Let be the
-dimensional vector space over, the unique finite field with elements; is necessarily a prime power. The set of all subspaces of is the projective space111This terminology is not standard. In other branches of mathematics the term projective space defines the collection of all lines passing through the origin of a vector space. which can be formally defined as
where signifies the usual vector space inclusion. The collection of all subspaces in with a fixed dimension is called the Grassmannian of dimension for all , and is denoted as . In terms of notation, . Clearly, . The subspace distance between two subspaces and in is defined as
where denotes the smallest subspace containing both and . It was proved in [KK, AAK] that the projective space is a metric space under the action of the subspace distance metric. A code in the projective space is a subset of .
Codes in projective spaces have recently gained attention since they were proved to be useful for error and erasure-correction in random network coding [KK]. An code in is a collection of number of subspaces of such that the minimum subspace distance between any two of them is . Kötter and Kschischang showed that an code can correct any combination of errors and erasures during the communication of packets through a volatile network as long as [KK]. Subsequently codes in were studied extensively [EV, HKK, BP, GR]. Such codes are also referred to as subspace codes.
However, designing and studying code structures in is considered relatively trickier than study of classical error-correction in the Hamming space . This is because unlike in , the volume of a sphere is not independent of the choice of its center in the projective space . Thus, standard geometric intuitions often do not hold in . In other words, is distance-regular while is not. This implies that a different framework is required to study codes in projective spaces than the approach taken for block codes in classical error-correction, e.g. in [MS]. The problem of lack of distance-regularity in is, however, tackled to some extent by considering codewords of a fixed dimension. Such class of subspace codes are commonly known as constant dimension codes. A constant dimension code in is a subset of for some . The fact that is distance-regular is exploited to construct various classes of constant dimension codes [SE, SE2, GY, XF, TMBR, KoK, ER].
A lattice framework for studying both binary block codes and subspace codes was discussed in [MB]. The authors of [BEV] established that codes in projective spaces are -analogs of binary block codes defined within Hamming spaces using a framework of lattices. A lattice is a partially ordered set wherein both least upper bound and greatest lower bound of any pair of elements exist within the set and are unique. We denote the set of all subsets of the canonical -set as , also known as the power set of . The lattices corresponding to the block codes in and the subspace codes in are the power set lattice and the linear lattice , respectively. Here the notation represents set inclusion.
A lattice is called modular if the modularity condition holds for all elements in it (see Def. 5). Many of the well-known coding spaces including both and are examples of a modular lattice. This motivated the work of Kendziorra and Schmidt where they generalized the model of subspace codes introduced in [KK] to codes in modular lattices [KS]. There are two significant types of semimodular lattices, viz. geometric lattices and distributive lattices that have inspired a rich variety of literature, e.g. [B, RS]. While is a geometric distributive lattice, is an example of a geometric lattice which is modular but non-distributive. There have been quite a few attempts to characterize distributive lattices, such as in [LS, S]. However, no known characterization of geometric distributive lattices exists to the best of our knowledge.
The notion of “linearity” and “complements” in are not as straightforward as they are in the Hamming space . This is owed to the fact that is a vector space with respect to the bitwise XOR-operation whereas or are not vector spaces with respect to the usual vector space addition. Therefore, the subspace distance metric is not translation invariant over or . Braun et al. addressed this problem in [BEV] and defined linearity and complements in subsets of by elucidating key features from the equivalent notion in .
The maximum size of a linear code in was conjectured to be by Braun et al. [BEV]. A particular case of this problem was proved by Pai and Rajan where the ambient space is included as a codeword [PS]. The maximal code achieving the upper bound was identified as a code derived from a fixed basis. The authors of [PS] observed that such a code is basically embedding of a distributive lattice into the linear lattice, which is geometric. This motivated them to conjecture a generalized statement which can already be found in literature, e.g. in [B, Ch. IX, Sec. 4, Ex. 1].
The size of the largest distributive sublattice of a gemetric lattice of height must be .
Lattice-theoretic connection of other classes of linear codes in was investigated thoroughly in [BK]. The findings of [BK] include the discovery that the only class of linear subspace codes that have a sublattice structure of the corresponding linear lattice must be geometric distributive. Thus it is an interesting problem to find out a unique characterization of geometric distributive lattices should it exist.
In this paper, we determine the unique criterion for a geometric lattice to be distributive. We in fact prove a more generalized version of this statement. This helps us to bring out the unique characterization of class of geometric distributive lattices. We then use this characterization to solve a few problems involving linear codes and complements in . Problem 1 is also solved by applying the said characterization.
The rest of the paper is organized as follows. In Section 2 we give a few requisite definitions concerning lattices and formally define linear codes and complements in the projective space . Section 3 concerns with the study of uniquely atomistic lattices; in particular we show that any such finite lattice is modular. The unique-decomposition theorem that gives the unique characterization of finite geometric distributive lattices is derived in Section 4 after proving a sequence of results regarding modular lattices and distributive lattices. Section 5 is attributed to various applications of the unique-decomposition theorem in lattice theory that include determining the maximum size of a distributive sublattice of a finite geometric lattice and counting the Whitney numbers of a geometric distributive lattice. In particular, we consider a few problems about linearity and complements in the linear lattice. An important finding is that any distributive sublattice of can be used to construct a linear code closed under intersection. Concluding remarks and interesting open problems are listed in Section 6.
represents the unique vector space of dimension over . The set of all subspaces of is denoted as . The usual vector space sum of two disjoint subspaces and , called the direct sum of and , is written as . For any subset , the collection of all -dimensional members of will be denoted as ; . The notation for any subset of vectors in will denote the linear span of all the vectors in . denotes the symmetric difference operator, which can be defined for two sets and as
2.1 An Overview of Lattices
We will go through some standard definitions and results concerning lattices that can be found in the existing literature, e.g. in [B].
For a set , the pair is called a poset if there exists a binary relation on , called the order relation, that satisfies the following for all :
(Antisymmetry) If and , then ; and
(Transitivity) If and , then .
The dual of a poset is the poset defined on the same set as such that in if and only if in .
The notation is read as “ is less than ” or “ is contained in ”. If such that , then we write . In the sequel a poset will be denoted as when the order relation is obvious from the context.
An upper bound (lower bound) of a subset of a poset is an element containing (contained in) every . A least upper bound (greatest lower bound) of is an element of contained in (containing) every upper bound (lower bound) of .
A least upper bound or a greatest lower bound of a poset, should it exist, is unique according to the antisymmetry property of the order relation . The least upper bound and the greatest lower bound of a poset are denoted as and , respectively.
A lattice is a poset such that and exist for all . The notation for the and the are (“ join ”) and (“ meet ”), respectively.
Once again, a lattice will be denoted as whenever the join and meet operations are obvious from the context. In this work we will consider only finite lattices, i.e. when the underlying poset is finite. The unique greatest element and the unique least element of a lattice will be denoted as and , respectively, unless specified otherwise.
A sublattice of a lattice is a subset such that for all .
The Hasse diagram of a finite poset completely describes the order relations of that poset. If in the poset such that there exists no satisfying , then is said to cover ; we denote this as . In the Hasse diagram of a lattice, two elements are joined if and only if one of them covers the other; is written above if covers . Hence, if and only if there exists a path from moving up to .
The power set of a finite set and the projective space are examples of lattices. The Hasse diagram associated with the lattice of is shown here (Fig. 1). This particular lattice is known as .
A finite lattice is semimodular if the following holds for all :
A lattice is modular if both the lattice and its dual are semimodular. It can be proved that a finite lattice is modular if for any the following holds:
The smallest finite lattice that is non-modular is called (Fig. 2). plays a crucial role in characterizing modular lattices as we will see later.
The elements of a lattice which cover the least element of the lattice are known as atoms. A lattice with a least element is atomic if for every non-zero element there exists an atom such that . An atomic lattice is called atomistic if any element is a join of atoms. A lattice that is uniquely atomistic is defined in the following way.
A lattice is uniquely atomistic if each element therein is uniquely expressible as join of its atoms. If is a uniquely atomistic lattice with as the set of all atoms in then for any there exists a unique subset such that . We denote this relation as when the choice of is clear from the context.
Atoms play an important role in defining geometric lattices.
A finite lattice that is both semimodular and atomistic is called geometric.
Both the linear lattice and the power set lattice are examples of a geometric lattice. From Section 5 onwards all lattices considered will be geometric. The variety of modular lattices that will play a key role in this work are the distributive lattices which are defined next.
A lattice is distributive if the following two equivalent conditions hold for any :
The power set lattice is an example of a distributive lattice. However, the linear lattice is modular but not distributive. In general, any distributive lattice is modular, and the lattice is pivotal in characterizing modular non-distributive lattices. Similarly a modular lattice can be defined by non-inclusion of the lattice . The following theorem is due to Dedekind and Birkhoff.
([G], Page 59) A lattice is modular if and only if it does not contain a sublattice isomorphic to . A modular lattice is non-distributive if and only if it contains a sublattice isomorphic to .
A real valued function on a lattice is called a positive isotone valuation if the following conditions hold for all :
The distance function induced by an isotone valuation is defined as .
([B]) For an isotone valuation defined on a lattice , the function is a metric if and only if is positive.
A totally ordered subset of a lattice is called a chain. Given two elements and in a lattice , a chain of between and is a chain such that . The length of this chain is . The height of an element is the maximum length of all chains between and , and is denoted by . We often use the notation when is obvious from the context. The height of the lattice is the number where is the greatest element in .
For modular lattices, the following is a consequence of Theorem 2.
Theorem 3 (Page 41, Theorem 16, [B]).
If is the height function defined on a finite modular lattice then is a positive isotone valuation and is a metric on .
By definition, if and only if is the least element in ; similarly, if and only if is an atom in .
The total number of elements with a given height of a lattice with a height function defined on is called the Whitney number, denoted as . In terms of notation, .
2.2 Complements and Linearity in Projective Spaces
The notions of complements and linearity in the projective space are not straightforward as they are in the Hamming space . Braun et al. introduced the definition of both in [BEV] by extracting key properties of the same in . We begin with a formal definition of the complement mapping in .
For any subset , a function is a complement on if satisfies the following conditions:
and for all ;
There exists a unique for each for all ;
for all ; and
for all .
It was proved before that a complement function does not exist in the entirety of [BEV, Theorem 10]. The largest size of a subset of wherein a complement can be defined still remains an open problem. However, we will tackle that question in Section 5 with the additional constraint that a subset has distributive sublattice structure. The following is an upper bound on the number of one-dimensional subspaces in a subset with a complement defined on it.
([BEV], Proposition 1) Suppose there exists a complement on the subset . Then .
We will later investigate the same for distributive sublattices of for all prime powers .
Braun et al. defined linearity in by identifying a subset that is a vector space over with respect to some randomly chosen linear operation such that the corresponding subspace distance metric is translation invariant within the chosen subset. Later this definition was generalized for all prime powers [PS, BK2].
A subset is called a linear code if and there exists a function such that
is an abelian group;
for all ;
for all ; and
for all .
The first three conditions stated in the above definition makes any linear code in a vector space over . It was conjectured in [BEV] that a linear code in can be as large as at most.
The linear addition of two disjoint codewords in a linear code yields their usual vector space sum.
([BEV], Lemma 8) For two codewords and of a linear code , if .
A linear code is said to be closed under intersection if it is closed with respect to vector space intersection: The subspace is a codeword of for any two codewords and if is a linear code closed under intersection. The following is a method to construct such class of linear codes.
([BK2], Theorem 7) Suppose there exists a linearly independent subset of over . Let be a partition of . Define for any nonempty subset and . The code is linear and closed under intersection.
A linear code thus constructed is also referred to as a code derived from a partition of a linearly independent set. The particular case when in Theorem 6 is referred to as a code derived from a fixed basis, and was first introduced in [PS]. It is known that any linear code closed under intersection can only be constructed in the way described in Theorem 6 [BK2, Theorem 8]. The lattice structure of such class of linear codes was studied in [BK].
([BK], Theorem 18) A linear code in that is closed under intersection forms a distributive sublattice of the linear lattice .
The maximum size of a linear code closed under intersection was investigated in [BK2] and it revealed that the maximal case is unique.
([BK2]) The maximum size of a linear code closed under intersection in is . The bound is reached if and only if the code is derived from a fixed basis.
We will exploit the lattice theoretic connection of linear codes further in Section 5 using the unique decomposition of geometric distributive lattices.
3 Uniquely Atomistic Lattices
We will prove modularity of uniquely atomistic lattices in this section via a series of results. First a few elementary lemmas follow from definition.
If and are two elements of a uniquely atomistic lattice , then .
By definition, if the set of all atoms in is then and . Since the join-operation is associative, it follows that . ∎
Suppose is an uniquely atomistic lattice. For any distinct we must have
where and .
By definition of a lattice, . We can write for some finite set as is uniquely atomistic. That and implies that by Lemma 9. By unique atomisticity, we get since . Similarly, . Thus .
Assume that , i.e. . But that means is a lower bound of and and , a contradiction. Thus the statement follows. ∎
Suppose and are elements of a uniquely atomistic lattice . If then .
As , we have . From unique atomisticity of and Lemma 10 it can be observed that . The rest follows because . ∎
We will now establish that modularity is inherent in any finite uniquely atomistic lattice.
A finite lattice is modular if is uniquely atomistic.
According to Theorem 1 it is enough to show that there exists no sublattice of which is isomorphic to . Let the set of all atoms in be . We proceed by contradiction.
Assume that there exists a sublattice of isomorphic to as shown in Fig. 2. By unique atomisticity of , we can write the following for some fixed subsets :
Any finite uniquely atomistic lattice is geometric.
Theorem 12 brings us to a position where we can prove the unique-decomposition theorem in the next section.
4 The Unique-decomposition Theorem
In this section we will establish the unique criterion needed for an atomistic lattice to be distributive and use that to characterize finite geometric distributive lattices. Atoms of geometric distributive lattices play an important part in the unique characterization. The first step towards that is observing that the greatest lower bound of two atoms in any lattice is the least element of that lattice.
For any two distinct atoms in a lattice , we have , where is the least element of .
Suppose . By definition, . As is an atom in , hence , which means by our supposition. Similarly we obtain . Since are distinct, this is a contradiction and the result follows. ∎
Next we will prove the generalization of the above lemma for any finite number of atoms in a distributive lattice.
Let be a set of atoms in a distributive lattice . Then for all ,
By distributivity in we can write,
According to Lemma 14, for , and the statement is proved. ∎
The height of join of two atoms in a modular lattice (if the height function is defined) is the sum of heights of the individual atoms, as illustrated in the following lemma.
Suppose is a modular lattice and is the height function defined on . For any two atoms and in the following holds true:
We are now going to generalize Lemma 16 for any number of atoms if the lattice is also distributive.
Consider a set of atoms in a distributive lattice . If is the height function defined on then
The proof is by induction. The base case for is covered by Lemma 16. Suppose the statement holds true for any atoms in , i.e., . Since the join operation is associative over the elements of , we can write . The height of can therefore be expressed as:
As according to Lemma 15, the rest follows from the induction hypothesis. ∎
Consequence of Lemma 17 is that the number of atoms in a distributive sublattice of a finite modular lattice of height cannot exceed . We formally state the result.
If is a finite modular lattice of height then the number of atoms in a distributive sublattice of can be at most . The maximum number of atoms is reached if and only if all atoms of are also atoms in and the greatest element of is join of the atoms in .
Suppose be the set of atoms in . If is the greatest element in then certainly . As is an isotone valuation, Definition 9(ii) implies that . Observe that for all . Since , by Lemma 17 we have the following inequality that proves the claim:
Since is positive isotone, it is evident from (3) that if and only if and . That ’s are atoms in for all if and only if is a maximal chain in proves the rest. ∎
Any element in a geometric lattice can be expressed as a join of atoms (Definition 7). However, such representation is not unique. To elaborate, if is the set of atoms in a geometric lattice then there may exist such that for two different subsets . We will now show that representation of elements as join of atoms is unique if and only if the lattice is also distributive. In fact we will prove the following which is a more generalized statement.
Theorem 19 (Unique-decomposition Theorem).
A finite atomistic lattice is distributive if and only if it is uniquely atomistic.
Let be an atomistic lattice with as its set of all atoms. First we prove that is uniquely atomistic, i.e. is uniquely determined by when is distributive.
The proof is by contradiction. Suppose the claim is false, i.e. there exist subsets such that and . Since and are distinct, at least one of them is not contained within the other. Without loss of generality, suppose . Then the difference set is nonempty, i.e. there exists an integer . Thus we can write as per our supposition:
The individual terms can be decomposed further. Since , by distributivity in we obtain . The penultimate step follows from Lemma 15. Similar technique yields as . Hence (4) suggests . However, this is a contradiction since , i.e. is an atom. We conclude that our initial assumption was wrong and the representation is uniquely determined by .
Now it remains to prove that is distributive if it is uniquely atomistic. Once again we proceed by contradiction. That is modular follows at once from Theorem 12. Suppose is modular non-distributive. By Theorem 1 must contain a sublattice isomorphic to . In other words there exist such that and , where for (See Fig. 3). Suppose and . By imposition of unique atmosticity, for , where uniquely determines . This implies the following:
On the other hand , where and . If uniquely determines , i.e. , then Lemma 10 implies that . Similarly we can deduce for and which indicates that .
We can now express as , i.e. . This means , which contradicts our initial assumption. Hence, is distributive. ∎
A geometric lattice is distributive if and only if it is uniquely atomistic.
A geometric lattice is finite atomistic by definition, which concludes the proof. ∎
The semimodularity of a geometric lattice is not required for it to be distributive.
The greatest element in a geometric distributive lattice is the join of all of its atoms.
Let be the set of all atoms in a geometric distributive lattice . We aim to show that the greatest element in is . To that end, say is the greatest element in . By Theorem 19 we can write for some . However, by associativity of the join operation in , can also be decomposed as , which implies ; thus the only possibility is , which concludes the proof. ∎
In the following section we will see a few applications of the Unique-decomposition theorem, mainly in the context of linearity and complements in .
5 Applications of the Unique-decomposition Theorem
The unique-decomposition theorem, akin to unique decomposition in context of linear subspace codes [BK2, Proposition 12], lays the path for determining the maximum size of a distributive sublattice of a finite geometric lattice. We also characterize the extremal case.
The size of any distributive sublattice of a finite geometric lattice of height can be at most . The bound is reached if and only if each of the atoms in the sublattice is also an atom in the geometric lattice and the greatest element of the geometric lattice belongs to the distributive sublattice.
Suppose is a distributive sublattice of a finite geometric lattice with the height function defined on such that , where is the greatest element of . We require to show that .
By supposition is geometric. Let be the set of all atoms in . We define a mapping from to as below:
By definition the map is well-defined. Suppose there exist such that , i.e., . As is geometric distributive, Corollary 20 dictates that ; thus is injective. To check that is also surjective, any can be expressed as for some as is geometric; the choice of is unique according to Corollary 20, hence . Therefore is a bijective map, which implies that . Combining this with Proposition 18 yields .
For the extremal case we must have . The rest then follows from Proposition 18. ∎
An atom in a sublattice of a geometric lattice may not be an atom in . E.g. consider the lattice , the set of all subsets of . It is a geometric lattice with atoms and . The sublattice of has atoms and , none of which is an atom in . On the other hand, a proper sublattice of a geometric lattice does not contain all atoms of .
Irrespective of the choice of the lattice , the mapping in Theorem 22 always maps the ground set to and the empty subset to .
Class of distributive sublattices of maximum size in a finite geometric lattice can be characterized in an alternative way.
The size of a distributive sublattice of a finite geometric lattice of height is if and only if contains atoms of .
Proof of Theorem 8.
We now consider the Whitney numbers of a geometric distributive lattice.
The Whitney numbers of a distributive sublattice of a finite geometric lattice of height are bounded by for all . Equality occurs if and only if contains atoms.
The statement in Corollary 24 is similar in nature to the main result in Pai and Rajan’s paper [PS, Theorem 2].
In the sequel we will consider the linear lattice instead of a geometric lattice in general. is non-distributive geometric. There remain a few unanswered questions regarding linearity and complements in that can be resolved by applying the unique-decomposition theorem. It was shown before that a linaer code in that is closed under intersection necessarily is a distributive sublattice of the geometric lattice [BK]. Using the unique-decomposition theorem we now prove the converse.
A subset is a distributive sublattice of the corresponding linear lattice if and only if is a linear code closed under intersection.
Suppose is a distributive sublattice of and is the set of all atoms in . Obviously is geometric distributive, which according to Lemma 15 implies that
Applying the unique-decomposition theorem it is easy to see that any can be uniquely expressed as for some fixed . If we choose arbitrary bases that span over for all then (6) implies that the set is a partition of , a linearly independent subset of over . Expressing any as where , we can say by Theorem 6 that is a linear code closed under intersection with linear addition of two codewords and defined as: