An Assmus-Mattson Theorem for Rank Metric Codes

A t-(n,d,λ) design over F_q, or a subspace design, is a collection of d-dimensional subspaces of F_q^n, called blocks, with the property that every t-dimensional subspace of F_q^n is contained in the same number λ of blocks. A collection of matrices in over F_q is said to hold a subspace design if the set of column spaces of its elements forms the blocks of a subspace design. We use notions of puncturing and shortening of rank metric codes and the rank-metric MacWilliams identities to establish conditions under which the words of a given rank in a linear rank metric code hold a subspace design.

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Introduction

The celebrated Assmus-Mattson Theorem [2] is among the best-known results in the theory of codes and designs. It has led to several constructions of -designs [2, 3, 14, 21, 26] and has been the subject of a number of generalizations [4, 15, 16, 27, 34].

The theorem specifies criteria under which the blocks of a combinatorial design are held by the words of a fixed weight in a linear code. It shows, in particular, that if the dual of a code has few non-zero weights in an interval, then the supports of codewords of fixed Hamming weight close enough to the minimum distance of the code form the blocks of a design.

The notion of a -design over (or a subspace design) has appeared in the literature since the 1970s [13], and various constructions of such objects were given in the past [22, 35, 32]. A - design over is a collection of -dimensional subspaces of , called blocks, with the property that every -dimensional subspace of is contained in the same number of blocks. There was a resurgence of interest in such designs during the last decade, due in part to the fact that some subspace designs are optimal as constant weight subspace codes. Subspace codes have been shown to have applications to error correction in network coding [24]. It was shown in [20] that non-trivial subspace designs exist over any finite field and for every value of , as long as the parameters are sufficiently large. There are now several papers showing the existence of such objects for various parameter sets, most of which rely on assumptions of their automorphism groups [6, 7, 8, 9, 10].

In this paper we give conditions under which a linear rank metric code yields a -design over . Explicitly, we show that if and its dual are and -invariant, respectively, and has at most non-zero weights in , then the codewords of and of ranks and , respectively, hold -designs over . This result offers a new approach to the difficult problem of constructing new subspace designs and a motivation to study new classes of rank metric codes. We also consider existence problems applying recent results on the covering radius and the external distance of codes. These arguments apply to both the rank and the Hamming metric.

In Section 2 we give preliminary results on rank metric codes and on subspace designs. In Section 3 we present our main result, the rank-metric analogue of the Assmus-Mattson Theorem. In Section 4 we restrict to the class of - linear codes, which are always -invariant. We show that the maximum rank distance (MRD) codes, those with parameters -, are characterized as precisely the codes whose minimal weight codewords hold trivial -designs. We show, furthermore, that the codes that are linear only over do not satisfy this property. In Section 5 we discuss asymptotic existence of codes satisfying the hypothesis of the Assmus-Mattson Theorem, for both the Hamming and the rank metric.

1. Preliminaries

1.1. Matrix Rank Metric Codes

We outline some preliminary results on rank metric codes. Throughout the paper, denotes a fixed prime power and are integers with . We let be the finite field with elements.

Given a pair of nonnegative integers and , the -binomial or Gaussian coefficient counts the number of -dimensional subspaces of an -dimensional subspace over and is given by:

 [NM]q:=M−1∏i=0qN−qiqM−qi.

For a subspace of we define to be the orthogonal space of with respect to the standard inner product .

Definition 1.

The rank distance between a pair of matrices is defined to be the rank of their difference:

 drk(X,Y):=rk(X−Y).
Definition 2.

A (linear) matrix code is an -subspace of . The minimum distance of is

 drk(C):=min{drk(X,Y):X,Y∈C, X≠Y}=min{rk(X):X∈C, X≠0}.

We say that is an - code if it has -dimension and minimum rank distance . The code denotes the dual code of with respect to the bilinear form on defined by , that is,

 C∗:={Y∈Fn×mq:Tr(XY⊤)=0 for all X∈C}.
Definition 3.

Let be an - matrix code. We define

 Wi(C):=|{X∈C:rk(X)=i}|,

which is the number of codewords of of rank . The (rank) weight distribution of is the sequence . We say that an integer is a non-zero weight of if .

The weight distributions of a linear matrix code and its dual are related by the rank metric MacWilliams identities [16]. We will use the following formulation from [28, Theorem 31].

Theorem 4.

Let be an - matrix code. Then for each we have:

 n−ℓ∑i=0Wi(C∗)[n−iℓ]q = (1)

The left-hand side of (1) corresponds to multiplication of the weight distribution of by the upper triangular -Pascal matrix

 M=([n−iℓ]q)0≤i,ℓ≤n.

The minors found in the first rows of can be expressed as

 det([rij−1]q)1≤i,j≤ℓ=q(ℓ2)∏1≤i

for some , (see, for example, [25]).

Clearly the determinant shown above is non-zero if the are all distinct and in particular if , for distinct . We will use this to establish invariance of the weight distribution of certain puncturings of a code whose dual code has few non-zero weights.

In the construction of a design over from a set or space of matrices, we will identify the blocks of the design with the column spaces of the matrices. This is the matrix analogue of the notion of the support of a vector.

Given a matrix over , we write to denote its column space, which we call the support of . In fact this concept can be viewed in the more general lattice-theoretic setting of additive codes in groups [29].

Definition 5.

Let be an - code. Let be a subspace of . We define

 C(U):={X∈C:σ(X)≤U} and C=(U):={X∈C:σ(X)=U}.

Note that is the subcode of consisting of all codewords of whose supports are contained in . The set comprises those codewords with support equal to .

Definition 6.

Let be an - code. Let be a -dimensional subspace of . We say that is a -support of if .

Definition 7.

Let be an - code, let and let . For each matrix , we denote by the matrix obtained from by deleting each th row of for . That is, is the projection of onto the rows indexed by . We define the projection map

 πs:Fn×mq→F(n−s)×mq,X↦X{s+1,...,n}.

We define . Let . The punctured code of with respect to and is

We define the shortened code of with respect to and by

 Σ(C,A,s):={πs(AX) : X∈C, (AX){1,...,s}=0}.

In the case that

is the identity matrix, the corresponding punctured code is found simply by deleting the first

rows and the shortened code is found by selecting the codewords whose first rows are all-zero and then deleting these rows from each selected codeword. We have the following duality result [11].

Lemma 8.

Let be an - code and let . Let be an integer. Then

 Π(C,A,s)∗=Σ(C∗,(A⊤)−1,s).

In particular, for any .

We will apply Lemma 8 a number of times throughout the paper.

1.2. Designs over Fq

We briefly recall the definition of -design and the known constructions. In this paper, we only treat simple -designs.

Definition 9.

Let be positive integers and let be a collection of -dimensional subspaces of . We say that forms (the blocks of) a -design over if every -dimensional subspace of is contained in the same number of elements of . In this case we say that is a - design over .

Designs over are also known as subspace designs and as designs over finite fields. A - subspace design is called a -Steiner system. The interested reader is referred to the survey [9] and the references therein for an outline of the state of the art on designs over finite fields.

We have the following notion of a dual subspace design [23, 33].

Lemma 10.

Let be positive integers and let be a - design over . Define . Then is a

 t−(n,n−r,λ[n−tr]q[n−tr−t]−1q)

design over .

The intersection numbers associated with a subspace design were given in [33]. These fundamental design invariants often play an important role in establishing the non-existence of a design for a given set of parameters.

Lemma 11.

Let be positive integers and let be a - design over . Let be dimensional subspaces of satisfying and . Then the number

 λi,j:=|{U∈D:I⊆U, J∩U={0}}|

depends only on and , and is given by the formula

 λi,j=qj(r−i)λ[n−i−jr−i]q[n−tr−t]−1q.

We list in Table 1 the known algebraic constructions of infinite families of - subspace designs that do not rely on the existence of other subspace designs. This list is still rather short, although many specific numerical parameter sets are known to be realizable [8, 9, 10]. In most cases a computer search is combined with an assumed automorphism group, often in fact the normalizer of a Singer cycle group. One of the first such examples is the construction of a - design over ; see [6].

The only known parameters of a -Steiner system with are given by - over . Subspace designs for these parameters were found by an application of the Kramer-Mesner method and required significant computation [7]. The existence of the -Fano plane, namely a - design over is still an open problem.

Other realizable parameter sets can be obtained on condition of the existence of other subspace designs. In [22], Itoh establishes the following construction.

Theorem 12.

Let . Suppose there exists a

 2-(n,3,q3[n−51]q)

design over that that is invariant under the action of a Singer cycle of . Then for each , there exists a

 2-(rn,3,q3[n−51]q)

design over that is invariant under the action of .

As an example, one of Suzuki’s - designs over yields a
- subspace design, whose blocks form the complement of the set of blocks of the original design. Moreover, both of these are invariant under the action of a Singer cycle in . Hence the infinite family of designs constructed in [32] yields a new infinite family via Itoh’s construction. Some further infinite families of - designs over have been constructed by Itoh’s result, based on sporadic examples of - Singer cycle invariant subspace designs found by computer search; see [9, Table 5].

Finally, in [8], the authors show how to construct new large sets of designs from existing ones. An large set is a partition of the set of -dimensional subspaces of into disjoint - designs. It is known, for example, that large sets exist for , such that and satisfying . In [9, Table 7] the authors list 5 large set parameters that are known to be realizable, and many more that are admissable but for which existence is not yet established. The interested reader is referred to [8] for further details.

2. An Assmus-Mattson Theorem for the Rank Metric

The main result of this section is Theorem 18, which gives criteria under which the -supports of an - form a design over . This is a rank-metric analogue of the Assmus-Mattson theorem, connecting codes and subspace designs.

Throughout this section, if is an - code and is a subspace, then we let

 C≥(U,d):={X∈C:rk(X)=d, σ(X)≥U}⊆C.

We also define

 πt(C)ℓ:={X∈πt(C) : rk(X)=ℓ}andΠ(C,A,t)ℓ:=πt(AC)ℓ.

We start with a series of preliminary results. In the sequel, we let denote the unit or standard basis vectors.

Lemma 13.

Let be an - code. Let be an integer, and let . There is a one-to-one correspondence between the words of rank in whose supports contain and the elements of the punctured code of rank . More precisely, the restriction of to is a bijection:

 πt:C≥(E,d)→πt(C)d−t.
Proof.

First observe that is an -linear homomorphism of onto and moreover is an injection, since for any non-zero , the rank of is at least . Let . Since

, there exists an invertible matrix

such that

 X=(It00Z)G,

for some matrix of rank . Therefore, and . In particular,

 rk(πt(X))=rk(Z)=d−t.

This shows that is a well-defined, injective map from to

 πt(C)d−t={X∈πt(C) : rk(X)=d−t}.

Let such that . By definition, there exists such that . Since , we have . Therefore

 d≤rk(X)≤rk(πt(X))+t=rk(Y)+t=(d−t)+t=d.

It follows that . We will show that . Towards a contradiction, suppose that does not contain . Define the matrix

 ¯¯¯¯¯X:=(Xe⊤1⋯e⊤t).

We then have , as the span of is not contained in the span of the columns of . Therefore

 rk(πt(¯¯¯¯¯X))≥rk(¯¯¯¯¯X)−t≥rk(X)+1−t=d+1−t.

On the other hand, by construction of we have

 rk(πt(¯¯¯¯¯X))=rk(πt(X))=%rk(Y)=d−t,

yielding a contradiction. We deduce that is a bijection onto . ∎

Corollary 14.

Let be an - code. Let be an integer, and let be a subspace of dimension . There exists such that the words of weight in whose support contains are in one-to-one correspondence with the words of weight in .

Proof.

Let be an -isomorphism with the property that . Let be a representation of . Define . Multiplication by from the left gives a bijection from onto . The result now follows since, by Lemma 13, we have

 |¯¯¯C≥(E,d)|=|πt(¯¯¯C)d−t|=Wd−t(πt(AC))=Wd−t(Π(C,A,t)).

We have shown that for an arbitrary -dimensional space , there is a bijection between and , where is an invertible matrix mapping a basis of  to . ∎

Proposition 15.

Let be an -. Let and let be non-zero for at most values of in . Let be an invertible matrix in . Then the weight distribution of is determined and is independent of .

Proof.

Let be the set of non-zero weights of in , for some . Let . The non-zero weights in of form a subset of and since every element of has rank at most , all of the weights of lie in . Moreover, since the first rows of each element of are all zero, the map

 Φ:C∗(E)→Σ(C∗,I,t) : X↦πt(X)

is a rank preserving bijection. As a consequence, the non-zero weights of the code form a subset of . Clearly, the non-zero weights of are precisely the non-zero weights of . Therefore, we may apply the same argument with in the place of to conclude that the non-zero weights of

 Π(C,A,t)∗=Σ(C∗,(A⊤)−1,t)=Σ((AC)∗,I,t)

form a subset of , the weights of in .

Now observe that for , since the minimum distance of is at least . In particular, at least of the values of the weight distribution of are known.

The MacWilliams duality theorem for rank metric codes [16, 28] then yields a system of equations in at most unknowns. In fact, using Theorem 4, for these equations are explicitly given by

 =

which is equivalent to the system,

 = (qm(n−ℓ)−k−1)[nℓ]q.

From (2), this system can be solved for unique . Furthermore, this solution is independent of , so the weight enumerator of , and hence , is independent of . ∎

We require one more definition before proving the main result of this section.

Definition 16.

Let be an - code and let be an integer satisfying . We say that is -invariant if depends only on for each -support . If is -invariant we define , where is any -support.

Lemma 17.

Let be an - -invariant code for some . Let . Then is also -invariant and .

Proof.

The map is a rank preserving bijection. Let be a -dimensional space and let . Then if and only if . The result follows. ∎

The following is a rank metric analogue of the Assmus-Mattson Theorem [2, Theorem 4.2]. As in the classical case, we show that if the dual code of a -invariant rank metric code has few weights, then the -supports of form the blocks of a subspace design.

Theorem 18.

Let be an - code. Let be an integer, and assume that

 |{1≤i≤n−t : Wi(C∗)≠0}|≤d−t.

Denote by the minimum distance of , and let be integers satisfying and . Suppose that (resp. ) is -invariant for each (resp. ). Then the -supports of (resp. ) form the blocks of a -design over for each (resp. ).

Proof.

Throughout this proof, let be a fixed -dimensional subspace of , let and let be a matrix representation of the -isomorphism that maps to . From Lemma 13 and the proof of Corollary 14, the words of weight in whose supports contain are in one-to-one correspondence with the words of weight in the punctured code . That is, there is a bijection from to .

We will count in two ways the elements of the set:

 P={(U,X) : U≤Fnq, dim(U)=d, T≤U, X∈C, σ(X)=U}.

On one hand, since is -invariant by hypothesis, we have:

 |P|=∑T≤U, dimU=d|C=(U)| = μ(C,d)|{U≤Fnq : U is a d-support of C,T≤U}|.

On the other hand, since each has a uniquely determined support we have:

 |P|=|{X∈C : rk(X)=d, T≤σ(X)}|=|C≥(T,d)|.

By Corollary 14, there exists such that . By Proposition 15, the value of only depends on , and . Now,

 Wd−t(Π(C,A,t))=∑U≥TdimU=d|C=(U)| = μ(C,d)|{U : T≤U is a d-support of C}|,

and hence the number of -supports of that contain is independent of the choice of of dimension . We deduce that the -supports of form the blocks of a -design over .

We now apply an inductive argument to show that the -supports of form the blocks of a -design for each . Suppose the result holds for . Let denote the number of -supports of satisfying and for a fixed -dimensional space and an -dimensional space . In other words, the are the intersection numbers of the designs held by the words of rank in . We will count the number of words of of weight . Let denote the number of -supports of that contain .

Let have rank . Since is an injection on , for a unique matrix of rank at most . If , then there exists an invertible matrix such that

 AXB=(M1|M20|M3),

for some matrices , where is a matrix of rank and is a matrix of rank , satisfying

 σ([M10])=σ(AX)∩E and σ(M3)=σ(Y),

in which case . Therefore every word of rank in corresponds to a unique word in of weight whose support meets in a space of dimension .

Now let be a -dimensional subspace of and let be an -dimensional subspace of satisfying . Clearly, for any with . Then

 |{X∈C : rk(πt(X))=w−ℓ, L∩σ(X)={0}, V⊂σ(X)}|=μ(C,w−ℓ)λt−ℓ,ℓ(w−ℓ).

It follows that

 Ww−t(Π(C,A,t))=λT(w)μ(C,w)+w−t∑ℓ=1[tℓ]qλt−ℓ,ℓ(w−ℓ)μ(C,w−ℓ).

Then is independent of and hence the -supports of are the blocks of a -design over .

Now consider the words of weight in . We claim the -supports of are the blocks of a -design over . For each -dimensional subspace define

 S(C∗,U) := {σ(X) : X∈C∗(U), rk(X)=d∗} = {σ(X) :X∈C∗, σ(X)≤U, %rk(X)=d∗}.

Furthermore, define . Note that is contained in every element of as if and only if for each .

Recall and so . We now compute , which is equal to .

The map

 Φ:C∗(E⊥)→Σ(C∗,I,t) : X↦πt(X),

is a rank preserving bijection, as any matrix with column space in has its first rows all-zeroes. In particular,

 Wd∗(C∗(E⊥))=Wd∗(Σ(C∗,I,t))=Wd∗(Π(C,I,t)∗)=Wd∗(πt(C)∗).

By assumption, and hence is -invariant, so that

 |S(C∗,E⊥)|μ(C∗,d∗)=Wd∗(C∗(E⊥))=Wd∗(Π(C,I,t)∗). (3)

We can now compute . Since if and only if , we have that

 S(C∗,E⊥) = {σ(X) : σ(AX)≤T⊥, rk(X)=d∗, X∈C∗} = {σ(A−1X) : σ(X)≤T⊥, rk(X)=d∗, X∈A(C∗)}

Substituting for in the above and the fact that for any GL yields

 S((A⊤C)∗,E⊥)={σ(A−1X) : σ(X)≤T⊥, rk(X)=d∗, X∈A(A⊤C)∗=C∗}.

It follows that .

Since from Lemma 17, again substituting for in (3), we get that

 |S⊥(C∗,T⊥)|=|S((A⊤C)∗,E⊥)|=Wd∗(Π(C∗,A⊤