . Let ϕ : F k q → F k−1 q be a surjective linear map and denote by D = ϕ(C) the subcode of C having as generator matrix G ϕ C . We have D ∈ C t (n, k − 1) if and only if there is a set Ω of t linearly independent columns in G C which is mapped into a set of linearly dependent columns in G ϕ C . In other words, this happens if and only if ker(ϕ) ∈ Ω . As ϕ varies in all possible ways, there are exactly (q k − 1)/(q − 1) possible subspaces for ker(ϕ). On the other hand the number of possible t-subspaces spanned by t columns of G C is n t and each of these contains at most (q t − 1)/(q − 1) distinct 1-dimensional subspaces. So if we have that there is at least one ϕ ∈ M such that the subcode of C determined by ϕ is in C t (n, k − 1) and, consequently, C ∈ I t (n, k). So if For t < k we have the approximation q k−t < q k −1 q t −1 since So, if we require q k−t > n t then q k −1 q t −1 > n t . Hence I t (n, k) = ∅.

Transparent embeddings
Given two point-line geometries G 1 and G 2 , we say that an injective map ε : G 1 → G 2 of the point-set of G 1 into the point-set of G 2 is an embedding of G 1 into G 2 if for any line ℓ of G 1 the image ε(ℓ) := {ε(p) : p ∈ ℓ} is a line of G 2 . If G 2 is a projective geometry we say that an embedding of G 1 into G 2 is a projective embedding into the projective subspace ε(G 1 ) .
In [3] the notion of transparency for an embedding has been introduced. An embedding ε : G 1 → G 2 is transparent if the preimage of any line of G 2 contained in the image of ε is a line of G 1 .
Let us focus now on the Grassmann geometry G (n, k) and the geometry P t (n, k). Since P t (n, k) is a subgeometry of G (n, k), the inclusion map ι : P t (n, k) → G (n, k) is an embedding.
There is a vast literature regarding projective embeddings of the Grassmann geometry. For the sake of our paper we stick only to the essential results referring the interested reader to, e.g., [2].
Proof of Theorem 1.11. By definition of P t (n, k), any line of P t (n, k) is also a line of G (n, k). In particular, the inclusion ι is an embedding ι : P t (n, k) → G (n, k).
To prove that ι is transparent we need to show that if a line ℓ of G (n, k) consists all of points of C t (n, k), then for any two X, Y ∈ ℓ we have X ∼ Λ Y . This is equivalent to saying that if X and Y are not on a line of Λ t (n, k), then ι(X) and ι(Y ) are not on a line contained in ι(C t (n, k)), i.e. either X and Y are not on a line of G (n, k) or X and Y are on a line of G (n, k) whose points are not all images of points in C t (n, k). If ι(X) and ι(Y ) are not collinear in G (n, k), then X and Y are not collinear in P t (n, k) as all lines of P t (n, k) are lines of G (n, k). So we have to deal only with the latter case. Observe that X and Y lie on a line of G (n, k) if and only if X ∼ ∆ Y .
So, suppose X ∼ ∆ Y but X ∼ Λ Y for X, Y ∈ C t (n, k). To obtain the result it is enough to show that there exists at least one Z on the line ℓ of G (n, k) determined by X and Y with Z ∈ C t (n, k). So, the theorem is a consequence of the following lemma. Lemma 4.6. Suppose X, Y ∈ C t (n, k), dim(X ∩ Y ) = k − 1 but X ∩ Y ∈ C t (n, k − 1). Then there exists Z such that X ∩ Y ⊂ Z ⊂ X, Y , dim(Z) = k and Z ∈ C t (n, k).
Proof. Put D = X ∩ Y . By hypothesis, D ∈ C t (n, k − 1). Suppose by contradiction that for all v ∈ X, Y \ D we have Z v := D, v ∈ C t (n, k).
Since D ∈ C t (n, k − 1), there is at least one set of t columns P i 1 , . . . , P it with 1 ≤ i 1 < · · · < i t ≤ n in any generator matrix G D of D which are linearly dependent. Denote by G D the (k − 1) × t submatrix of G D having P i 1 , . . . , P it as columns (and R 1 , . . . , R k−1 as rows). By construction, we have that rank(G D ) ≤ t − 1.
The matrix G α,β can be regarded as the submatrix of a generator matrix of the generic code Z αx+βy with D ⊂ Z αx+βy ⊂ X, Y . If Z αx+βy ∈ C t (n, k), then necessarily rank(G α,β ) = t.
On the other hand, since of a generator matrix of the code X and since X ∈ C t (n, k), we have that Thus, the vector (y i 1 , . . . , y it ) is a linear combination of the rows R 1 , . . . R k−1 of G D and of the vector (x i 1 , . . . , x it ), where (x i 1 , . . . , x it ) appears with a nonzero coefficient (otherwise (y i 1 , . . . , y it ) ∈ R 1 , . . . , R k−1 and consequently the columns P i 1 y i 1 , P i 2 y i 2 , . . . , P it y it of the generator matrix G D y of Y would be linearly dependent and thus Y ∈ C t (n, k), against the hypothesis). Hence, we can write λ(y i 1 , . . . , y it ) = (x i 1 , . . . , x it ) + k−1 j=1 θ j R j for suitable θ j , λ ∈ F q and λ = 0. So, λ(y i 1 , . . . , y it ) − (x i 1 , . . . , x it ) = k−1 j=1 θ j R j , and the rank of the matrix G −1,λ (obtained from (2) with α = −1 and β = λ) is at most t − 1. It follows that Z −x+λy ∈ C t (n, k).