Generalized LMRD Code Bounds for Constant Dimension Codes

In random network coding so-called constant dimension codes (CDCs) are used for error correction and detection. Most of the largest known codes contain a lifted maximum rank distance (LMRD) code as a subset. For some special cases, Etzion and Silberstein have demonstrated that one can obtain tighter upper bounds on the maximum possible cardinality of CDCs if we assume that an LMRD code is contained. The range of applicable parameters was partially extended by Heinlein. Here we fully generalize those bounds, which also sheds some light on recent constructions.


I. INTRODUCTION
L ET V ∼ = F v q be a v-dimensional vector space over the finite field F q with q elements. By V k we denote the set of all k-dimensional subspaces in V , where 0 ≤ k ≤ v. The size of the so-called Grassmannian V k is given by the More generally, the set P (V ) of all subspaces of V forms a metric space with respect to the subspace distance defined by Coding theory on P (V ) is motivated by Kötter and Kschischang [8] via random network coding. For C ⊆ V k we speak of a constant dimension code (CDC). By a (v, N, d; k) q code we denote a CDC in V with minimum (subspace) distance d and cardinality N . The corresponding maximum size is denoted by A q (v, d; k). In geometrical terms, a (v, N, d; k) q code C is a set of N k-dimensional subspaces of V , k-spaces for short, such that any (k − d/2 + 1)-space is contained in at most one element of C. In other words, each two different codewords intersect in a subspace of dimension at most k − d/2. For two k-spaces U and W that have an intersection of dimension zero, we will say that they intersect trivially or are disjoint (since they do not share a common point, i.e., a 1-space). For the known lower and upper bounds on A q (v, d; k) we refer to the online tables http://subspacecodes.uni-bayreuth.de associated with the survey [5].
If a CDC contains an LMRD, see Section II for the definition, then the best known upper bound on the cardinality for the general case can be improved. Corresponding results have been obtained in [2], [4] for a restricted range of parameters. Here we remove the restriction and generalize those bounds to all parameters. To this end, we consider the so-called Anticode bound, which counts t-spaces that are contained in at most one codeword. We refine the approach by splitting the counts according by the dimension of the intersection with the special subspace that is disjoint to all codewords of the LMRD. This gives an integer linear programming problem, see Lemma 6, from which we conclude an explicit upper bound, see Corollary 7. Technically, we prove those results for the maximum number B q (v 1 , v 2 , d; k) of k-spaces in F v1 q with minimum subspace distance d such that there exists a v 2 -space W which intersects every chosen k-space in dimension at least d/2, which is more general.

II. PRELIMINARIES
In the following we will mainly consider the case V = F v q in order to simplify notation. We associate with a subspace U ∈ V k a unique k × v matrix X U in row reduced echelon form (rref) having the property that X U = U and denote the corresponding bijection For two matrices A, B ∈ F m×n q we define the rank distance Theorem 1 (See [3]): Let m, n ≥ d be positive integers, q a prime power, and M ⊆ F m×n q be a rank metric code with minimum rank distance d . Then, #M ≤ q max{n,m}·(min{n,m}−d +1) . Codes attaining this upper bound are called maximum rank distance (MRD) codes. They exist for all choices of parameters. Using an m × m identity matrix I m×m as a prefix one obtains the so-called lifted MRD (LMRD) codes, i.e., the CDC where (B|A) denotes the concatenation of the matrices B and A.
which is known as the Anticode bound. Analyzing the right hand side we obtain We will also need to count the number of subspaces with certain intersection properties, see e.g. [6,Lemma 2]: and As shown in e.g. [8,Lemma 4] we have III. BOUNDS FOR CDCS CONTAINING AN LMRD SUBCODE Before we consider upper bounds we start with the constructive point of view.
The (v − m)-space W whose pivots in τ (W ) are in the last v − m coordinates is disjoint from all elements from C . Now let C ⊆ F v q k be a CDC with minimum subspace distance d such that every codeword intersects W in dimension at least d/2, which has the maximum possible cardinality.
For each U ∈ C and each U ∈ C we have We remark that the construction of CDC C is called and d > 2k, then we also have B q (v 1 , v 2 , d; k) = 0. We will call those parameters trivial. For (implicit) lower bounds for B q (v 1 , v 2 , d; k) we refer to [1], [11] and the references cited therein.
By refining the counting of (k − d/2 + 1)-spaces contained in codewords, underlying the presented argument for the Anticode bound, we obtain: Lemma 6: As an abbreviation we set For non-trivial parameters we have where the a i are non-negative integers satisfying the constraints for all 1 ≤ j ≤ min{t, v 2 } and min{k,v2} and C be a set of k-spaces in V that intersect W in dimension at least d 2 and has minimum subspace distance d. By a i we denote the number of elements in C that have an intersection of dimension exactly i with W , so that We note that every t-space is contained in at most one element from C.
Let 1 ≤ j ≤ min{t, v 2 } be arbitrary. First we count the number of t-spaces T in V such that dim(T ∩ W ) = j.
possibilities, which is the right hand side of Inequality (5). Now, consider a codeword U ∈ C with intersection dimension i = dim(U ∩ W ). Next we want to count those t-spaces T ≤ min{i, t}). Since each such t-space T is contained in at most one codeword U ∈ C, we obtain Inequality (5).
Given an integer max{t, d/2} ≤ h ≤ min{k, v 2 } we construct a CDC consisting of h-spaces from C. To this end, ; h) q code and we obtain (6).
For given parameters v 1 , v 2 , d, and k we can easily turn Lemma 6 into an integer linear programming formulation and solve it numerically. We can also conclude an explicit parametric upper bound: Corollary 7: For non-trivial parameters we have Proof: We apply Lemma 6 and use the corresponding notation, i.e., we will use and upper bound the right hand side. For k < d we have d/2 ≥ k − d/2 + 1 = t so that we can apply Inequality (6) with h = d/2 to conclude the proposed upper bound for Λ = 1.
In the following we assume k ≥ d, i.e., Λ ≥ 2. From Equation (2) and Equation (3) we conclude i.e., the sequence (b i,j ) i is weakly monotonic increasing. Next we want to apply Inequality (5) for special values of j. To this end, we use the parameterization j = ld/2 for 1 ≤ l < Λ. Here we note that max{d/2, j} = ld/2, due to l ≥ 1, and min{k, d/2 − 1 + j} = (l + 1)d/2 − 1, due to l ≤ 2k/d − 1. With this, we have for j = ld/2, where the latter inequality follows from a i ≥ 0 and the monotonicity of (b i,j ) i . Thus, we conclude from Inequality (5) for 1 ≤ l < Λ and j = ld/2. Dividing Inequality (7) by b(ld/2, ld/2) gives we can add the right hand side of Inequality (9) to the sum over the right hand side of Inequality (8) for 1 ≤ l < Λ to conclude the proposed upper bound. Note that the sum of the corresponding left hand sides equals Applying Theorem 5 with m = k gives a (v, , d; k) q code C with cardinality Under the assumption that C contain a lifted MRD code as a subcode this is indeed the maximum possible cardinality: Proposition 8: Let v, k, and d/2 be positive integers with d ≤ 2k ≤ v and C be a (v, , d; k) q code that contains a lifted MRD code C of cardinality q (v−k)·(k−d/2+1) as a subcode. Then, we have Proof: Let W be the (v − k)-space that is disjoint from all codewords of C . From e.g. [2,Lemma 4] we know that every (k − d/2 + 1)-space that is disjoint to W is contained in a codeword from C . Thus, the codewords in C\C have to intersect W in dimension at least d/2.
Corollary 9: Let v, k, and d/2 be positive integers with d ≤ 2k ≤ v and C be a (v, , d; k) q code that contains a lifted MRD code C of cardinality q (v−k)·(k−d/2+1) as a subcode. Next we show that the upper bound of Corollary 7 for B q (v 1 , v 2 , d; k) is tight for k < d, i.e., those cases where the bound does not depend on v 1 , provided that v 1 is sufficiently large.
Since d ≥ 4 and k ≥ 1 the right hand side of (10) is at most v 2 k, so that #P ≥ #F . For each U ∈ F we can choose a different element f (U ) ∈ P and set C = {U × f (U ) | U ∈ F}, which has the desired properties of Definition 4 by construction.