Coset Construction for Subspace Codes

One of the main problems of the research area of network coding is to compute good lower and upper bounds of the achievable cardinality of so-called subspace codes in $\operatorname{PG}(n,q)$, i.e., the set of subspaces of $\mathbb{F}_q^n$, for a given minimal distance. Here we generalize a construction of Etzion and Silberstein to a wide range of parameters. This construction, named coset construction, improves or attains several of the previously best-known subspace code sizes and attains the MRD bound for an infinite family of parameters.


Introduction
Let F q be the finite field of order q and V be a vector space of dimension n over F q .Since V is isomorphic to F n q , we will assume V = F n q in the following.By G q (n, k) we denote the set of all k-dimensional subspaces of F n q , where 0 ≤ k ≤ n.The projective space of order n over F q is given by P q (n) = ∪ 0≤k≤n G q (n, k).It is well known that is a metric on P q (n) [1].Thus, one can define codes on P q (n) and G q (n, k), which are called subspace codes and constant dimension codes, respectively.The distance function d S is known as subspace distance and one of the two distance functions that can be motivated by an information-theoretic analysis of the so-called Silva-Kschischang-Kötter model [24].The second distance function is the so-called injection distance d I (U, V ) := max {dim U, dim V } − dim(U ∩ V ).For two subspaces of the same dimension we have d S (U, V ) = 2d I (U, V ), i.e., the two metrics are equivalent on G q (n, k), and d I (U, V ) ≤ d S (U, V ) ≤ 2d I (U, V ) in general.We say that C ⊆ P q (n) is an (n, M, d) q code in the projective space if |C| = M and d(U, V ) ≥ d for all U, V ∈ C. If C ⊆ G q (n, k) for some k, we speak of an (n, M, d; k) q code.The minimum distance of a code C ⊆ P q (n) is denoted by D S (C) := min U =V ∈C d S (U, V ).One major problem is the determination of the maximum size A q (n, d) of an (n, M, d) q code in P q (n) and the maximum size A q (n, d; k) of an (n, M, d; k) q code in G q (n, k).Bounds for A q (n, d) and A q (n, d; k) have been heavily studied, see e.g. the survey [13] or the new online database at http://subspacecodes.uni-bayreuth.de[15].
With respect to lower bounds on A q (n, d; k), an asymptotically optimal construction is given by lifted maximum-rank-distance codes [14,24].To be more precise, the rate of the transmission log q |C| n•max U ∈C dim(U ) is asymptotically optimal up to a constant factor [17].A rough estimation between |C| and the Singleton bound yields an approximation factor of at most 4. The concept of maximum-rank-distance codes was generalized from arbitrary rectangular matrices to matrices with a (structured) 1991 Mathematics Subject Classification.Primary 05B25, 51E20; Secondary 51E22, 51E23.Key words and phrases.Constant dimension codes, subspace codes, subspace distance, Echelon-Ferrers construction.
The work of the authors was supported by the grant KU 2430/3-1 and WA 1666/9-1 "Integer Linear Programming Models for Subspace Codes and Finite Geometry" from the German Research Foundation and the COST Action IC1104 "Random Network Coding and Designs over GF(q)".
set of prescribed zeros in [11] and used to combine several maximum-rank-distance codes to generate a constant dimension code -the so-called multilevel or Echelon-Ferrers construction.Many of the best-known lower bounds on A q (n, d; k) arise from this construction.However, it is rather general and involves several search spaces or optimization problems in order to be evaluated optimally.For special subclasses explicit variants of the construction and indeed explicit formulas for the sizes of the corresponding codes have been obtained, see [25].We remark that additional refinements of the Echelon-Ferrers construction have been proposed recently, see [10,12,23].
An improvement beyond the Echelon-Ferrers construction was Construction III in [12] giving A 2 (8,4; 4) ≥ 4797.The authors conjecture that the underlying idea can be generalized to further parameters assuming the existence of a corresponding parallelism.In Theorem 9 we will show that this is indeed the case.Moreover, there is a more general underlying construction for (n, M, d; k) q and (n, M, d) q codes that is capable of improving some of the so far best-known lower bounds on A q (n, d; k), which is the core of this paper.To this end, we will give several infinite, parametric families of constructions as well as sporadic examples.
The remaining part of the paper is organized as follows.In Section 2 we collect some facts about representations of subspaces, MRD codes, parallelisms, and the Echelon-Ferrers construction.The main idea of the coset construction is described in Section 3. Since this construction has several degrees of freedom, we present some first insights on the choice of "good" parameters in Section 4. After listing some examples improving or attaining several lower bounds on A q (n, d; k) in Section 5, we conclude with open questions in Section 6.

Preliminaries
In this section we summarize some notation and well-known insights that will be used in the later parts of the paper.

2.1.
Gaussian elimination and representations of subspaces.Let A ∈ F k×n q be a matrix of (full) rank k.The row-space of A forms a k-dimensional subspace of F n q .The matrix A is called generator matrix of a given element of G q (n, k).Since the application of the Gaussian elimination algorithm on a generator matrix A does not change the row-space, we can restrict ourselves on generator matrices which are in reduced row echelon form (RRE form), i.e., the matrix has the shape resulting from Gaussian elimination.The representation is unique and does not depend on the elimination algorithm.This well-known connection is indeed a bijection, which we denote by This observation is capable of easily explaining many properties of G q (n, k) so that we commonly identify the elements of G q (n, k) with their corresponding generator matrices in reduced row echelon form.Given a matrix A ∈ F k×n q of full rank we denote by p(A) ∈ F n 2 the binary vector whose 1-entries coincide with the pivot columns of A. For each v ∈ F n 2 let EF q (v) denote the set of all k × n matrices over F q that are in reduced row echelon form with pivot columns described by v, where k is the weight of v.
where the s represent arbitrary elements of F q , i.e., |EF q (v)| = q 4 .In general we have and the structure of the corresponding matrices can be read off from the corresponding (Echelon)-Ferrers diagram where n k q is called Gaussian binomial coefficient.Later on we will use the inverse operation of deleting the pivot columns of a matrix in RRE form: Definition 2. Let B ∈ F k×n q be a full-rank matrix in RRE form and F ∈ F k ×(n−k) q be arbitrary, where k, k , n ∈ N and k ≤ n.Let further f i denote the ith column of F for i ∈ {1, . . ., n}.Then, G = ϕ B (F ) denotes the k × n matrix over F q whose columns are given by g i = 0 ∈ F k q if v i = 1 and g i = f i−si otherwise, where (v 1 , . . ., v n ) = p(B) and

MRD codes and the Echelon-Ferrers construction.
For matrices A, B ∈ F m×n q the rank distance is defined via d R (A, B) := rk(A − B).It is indeed a metric, as observed in [14].The maximum possible cardinality of a rank-metric code with given minimum rank distance is exactly determined in all cases.
Theorem 4. (see [14]) Let m, n ≥ d be positive integers, q a prime power, and C ⊆ F m×n q be a rank-metric code with minimum rank distance d.Then, |C| ≤ q max(n,m)•(min(n,m)−d+1) .Codes attaining this upper bound are called maximumrank distance (MRD) codes.They exist for all (suitable) choices of parameters.
If m < d or n < d, then only |C| = 1 is possible, which can be combined to give a single upper bound |C| ≤ q max(n,m)•(min(n,m)−d+1) .Using an m × m identity matrix as a prefix, one obtains the corresponding subspace codes known as lifted MRD codes.
The subspace distance between two subspaces with the same pivots can be computed by the rank distance of the corresponding generator matrices.Using τ from (1), we have: So, in order to construct an (n, M, 2δ; k) code, it suffices to select a subset of EF q (v) with minimum rank distance δ.Additionally, we can further expand such a code by introducing codewords with different pivot columns as long as the sets of pivot columns are sufficiently apart.Let Having Lemma 1 and Lemma 2 at hand, the Echelon-Ferrers construction from [11] works as follows: For two integers k and δ choose a binary constant weight code S of length n, weight k, and minimum Hamming distance 2δ as a so-called skeleton code.For each s ∈ S construct a code C s ⊆ EF q (s) having a minimum rank distance of at least δ.Setting C = ∪ s∈S C s yields an (n, M, 2δ; k) code, where M = s∈S |C s |.We remark that Lemma 2 does not need two binary vectors v, v of the same weight, i.e., the very same approach can be used to construct general subspace codes in which the codewords may have different dimensions.The only necessary modification is to choose a general binary code S of length n and minimum Hamming distance d as skeleton code.The codes C s need to have a rank distance of at least d/2.
For a given binary vector v ∈ F n 2 and an integer 1 ≤ δ ≤ n let q dim(v,δ) be the largest cardinality of a linear rank-metric code over EF q (v) with rank distance at least δ.Theorem 6. ([11, Theorem 1]) For a given i, 0 ≤ i ≤ δ − 1, if ν i is the number of dots in the Echelon-Ferrers diagram corresponding to v, which are not contained in the first i rows and not contained in the rightmost δ − 1 − i columns, then min i {ν i } is an upper bound of dim(v, δ).
The conjecture that the upper bound of Theorem 6 can be obtained for all parameters is still unrefuted and valid in many cases, see [10].Several of the currently best known lower bounds for constant dimension codes are obtained via the Echelon-Ferrers construction.We remark that for the special binary vector v = (1, . . ., 1, 0, . . ., 0) of length n and weight k, the rank-metric codes of maximum cardinality in EF q (v) are given by lifted MRD codes, see Theorem 5. So, the Echelon-Ferrers construction uses building blocks that are lifted MRD codes with a prescribed structure.It is possible to improve the best currently known upper bounds on A q (n, d; k) for constant dimension codes that contain a lifted MRD code.Theorem 7. (see [12,Theorems 10 and 11]) Let C ⊆ G q (n, k), where n ≥ 2k, with minimum subspace distance d contain a lifted MRD code. •

2.3.
Parallelisms and packings of G q (n, k).Let X be a set.A packing P = {P 1 , . . ., P l } of X is a set of subsets P i ⊆ X such that P i ∩ P j = ∅ for all 1 ≤ i < j ≤ l, i.e., the subsets P i are pairwise disjoint.A point is an element of G q (n, 1) and a spread is a subset of G q (n, k) that partitions the corresponding set of points, i.e., the elements have a pairwise trivial intersection.Counting the points yields that the size of a spread is Parallelisms in G q (n, k) are known to exist for: (1) q = 2, k = 2 and n even; (2) k = 2, all q and n = 2 m for m ≥ 2; (3) n = 4, k = 2, and q ≡ 2 (mod 3); (4) q = 2, k = 3, n = 6, see e.g.[13].

The coset construction
Construction III in [12] gives A 2 (8, 4; 4) ≥ 4797.While this specific construction does not involve parameters, the authors conjecture that the underlying idea can be generalized to further parameters assuming the existence of a corresponding parallelism.In Theorem 9 in Subsection 5.1 we will show that this is indeed the case.Moreover, there is a more general underlying construction, introduced as coset construction in this paper, that yields improvements of the best-known lower bounds for constant dimension codes, see Section 5.
The main idea of the coset construction is to use a collection of codewords which will be part of a subspace code such that τ form (1), i.e., the corresponding RRE form, of each element of this collection is of the form Here, A is the RRE form of a k -dimensional subspace in F n q and B is the RRE form of a k − k -dimensional subspace in F n−n q , so that we obtain a RRE form of a k-dimensional subspace C(A, B, F ) of F n q .Note that the integers k and n are respectively the same for any codeword in this collection although Lemma 6 allows to combine multiple such collections.F is an arbitrary k × (n − n − k + k ) matrix over F q , in which ϕ B inserts zero columns at the pivot positions of B, see Subsection 2.1 for the precise definition of ϕ B and an example.In C(A, B, F ), the vectors have the shape (λ • A, λ • F + µ • B).So λ • F is the offset for the coset of the suffixes, i.e., the vector λ • A is prefix for every vector in the coset λ • F + B, explaining the naming of our construction.In order to obtain a constant dimension code with large minimum subspace distance, the matrices A, B, and F , as well as their combinations, are chosen from certain sets.Using τ from (1), we have: Lemma 3. (Coset construction) Let q be a prime power and n, k, n , k ∈ N satisfy is a subset of G q (n, k), i.e., a constant dimension code where the codewords have dimension k.
Proof.For an arbitrary but fixed index 1 ≤ i ≤ l let A, B be matrices with . The dimensions fit so that Moreover ϕ B (F ) has zero columns at the positions of the pivot columns of B. Since A has k and B has k − k pivot columns, M has exactly k pivot columns and full rank.Thus, The number l of disjoint subsets for A and B is called the length of the specific coset construction.We remark that we have excluded the ranges for the parameters k , n where the construction would be degenerated in the sense that either A or B have to be empty matrices.Nevertheless, the degenerate case k = k has a nice interpretation.Here B is an empty matrix and A is a k × n matrix.If additionally n = k then A is an identity matrix and we are in the case of lifted MRD codes.Using τ from (1), we have: Lemma 4. Let q, n, k, n , k be parameters satisfying the conditions from Lemma 3, A, A ∈ F k ×n q and B, B ∈ F be full-rank matrices in RRE form.Let further d be a positive integer and The proof is rather technical and can be found in the appendix.We remark that condition (2) of Lemma 4 is trivially satisfied for the special case of distance d = 4, if A = A and B = B .
Next we demonstrate that the coset construction from Lemma 3 can in general not be obtained by an application of the Echelon-Ferrers construction.(For a more explicit example, see Theorem 13 in Subsection 5.3.)It is easy to construct a family of examples with subspace distance d but whose pivot vectors have Hamming distance 2, so that they cannot be used in the Echelon-Ferrers construction.To this end, let q be an arbitrary prime power, d an even integer ≥ 2, and n, k, n , k For the sake of this example we use: , where I denotes the identity matrix.Then, for arbitrary

3.1.
A multilevel coset construction.In this subsection we want to use the coset construction in combination with other constructions.At first we show that it is compatible with the Echelon-Ferrers construction.Using τ from (1), we have: Lemma 5.For a prime power q and n, k, n , k , , and X ∈ G q (n, k).Let s be the sum of the first n entries in the pivot vector p(X) of X, i.e., s := Proof.Let x := p(X) and w := p(W ) be the pivot vectors of X and W , respectively.
From the construction we know Applying Lemma 2 yields the stated lower bound on the subspace distance.
For the special case k = k, i.e., the constant dimension case, we have There is also an easy-to-check sufficient criterion to determine whether the union of two codes constructed by the coset construction have a subspace distance of at least d.Lemma 6.Let C i be codes having subspace distance at least d and that are obtained from the coset construction with suitable parameters n, k i , n i , and k i for i = 1, 2, where we assume Proof.At first we observe that we have We set x := p (W 1 ) and y := p (W 2 ), where the W i are matrices corresponding to an arbitrary but fixed codeword from C i , see the formulation of Lemma 5.
Let x 1 consist of the first n 1 entries of x, y 1 consist of the first n 1 entries of y, x 2 consist of the last n − n 1 entries of x, and y 2 consist of the last n − n 1 entries of y.For m := y 1  1 , where Applying Lemma 2 yields the stated lower bound on the subspace distance.
Considering the exemplary parameters n = 6, This lower bounds any two codewords from two coset constructed parts having these parameters, whereas the Hamming distance of the depicted pivot vectors is 2.
We remark that Lemma 6 is best possible in the sense that the estimations on the Hamming distance of two binary vectors with known weights and weights of two suffixes, of possibly different lengths, is tight.Performing similar analyses on generalized structures like may have the potential to yield stronger bounds.

4.
Optimal choices for the parameters of the coset construction 4.1.General reasoning.Like the Echelon-Ferrers construction, the coset construction from the previous section is far from being explicit, i.e., there are several degrees of freedom.In this section we give several lower and upper bounds for the sizes of the codes obtained from the coset construction, which allow to minimize the range of choices of the parameters that can lead to improvements of the best-known bounds.
The cardinality of a subspace code obtained from the coset construction with length l is given by Given q, n, and the desired even subspace distance d, the aim is to maximize (3) under the restrictions of Lemma 4. Obviously, this term is maximal if both F and the sum are maximal.Thus, we may choose an MRD code, with appropriate parameters, for F , so that is optimal by Theorem 4.
The sets A i and B i need to have additional structure.
Lemma 7.For a code obtained from the construction of Lemma 3 with d := D S C (A i ) i , (B i ) i , F , length l, and parameters q, n, k, n , k we have A similar conclusion can be drawn for the elements in B i .
From this we can conclude an upper bound on Λ.
Corollary 1.Using the notation from Lemma 3 and Equation (3) we have Proof.Due to Lemma 7 we have Interchanging the roles of the A i and B i yields the other stated upper bound.
Corollary 2. The upper bound of Corollary 1 can be attained if d ≤ 4 and both G q (n , k ) and G q (n − n , k − k ) admit parallelisms, e.g., the corresponding parameters are in the list in Subsection 2.3.
The dependency between the cardinalities of the A i and B i in optimal solutions of ( 3) is already decoupled to some extent, but we can even do more.Lemma 9.For a code obtained from the construction of Lemma 3 with d := D S C (A i ) i , (B i ) i , F , length l, and parameters q, n, k, n , k , then for each permutation σ : {1, . . ., l} → {1, . . ., l} we have Proof.Apply Lemma 4.
The question which permutation σ of Lemma 9 maximizes the crucial parameter Λ can be answered easily.
For each permutation σ : {1, . . ., l} → {1, . . ., l}, we have Proof.For integers a > a and b < b we have Having these ingredients at hand we can generalize and improve the upper bound from Corollary 1 using the analytical solution of another optimization problem.Lemma 11.Let α, β, α, β, and l be positive integers with α, β ≥ l.An optimal solution of the non-linear integer programming problem This is done in the formula of case (2).The underlying idea is the following: Start with b i = 1 for all 1 ≤ i ≤ l; observe β ≥ l.Then fill up the b i with increasing indices up to β as long as the sum does not violate β.Observe that every (integer) vector (b i ) with l i=1 b i = β gives the same target value.Case (3) describes the symmetric situation.It remains to assume α • l > α and β • l > β.Let âi , bi be an optimal solution of our initial optimization problem where we assume â1 ≥ • • • ≥ âl and b1 ≥ • • • ≥ bl .Let further f be the smallest index such that âf < α and r be the largest index such that âr > 1.If either r does not exist or f = r (f exists due to ᾱ • l > α), then the solution âi has the shape described in case (4).But, for f < r we could improve the target value by so that such a case could never produce an optimal value and so our solution must have the shape described in case (4).The same reasoning applies for the bi .
Lemma 12. Using the notation from Lemma 3 and Equation (3) we have where the a i , b i are given by Lemma 11 for Proof.From Lemma 8 we conclude |A| ≤ A q (n , d ; k ) and The possible values for the length l are part of the stated optimization formulation.For each index 1 It remains to check that we can apply Lemma 11.
Fixing the parameter d from Lemma 8 one can state a lower bound on the maximal value of Λ in terms of the sizes of lifted MRD codes (cf.Theorem 5).
for, with respect to Lemma 3, feasible parameters q, n, k, n , k , d. M (q,k ,n ,d) cosets and for S B there are exactly M (q,k−k ,n−n ,d−d ) Combining a lifted MRD code with a code constructed from Lemma 13 yields a (9, 1032, 6; 4) 2 code, which improves on the previously best-known codes, see Subsection 5.2.
We can formulate the following greedy-type algorithm to construct sequences A i and B i that yield a "reasonable" lower bound on Λ.
Unfortunately, this algorithm is not capable of determining the optimal Λ in general.If we use E := {all constant dimension codes in G q (ñ, k) with subspace distance d} as ground set and I := {disjoint subsets of E} as independent sets, then this does not form a matroid and hence a greedy algorithm will not yield an optimal solution in general, see e.g.[8].To be more precise, the independent set exchange property fails: Use for example U = V ∈ G q (ñ, k) with d S (U, V ) ≥ d, A := {{U }, {V }} ∈ I and B := {{U, V }} ∈ I.Although A is larger than B we cannot add an element of A to B without losing the independence.
Setting the length l of the coset construction to l := min {l A , l B }, we observe that trying to maximize the cardinalities |A i | or |B i | for i > l has no benefit, so that we may simply complete a given packing by singletons.Or, in other words, we directly start from packings within A and B.
However, the design of suitable A i is not that obvious since the Λ-part of the target function (3) comprises a non-linear integer optimization problem.Ignoring almost all of the geometric restrictions from P q (n), we are able to exactly solve the mentioned optimization problem in Lemma 11.In general this gives us an upper bound only.To obtain tighter bounds one has to go a bit more into the details.In Lemma 12 we have only used the implication For a given A we may be able to determine tighter bounds on the cardinalities of the A i s.Since the only change in the setting is the exclusion of the possible codewords in G q (n , k )\A this subproblem can be formulated as an independent set problem and be solved using several algorithmic approaches, see e.g.[18].We will present an explicit example of this technique in Subsection 5.3.
Having candidates for the A i at hand, it still remains to select a subset of the candidates that are pairwise disjoint.This subproblem can also be formulated as a (restricted) independent set problem of a, possibly large, graph G = (V, E).To this end, let κ be a suitable upper bound on the cardinalities of the |A i | and S i be the set of subsets of A of cardinality i having a subspace distance of at least d.Setting S = ∪ 1≤i≤κ S i one can consider the optimization problem max for a given number l of parts of the desired packing.Notwithstanding that the target function of ILP formulation (4) completely ignores the correlation with the sizes of the items of the second packing on Λ, it can be used to determine the exact value of Λ in special cases, see Subsection 5.3.Setting the vertex set of our graph G to V = S and taking edges e = {s 1 , s 2 } ∈ E iff s 1 ∩ s 2 = ∅, this corresponds to a vertex-weighted independent set problem with an additional restriction on the number of chosen vertices.The algorithmic approaches described in [18] can be adopted easily for these extra requirements.
Since the two subproblems from this subsection on their own even might be too hard, we may apply heuristic approaches only.The very successful approach of prescribing automorphisms can also be applied here.Here the prescribed subgroup of automorphisms has to be a subgroup of the automorphism group of A which typically is much smaller than GL(n, q).However, "good" codes often have nontrivial automorphism groups.
As already observed in [12], the crucial ingredient for the feasibility of the above construction is the existence of a parallelism in G q (4, 2).Performing the above cardinality computations for arbitrary q we obtain A q (8, 4; 4) ≥ q 12 + 4 2 q (q 2 + 1)q 2 + 1, which also attains the MRD bound from Theorem 7.
The authors of [12] have remarked that they believe that their construction from their Construction III can be generalized to further parameters assuming the existence of a corresponding parallelism.This is indeed the case.Theorem 9.If P 1 is a parallelism in G q (n , k ) and P 2 a parallelism in G q (n − n , k−k ), then we can choose A = P 1 , B = P 2 , and d = 4 in the coset construction.The corresponding code C attains the upper bound of Corollary 1.If additionally k − k ≥ 2 and n − k ≥ 2, then C is compatible with the lifted MRD code having pivot vector (1, . . ., 1 k , 0, . . ., 0). 5.2.n = 9, d = 6, k = 4, and general field sizes q.Since the combination of the MRD code C 1 with pivot vector v = (1, 1, 1, 1, 0, 0, 0, 0, 0) and cardinality 1024 with the code C 2 obtained from the explicit construction of Lemma 13 of cardinality 8 yields a (9, 1032, 6; 4) 2 constant dimension code whose cardinality is one less than the MRD bound from Theorem 7, we were motivated to look for a coset construction yielding a larger addendum than 8.
Proof.We choose n = 4, k = 1, and d = 2 in the coset construction.For the choice of A and B we observe A q (4, 2; 1) = q 3 + q 2 + q + 1 and A q (5, 4; 3) = A q (5, 4; 2) = q 3 + 1, see e.g.[5].Choose A and B as arbitrary codes attaining the mentioned upper bounds.Choosing a trivial packing of B into singletons yields a code C of cardinality q 3 +1.Adding the lifted MRD code of size q 10 gives the stated upper bound.
We remark that the codes from Theorem 10 meet the MRD bound from Theorem 7. The underlying construction can be generalized even more.Theorem 11.For each k ≥ 4 and arbitrary q we have A q (3k − 3, 2k − 2; k) ≥ q 4k−6 + q k−1 + 1.
Proof.We choose n = k, k = 1, and d = 2 in the coset construction.For the choice of A and B we observe A q (k, 2; 1) = k 1 q and A q (2k − 3, 2k − 4; k − 1) = A q (2k − 3, 2k − 4; k − 2) where the first equality is true by considering the so-called complementary subspace code C ⊥ = {U ⊥ | U ∈ C}, cf.[17].Choose A and B as arbitrary codes attaining the mentioned upper bounds.Choosing a trivial packing of B into singletons yields a code C of cardinality q k−1 + 1. Adding a (k × (3k − 3)) lifted MRD code gives the stated lower bound.
We remark that the codes from Theorem 11 meet the MRD bound from Theorem 7. 5.3.n = 10, d = 6, k = 4, and q = 2.For the coset construction we choose n = 4 and k = 1.Since A ⊆ G 2 (4, 1) we can only have D S (A i ) = 2, so that we must choose d = 2.Then, we can choose A = G 2 (4, 1) and 4  1 2 = 15 singletons A i , which is obviously best possible.For B ⊆ G 2 (6, 3) we have the condition D S (B) ≥ 4. Reasonable candidates for B might be the five isomorphism types of (6, 77, 4; 3) 2 codes attaining the maximum cardinality A 2 (6, 4; 3) = 77, see [16].Using the first subproblem from Subsection 4.2 we computationally obtain the upper bound |B i | ≤ 5 =: κ for four out of the five isomorphism types.This information is enough to conclude the upper bound Λ(B) ≤ 15 • 5 = 75.For the remaining isomorphism type, i.e., the self-dual code having 168 automorphisms which was labeled as "type A", we have |B i | ≤ 7 =: κ.So, we solve the optimization problem (4) for l = 15.The sizes of the requested sets S I are stated in Table 2.The optimal target value is 76 and there exists a solution where the sizes of the elements in the packing are given by 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7. Since in our situation we have |A i | = 1 for all i, the target function of (4) coincides with the expression for Λ.Also the predefinition of l = 15 results in the maximum possible value, since we have l ≤ 15 from the A-part and the existence of a packing of B into l sets implies the existence of packings into l ≥ l sets.In general it is far from being obvious that we obtain the best possible codes from the coset construction by choosing codes for B that have the maximal possible cardinality A q (n − n , d; k − k ).However, in our situation each choice for B different from the five considered isomorphism types of (6, 77, 4; 3) 2 codes has a cardinality of at most 76, so that i |A i | • |B i | ≤ 76.
We remark that the code from Theorem 13 meets the MRD bound from Theorem 7. By an exhaustive search we have verified that the general Echelon-Ferrers construction yields only codes with |C| ≤ 4167.

Conclusion
The arguably most successful generally applicable construction for both constant dimension and subspace codes of large minimum subspace distance is the Echelon-Ferrers construction from [11].Here, we have introduced a generalization of [12, Construction III], which we call coset construction.It turned out that the new construction is provably superior to the Echelon-Ferrers construction for some parameters, see Subsection 5.3.We were able to apply the coset construction to an infinite family of constant dimension codes that attain the MRD bound from [12,Theorem 11].So far all improvements include the usage of a lifted MRD code of maximal shape, so that these approaches are all limited by the MRD bound from Theorem 7.For the relatively small addendums constructed by the coset construction, we may utilize subcodes that have a larger cardinality than the corresponding value of the MRD bound, see Subsection 5.3.The constructions of subspace codes based on the coset construction typically should yield many non-isomorphic codes, since there are already many non-isomorphic MRD codes, see e.g.[4,20].In Section 4 we have obtained some first insights on the optimal choice of parameters for the coset construction and related optimization problems.However, we are rather

Lemma 8 .
For a code obtained from the construction of Lemma 3 with d := D S C (A i ) i , (B i ) i , F , length l, and parameters q, n, k, n , k , there exists an integer d such that D S (A) ≥ d and D S (B) ≥ d − d , where A = ∪ i A i and B = ∪B i .Proof.Let U, U ∈ A with d S (U, U ) = D S (A) =: d and V, V ∈ B with d S (V, V ) = D S (B) =: d .W.l.o.g.we can assume that F contains the zero matrix, since the rank distance is invariant with respect to translations.Choosing F = F = 0 we can conclude d ≥ d − d from Inequality (2).In later applications we will commonly assume 2 ≤ d ≤ d − 2, since the other values lead to trivial cases where either |A| = 1 or |B| = 1.

+ a 2 • b + b 2 =
Proof.W.l.o.g.we can additionally assume a 1 ≥ • • • ≥ a l and b 1 ≥ • • • ≥ b l without decreasing the maximal target value of the optimization problem.Let us allow a i , b i ∈ R for a moment, i.e., we consider the standard relaxation, and denote a corresponding optimal solution by ãi , bi ∈ R ≥1 .For non-negative real numbers a ≥ a and b ≥ b we have (a b + a b ) − 2 • a (a − a ) • (b − b ) 2 ≥ 0, so that we can assume ãi = ãj =: ã and bi = bj =: b, for all 1 ≤ i, j ≤ l, w.l.o.g.Either we have lã = α or ã = α, since otherwise we could slightly increase ã and improve the target value.The same reasoning applies to b.If ã = α and b = β, then we are in case (1).Next we consider the case where ã = α and b < β so that b = β/l.Since l i=1

Proof.
Similar to the proof of [12, Lemma 5], we consider A as a linear MRD code with parameters k × n with distance d and B as a linear MRD code with parameters (k − k ) × (n − n ) with distance d − d .Let S A be a linear MRD code with parameters k × n with distance d > d and S B be a linear MRD code with parameters (k − k ) × (n − n ) with distance d > d − d .We choose the A i as the cosets of S A in A and B i as the cosets of S B in B. For S A there are exactly M (q,k ,n ,d )

4. 2 .
Decomposing constant dimension codes.Due to Lemma 8 we can construct the necessary parts of the coset construction of Lemma 3 starting from constant dimension codes A and B with D S (A) ≥ d and D S (B) ≥ d − d .The aim is to partition the codewords of A into subcodes |S i | = 77 840 2240 1792 560 112 16 In the case when A = A and B = B we concluded S τ −1 A ϕ B ((A) + rk ϕ B (F ) − ϕ B (F ) B − kSince the pivot columns of B in ϕ B (F ) − ϕ B (F ) consists solely of zeros, we have2 rk(A) + rk ϕ B (F ) − ϕ B (F ) B − k = 2(rk(A) + rk(ϕ B (F ) − ϕ B (F )) + rk(B) − k) = 2(k + rk(F − F ) + k − k − k) = 2 rk(F − F ) = 2d R (F, F ).For A = A or B = B we similarly concluded S τ −1 A ϕ B (rk(X) + rk(Z)with equality if Y is zero and swapping rows or columns, respectively, does not change the rank.d S (B, B ) 2 + k − k − k = d S (A, A ) + d S (B, B ).
where the pivot columns and zeros are omitted and the stars are replaced by solid black circles.A Ferrers diagram represents partitions as patterns of dots, with the ith row having the same number of dots as the ith term s i in the partition n = s 1 + • • • + s l , where s 1 ≥ • • • ≥ s l and s [3] N >0 , cf.[3].Usually a Ferrers diagram is depicted in such a way that it is the vertically mirrored version of the above constructed (Echelon)-Ferrers diagram.In the special case of Echelon-Ferrers diagrams, we have n= n i=1 (1 − v i ) • i j=1 v j .By summing over all binary vectors of weight k in F n 2 one can compute