Network coding with modular lattices
In , Kötter and Kschischang presented a new model for error correcting codes in network coding. The alphabet in this model is the subspace lattice of a given vector space, a code is a subset of this lattice and the used metric on this alphabet is the map . In this paper we generalize this model to arbitrary modular lattices, i.e. we consider codes, which are subsets of modular lattices. The used metric in this general case is the map , where is the height function of the lattice. We apply this model to submodule lattices. Moreover, we show a method to compute the size of spheres in certain modular lattices and present a sphere packing bound, a sphere covering bound, and a singleton bound for codes, which are subsets of modular lattices.
2010 Mathematics Subject Classification: 06C05, 68P30, 94B65, 05A15, 20K27.
Network coding is a tool for information transmission in networks. A network is considered to be a directed graph, where an edge from a vertex to a vertex is drawn, if is able to send information directly to (cf. ). A subset of the vertices is the set of senders and another subset is the set of receivers. Each sender is interested in sending his information to every receiver (broadcasting). The information is transmitted over several vertices to the receivers. With network coding a vertex is allowed to combine received information and forward these combinations. Usually the information is represented by vectors of the -vector space for a prime power and a positive integer (cf. ). The combinations are then -linear combinations. In random network coding the coefficients of these linear combinations are randomly chosen. For basic properties, advantages and further information on random network coding the reader is referred to [10, 11, 13]. Regarding general network coding see .
Kötter and Kschischang presented in  a new model for error correcting codes in random network coding. A sender transmits vectors of the , spanning a subspace of . A receiver receives vectors, which will span a subspace of . In the error free case thes subspaces are equal. Thus the alphabet in this model is the subspace lattice of the -vector space and a code is a subset of this lattice. To transmit a codeword a sender injects a basis of this codeword. The metric on this alphabet is the map .
In this paper we generalize this model to modular lattices. So we will consider codes as subsets of modular lattices with finite length and we use the metric , where is the height function of the lattice. This generalization is used to apply submodule lattices for random network coding. As in coding theory codes over (see e.g. [5, 6, 14]) came out to be useful we place emphasis to -modules of the form for a prime and positive integers and . We introduce so called enumerable lattices, which are a generalization of the submodule lattices of these modules with certain combinatorial properties. We derive a method to compute the cardinalities of spheres in these lattices. We present a sphere packing, a sphere covering, and a singleton bound for codes in modular lattices. These bounds are stated for arbitrary (finite) modular lattices and for enumerable lattices. In the latter case the bounds can be computed explicitly.
This paper is not meant to present concrete code constructions with encoding and decoding algorithms. It is rather a beginning or an introduction into a research topic. Basically we wish to explore modular lattices as metric spaces. Furthermore, we want to show, that the model presented in  is also applicable to submodule lattices of arbitrary finite modules and not only to subspace lattices. For concrete codes and algorithms further research will be required.
The outline of the paper is as follows. In chapter 2 we give all necessary definitions. Chapter 3 describes how network coding with modular lattices and especially submodule lattices can work. In chapter 4 we introduce enumerable lattices. The main part of this chapter describes a method to compute sizes of spheres in enumerable lattices. Bounds for codes in modular lattices are presented in chapter 5.
For basic notations in lattice theory the reader is referred to  and . For technical reasons we will consider a lattice mostly as an algebraic structure, instead as an ordered set. So a lattice is an algebraic structure with a set and two binary operations (join) and (meet), which are both associative, commutative and satisfy the absorption laws
for all . Every lattice gives rise to an ordered set where for .
For the set is called the interval between and . Note that it is again a lattice.
If the lattice is bounded then we denote the least element by (zero) and the greatest element by (one).
A totally ordered set is called a chain. The length of a chain is its cardinality minus one. The length of a lattice is the least upper bound of the lengths of chains in . If is finite, then is said to be of finite length. A lattice of finite length is complete, thus it has a zero and an one. If is finite, then has finite length.
In a lattice of finite length the height function gives for an element the greatest length of the chains between and . is called the height of . For we denote by the set of elements in with height .
2.1 Modular lattices and submodule lattices
A lattice is called modular if for all holds:
For a modular lattice of finite length the map
is a metric (see  chapter X §1 and §2). Further, the height function satisfies the equality
for every (see  chapter IV §4). For this reason one obtains for the metric also
for every .
We briefly recall the definitions of ring and module, which we take from .
A ring is an algebra consisting of a set , two binary operations and and two elements of such that is an abelian group, is a monoid (i.e. a semigroup with identity ) and is both left and right distributive over .
Let be a ring. An abelian group together with a map (”left scalar multiplication”) via is called a left -module if for all and the equations
hold. A subgroup of is called left -submodule of if holds for every and . For the submodule of generated by is denoted by .
Accordingly, one can define right -module and right -submodule by a ”right scalar multiplication”. If is commutative this distinction will be obsolete. We will consider from now on just left -modules and we will say just “-modules” instead of “left -modules”. For further information on modules see e.g. .
For any ring and a -module we will denote the set of all -submodules by . This set with the operations , which is defined by , and is a modular lattice (see  chapter VII §1 Theorem 1; note that this Theorem uses a more general definition of module, which covers the definition used here). We will call this lattice the submodule lattice of and denote it by . Because of the modularity of this lattice, we have the metric
For a prime power and a positive integer the submodule lattice (here the subspace lattice) of the -vector space is a finite modular lattice. The height of a subspace is exactly the dimension of . The metric on this lattice is which was presented in .
Consider the abelian -group for a prime and positive integers . This group is a -module. The set of all -submodules equals the set of all subgroups of . If , then there exists such that is isomorphic to . For the height function there holds . With this height function one obtains again a metric with the function defined in (3).
2.2 Partitions of nonnegative integers
We will shortly introduce partitions of nonnegative integers. The notations are done as in . We will use partitions later for semi-primary lattices.
A partition of a nonnegative integer is a finite monotonically decreasing sequence of nonnegative integers with . Zeros in this sequences are permitted and if two partitions differ only in the number of zeros, then they are considered to be equal. If is a partition of , then it is denoted by . With we denote the set of all partitions of . For a partition with times the entry we write also .
One can define an order on the set of all partitions by
for two partitions and .
For a partition the set is called the Ferrers diagram of . The partition with the Ferrers diagram is called the conjugated partition of and is denoted by . Note that is the number of sequence elements in , which are distinct from zero.
For the partitions and we define the partition , where we set for . Note that implies .
2.3 Semi-primary lattices
Definitions and results in this chapter are mostly taken from .
An element in a lattice is called cycle if the interval is a chain and dual cycle if the interval is a chain.
A modular lattice of finite length is called semi-primary if every element in is the join of cycles and the meet of dual cycles.
The elements of a modular lattice of finite length are called independent if the equation
holds for every . If are independent, then we write also instead of .
For semi-primary lattices we now state Theorem 4.9. of .
Every element of a semi-primary lattice is the join of independent cycles. Moreover, if has the two representations
with cycles , and , which are distinct from , then , and there exists a permutation such that for .
Because of this theorem we can agree on the following definition.
Let be a semi-primary lattice, , cycles distinct from with , and , such that . Then is called the type of . The type of is also called the type of and also denoted by .
Types of elements in semi-primary lattices can be considered as partitions of nonnegative integers. For a partition and a semi-primary lattice we denote by the set of elements in , which have type . If an element of a semi-primary lattice has type , then it is easy to see, that this element has height (see Lemma 4.3).
Note that if is a semi-primary lattice and an interval in , then is also semi-primary (see  Corollary 4.4.) and it holds (see  Lemma 2.4.). It follows for every the implication , because is an interval in .
As in  we call a Ring completely primary uniserial if there exists a two-sided ideal of such that every left or right ideal of is of the form (where ). Theorem 6.7. of  says, that every submodule lattice of a finitely generated module over a completely primary uniserial ring is semi-primary (in fact the theorem says more than that).
The field is completely primary uniserial, because the only ideals of are and . So the subspace lattice of the -vector space is semi-primary. If has dimension , then has the type .
The Ring is completely primary uniserial, because every ideal is of the form for some and with we have . So the submodule lattice of the -module is semi-primary. If , then there exists with such that is isomorphic to . Then has the type .
3 Network coding with modular lattices
In this chapter, we will generalize the notion of operator channel, which was presented in . Similar to the discussion in  we can here decompose the metric distance between two elements in an error and an erasure part. We consider the signal transmission from a single sender to a single receiver with an arbitrary finite modular lattice as the alphabet. In this context it is not important, whether the channel is a network or not. For an input the channel will deliver an output . The metric on this alphabet is the function defined in (1). We define the functions
It is easy to see that
holds for every . For an input and an output we call the erasure and the error from to . Roughly speaking is a measure for the information, which was contained in but after the transmission not anymore in , and is a measure for the information, which was not contained in but after the transmission is contained in .
If for there exists such that has the representation
then we have and and so the intervals and are isomorphic (see  chapter I, §7, corollary 2). If follows
If the chosen lattice is the subspace lattice of the -vector space , then such an e exists for every and corresponds to the definition of errors in Definition 1 in . corresponds also to the definition of erasures in Definition 1 in , independently of the existence of such an .
Such an does not exist in general for modular lattices. More precisely: For every exists an , such that (choose for example ), but an , such that and holds, does not exist in general. Let now , such that . Then . Because of holds . Thus, the intervals and are isomorphic (see again  chapter I, §7, corollary 2). It follows
Roughly speaking is also a measure for the information which is contained in , but not in .
A code is in this paper a subset of a finite modular lattice . We denote the minimum distance of by . If every codeword in has the same height, then we call a constant height code. If is moreover semi-primary and every codeword in has the same type, then we call a constant type code. Clearly every constant type code is a constant height code.
3.1 Random network coding with submodule lattices
Now we consider the case that the information is transmitted through a network and that the alphabet for the signal transmission is a submodule lattice of a finite -module for a ring . As in  we consider the case of the communication between a single sender and a single receiver (single unicast). The generalization to multicast is straightforward. If the sender wishes to transmit a submodule , then he sends a generating set of into the network. A node in the network, which receives module elements , sends to the node a -linear combination
with random ring elements for if there is a link from to . If the sender sends the generating set into the network and a receiver receives the elements , then has in the error free case the representation
for some elements for and . is a submodule of . If the receiver collects sufficiently many module elements, then equals . In the case that errors appear, that means that module elements , which are not contained in , are transmitted through the network, then has the representation
for some elements for , and . Let , and . Then there exists a submodule of , such that has the representation
The intersection of and must not necessarily be trivial. The erasure in this case is . The error is , or if we wish to express it in terms of , it is . If the intersection of and is trivial (and so the intersection of and as well), then the error is .
4 Enumerable lattices and spheres
Let be the subgroup lattice of a finite abelian -group and a partition. If two subgroups in this lattice are isomorphic, i.e. they have the same type, then they have the same number of subgroups of type . More precisely, if have the same type, then holds. But if we consider the number of groups in this lattice, which are greater or equal than or instead of less or equal, then the statement does not hold in general. More precisely, if have the same type, then does not necessarily follow. E.g. if we consider the subgroup lattice of in Figure 1, then the black colored element of type is covered by two elements of type and the other two elements of type by none. If we define for and the sphere with radius centered at , then we have as a consequence that the spheres with radius 1 centered at the elements of type have not the same cardinality. The sphere centered at the black colored element has the cardinality and the other two spheres have cardinality . More general, if have the same type, then does not necessarily follows. But that might be a desired property. If we restrict now to be of the form for some integers and , then for follows if and have the same type (see Theorem 4.2 and Example 6). For example, this can be seen in the subgroup lattice of in Figure 1. Consequently for follows if and have the same type (see chapter 4.2).
In the following we will generalize the subgroup lattices of finite abelian -groups to down-enumerable lattices and subgroup lattices of finite abelian -groups of the form to enumerable lattices. Enumerable lattices are semi-primary lattices with the desired property described above. In chapter 4.1 we will present a result, which shows that down-enumerable lattices are under certain circumstances even enumerable, which is a generalization of the group case described above. Since we know that two spheres in an enumerable lattice with same radius and centered at two elements with the same type have the same cardinality, we would like to compute the size of these spheres dependent on the radius and the type of the element in the center. This will be described in chapter 4.2.
A finite semi-primary lattice is called down-enumerable if for every and every partition the implication
holds. Then for an element of type and a partition we denote . is called up-enumerable if for every and every partition the implication
holds. Then for an element of type and a partition we denote . If is down-enumerable and up-enumerable, then it is called enumerable.
4.1 A duality result
This section is devoted to a proof of the following theorem.
Let be a self-dual down-enumerable lattice and for some positive integers . Assume further, that for every cycle there exists a cycle with and . Then is enumerable and for every two partitions holds
Let be a modular lattice of finite length and . Then there holds
Furthermore we have the equivalence:
See  chapter IV §1 and §4. ∎
The type of a semi-primary lattice is equal to the type of its dual lattice.
See  Corollary 4.11. ∎
For the next Lemma, we need another notation from . Let be a semi-primary lattice, and a positive integer. The join of all cycles with and is denoted by .
Let be a semi-primary lattice and . Then the following equivalence holds:
See  Theorem 4.14. ∎
Let be a semi-primary lattice, and . Then there exists an element with .
Let . There exist independent cycles distinct from zero with , such that is the join of . Since , there exists a cycle with for . Because are independent cycles, also the cycles must be independent. Hence, we conclude and . ∎
An element of a bounded lattice is called atom if it has height 1.
Let be a semi-primary lattice, , and with and , such that . Furthermore let be independent cycles distinct from zero, such that is the join of . Then there exists an atom , such that are independent.
Let be the uniquely determined atom with for . Let be an atom such that are not independent. By Lemma 4.3 it follows that
So we have and finally . Let be the set of atoms in . Assume that for every the elements are not independent. Then it follows that
and so . It follows that there exists no element in with type and . But that is a contradiction to Lemma 4.6, because holds for . It follows, that there exists an atom , such that are independent. With Lemma 4.5, it follows that are independent, because of for and . ∎
Let be a semi-primary lattice, , , and independent cycles distinct from zero, such that is the join of . If , then there exist atoms , such that are independent.
Let be a semi-primary lattice and . With we denote the type of in the dual lattice of and call it the dual type of .
Let be a semi-primary lattice, for some positive integers , and . Further assume, that for every cycle , there exists a cycle with and . Then there holds
Let be independent cycles distinct from zero, such that is the join of . If then there exist by Corollary 4.8 atoms , such that are independent. By our premise, there exist cycles with for , for and for . If is the uniquely determined atom with for , then are independent. By Lemma 4.5, it follows that are independent, because of . It follows , because has type and is the only element in with type . We define for (it holds ) and . So is a cycle in . We will show, that are independent in . That means, that holds for every . We define for and so are independent and is the join of . Let be fixed. Then we have
The last equality holds because of for . are independent. It follows
For the second equality we used Lemma 4.3. Because of , we have
From this, we obtain , because of and . So, are independent cycles in and there holds . We denote by the height of in . There holds for . It follows that has in the type and since we have . By Lemma 4.4, it follows that also the dual lattice of has type . The one-element of this dual lattice is exactly , and it follows, that has dual type in and so in . ∎
Now we can state the proof of Theorem 4.2.