PartitionBalanced Families of Codes
and Asymptotic Enumeration in Coding Theory
Abstract.
We introduce the class of partitionbalanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise asymptotic estimates on the density functions of several classes of codes that are extremal with respect to minimum distance, covering radius, and maximality. The techniques developed in this paper apply to various distance functions, including the Hamming and the rank metric distances. Applications of our results show that, unlike the linear MRD codes, the linear MRD codes are not dense in the family of codes of the same dimension. More precisely, we show that the density of linear MRD codes in in the set of all matrix codes of the same dimension is asymptotically at most , both as and as . We also prove that MDS and linear MRD codes are dense in the family of maximal codes. Although there does not exist a direct analogue of the redundancy bound for the covering radius of linear rank metric codes, we show that a similar bound is satisfied by a uniformly random matrix code with high probability. In particular, we prove that codes meeting this bound are dense. Finally, we compute the average weight distribution of linear codes in the rank metric, and other parameters that generalize the total weight of a linear code.
Key words and phrases:
Asymptotic enumeration, partitionbalanced family, errorcorrecting code, density function, Hamming metric, rank metric, MDS code, MRD code.2010 Mathematics Subject Classification:
05A16, 11T71Introduction
Linear codes over finite fields have been extensively studied as combinatorial objects, with connections to many areas in mathematics such as graph theory, curves over finite fields, finite geometry, lattice theory and numerous topics in algebraic combinatorics. See, for example [8, 9, 16, 28, 38] and the references therein.
There are several fundamental parameters and invariants associated with a linear code, such as its dimension, minimum distance, covering radius and weight distribution. Determination of some or all of these parameters is a nontrivial problem for an arbitrary code, especially as the dimension of its ambient space increases. For this reason, constructions of particular classes and families of codes with prescribed parameter sets are often sought. Much research has been spent on developing coding theoretic bounds as functions of some of the code parameters. Codes that meet such bounds are extremal and highly interesting from a combinatorial point of view, as they often have remarkable rigidity properties.
In this paper, we offer a new perspective on extremal codes. We obtain upper and lower bounds on the density functions of a number of families of codes within a larger family, and give precise asymptotic estimates of these. We introduce the idea of a partitionbalanced family of codes, and show how the combinatorial invariants of such families can be used to obtain estimates on the number of codes satisfying a particular property.
As we will show, these techniques can be applied in different contexts to establish the density or sparsity of families of codes that are extremal with respect to minimum distance, covering radius, and the related concept of code maximality. Our methods can be used to study codes whose ambient space is endowed with a large class of distance functions. In this paper, we focus on two major distance functions as applications of our results, namely the Hamming and the rank metric.
The Hamming metric is the classical distance function associated with coding theory. The linear maximum distance separable (MDS) codes are those dimensional subpaces of meeting the Singleton bound, namely those whose minimum Hamming distance is exactly . There is still substantial activity around such codes, which form a central topic in coding theory.
Another important distance function of coding theory is the rank metric, which measures the rank of the difference between a pair of matrices with entries from a finite field . Rank metric codes have seen a recent resurgence of interest both for their potential use in code based cryptography and as errorcorrecting codes in network communications [19, 26, 27, 36, 39, 40]. They are also intriguing as mathematical objects in their own right, and several researchers have sought to describe their structural properties [1, 4, 7, 12, 13, 14, 20, 21, 25, 32, 35]. However, the general theory of rank metric codes is still rather unexplored. The rank metric analogue of the Singleton bound yields the class of maximum rank distance (MRD) codes, which exist for all choices of and minimum rank , both for linear subspaces of (which we will refer to as vector rank metric codes) and the larger class of linear subspaces of (which we will refer to as matrix rank metric codes).
Concrete realisations of linear MRD codes have been known since the 1970s, having been independently introduced by Delsarte, Gabidulin and Roth who studied them from different perspectives [14, 17, 33]. On the other hand, general classes of linear MRD codes that are not linear were unknown until Sheekey [35] introduced the family of twisted Gabidulin codes.
While the vector rank metric codes often exhibit a behaviour similar to block codes with the Hamming distance, there is considerable divergence between these families and the class of matrix rank metric codes: if similar techniques for Hamming metric codes can be applied to make statements on vector rank metric codes, such methods often fail for matrix rank metric codes. Several examples of this can be observed in this work.
Perhaps the most profound difference is to be seen in the behaviour of the density functions of codes that are extremal with respect to the minimum distance. As the reader will see, while both MDS and vector rank metric MRD codes are dense among codes having the same dimension, the matrix MRD codes are never dense in this sense, both as and as ^{1}^{1}1In the very final stages of writing this paper, we became aware of the preprint [1], in which the authors independently show, by a different argument, that the MRD matrix codes are not dense in the set of codes with the same dimension as .. More precisely, one of the results of this paper is the following (see Theorem 6.1 and its corollaries).
Theorem.
Fix integers . Given with and a prime power , denote by the number of rank metric codes in of dimension , and by the number of such codes of minimum distance at most . Then for every positive real number there exists such that
Moreover, for every positive real number there exists such that
We obtain several other estimates on the density of codes that are extremal with respect to minimum distance, covering radius and maximality, which we outline below. Standard methods attempting to address density questions in coding theory often rely on the SchwartzZippel Lemma [34, 41] to obtain lower bounds on density functions. However, as the reader will see, in some important cases these methods fail. Our techniques offer an alternative general approach to asymptotic enumeration problems in coding theory.
Outline.
In Section 1 we define basic concepts of distanceregular spaces and their codes. In Section 2, we introduce the concept of a partitionbalanced family of codes, with respect to an arbitrary partition of the ambient space. We compute the invariants associated with some of these families, which will be used several times throughout the paper. In Section 3 we define the density functions associated with a family of codes, and give asymptotic estimates of functions that are used in later sections.
In Section 5 we give precise asymptotic estimates for the number of codes with the Hamming and the rank metric having given dimension and minimum distance. As immediate corollaries, we obtain the density of MDS and MRD vector rank metric codes in their respective ambient spaces. In Section 6 we show that the matrix MRD codes are not dense in the family of matrix codes of the same dimension. In particular, we show that the density function of nonMRD matrix codes of dimension in is asymptotically lower bounded by .
In Section 7, we show that the MDS and vector rank metric MRD codes are dense in the family of maximal codes for the same dimension. In Section 8 we show that Hamming and vector rank metric codes meeting the redundancy bound are dense in the family of codes of the same dimension. We introduce a new upper bound on the covering radius of matrix rank metric codes, which is not in fact satisfied by all rank metric codes, but rather by a uniformly random code with high probability. We then show that the matrix codes satisfying this bound are dense in the family of matrix codes of the same dimension.
Finally, in Section 9 we compute the average weight distributions of Hamming metric, vector rank metric and matrix rank metric codes. We obtain asymptotic estimates of these values and observe the interesting fact that, although the MDS and vector rank metric MRD codes are dense as the field size grows, the number of words of weight in a uniformly random linear code converges to a nonzero constant.
1. DistanceRegular Spaces and Codes
We start by describing the class of metric spaces for which our methods apply. These are linear spaces defined over finite fields that exhibit certain regularity properties with respect to their distance functions. They include important examples from coding theory, such as the Hamming metric and rank metric spaces.
Definition 1.1.
Let be a prime power, and let be the finite field with elements. Let be a finitedimensional vector space over and let be an integervalued distance function on . We say that is a ary distanceregular space if:

for all and all we have ,

for all we have ,

for all and , only depends on .
Denote by the weight induced by , i.e., the function defined by for all . To simplify the discussion in the sequel, we also assume that a ary distanceregular space with weight satisfies:

,
where is the image of under .
Note that a ary distanceregular space is an example of a symmetric association scheme. Therefore, for each triple there is an associated intersection number of , defined to be the integer
where are any vectors with . The properties of the ’s are wellstudied. The interested reader is referred to [5] for further details.
Remark 1.2.
Notation 1.3.
For the remainder, denotes a prime power, and a fixed ary distanceregular space of dimension over .
We are interested in the combinatorial properties and invariants of the subsets of . Our focus will be mostly on the subspaces. We define several of the coding theoretic invariants that we will consider in this paper.
Definition 1.4.
A code is a nonempty subset , and its elements are its codewords. We say that is linear if it is an linear subspace of . In this case we write .
Let be a (not necessarily linear) code. If , then the minimum distance of is the integer . We also set . The weight distribution of is the sequence , where for all . Finally, the covering radius of is the integer .
Note that if is a nonzero linear code, then it follows immediately from the definition of minimum distance and the linearity of that .
A ball in of given radius, say , is an example of a code that contain only codewords of weight at most . Clearly, a code will intersect a ball of radius centred at zero only if the minimum distance of is at most . We will apply this observation later to obtain estimates on the density of families of codes characterized as having certain properties.
Definition 1.5.
Let and . The ball of radius and center is the set
The size of the ball only depends on . This follows easily from the definition of and Property 3 of Definition 1.1. More precisely, for all and we have
Notation 1.6.
For , we denote by the cardinality of , for any vector .
2. PartitionBalanced Families of Codes
We describe families of codes that exhibit regularity properties with respect to a given partition of the ambient space .
Notation 2.1.
Let be a family of codes in , i.e., a collection of nonempty subsets of . For , we let
We are interested in families such that the cardinality of depends only on the class of with respect to a given partition, say , of the ambient space . This motivates the following definition.
Definition 2.2.
Let be a partition of of size . A nonempty family of codes in is called balanced if depends only on the integer such that , for all . In words, the family is balanced if the number of codes containing only depends on the class of containing . If is balanced, then the invariants of the pair are the integers defined by
where is any element with .
In [31], the authors give a definition of a balanced family of codes that is a special case of Definition 2.2. In particular, it is defined with respect to the partition of into two classes, namely, and . As the reader will see, for our results we need to consider more general partitions on .
A useful property of partitionbalanced families is the following simple identity, which will play a crucial role throughout the paper.
Lemma 2.3.
Let be a partition of of size , and let be a balanced family of codes in . Then for all functions we have
Proof.
Exchanging the order of summation we obtain
as desired. ∎
Remark 2.4.
In the case that the function of Lemma 2.3 is the characteristic function of a set , Lemma 2.3 yields
In particular, it expresses the average intersection between and a code in in terms of , the invariants of , and the intersections between with the classes of . We will apply often arguments of this type.
We now consider specific partitions and families of codes that are balanced with respect to these partitions. We explicitly compute the invariants of such partitionfamily pairs. These invariants will be required in later in order to make statements on the density of certain classes of codes. More precisely, Proposition 2.5 will be used in Section 5 to study the density of the MDS codes and MRD vector rank metric codes, and again in Sections 7 and 9. Proposition 2.8 will be applied in Section 6 to establish the remarkable fact that linear MRD codes are not dense in , both as and as .
Proposition 2.5.
Let be a linear code of dimension . Construct a partition of of size two via and . Fix an integer with , and define the family . Then is balanced. Moreover,
Proof.
We first show that is balanced. Assume that are in the same class of . If , then all codes contain both and . If , then it is easy to see that there exists an linear isomorphism such that and . Such a map induces a bijection between codes in containing and codes in containing .
The formulas for and are immediate. To compute , it suffices to count the elements of the set in two different ways, which gives the identity . ∎
A natural partition of is the one induced by the weight function .
Definition 2.6.
The invariants of a weight partition balanced family can be computed as follows.
Proposition 2.7.
Assume that is a balanced family of linear codes in . We have
Proof.
It suffices to doublecount the elements of and then use Remark 1.2, which guarantees that for all . ∎
We conclude this section with another class of partition balanced families.
Proposition 2.8.
Let be an linear code of dimension . Let be the partition of of size three given by , , and . Fix an integer , and define the family . Then is balanced. Moreover,
Proof.
By definition, to see that is balanced we need to show that, for every class of and for all , the number of codes containing is the same as the number of codes containing . We only show this for .
Let be arbitrary. Fix a basis of , and let and be bases of . Denote by the unique linear isomorphism that sends to . Note that preserves and sends to . Now let be an arbitrary code that contains . Since by assumption and is an isomorphism, we have . Thus . Moreover, as , we have . All of this shows that induces a bijection between codes in containing and codes in containing .
We can now compute the invariants of . It is immediate that . To compute , it suffices to use the fact that is balanced, and double count the elements of the set . The value of can be obtained similarly. ∎
3. Density Functions and Their Asymptotics
We formally define density functions and what it means for a family to be sparse or dense within a larger family. Such notions have been used for some decades in number theory [30]. The following definition is most apt in the context of our work, namely describing the asymptotic behaviour of density functions of families of errorcorrecting codes.
Definition 3.1.
Let be an infinite subset of the natural numbers. Let be a sequence of finite nonempty sets indexed by , and let be a sequence of sets with for all . The density function of in is given by
When exists and equals , then we say that has density in . If has density in , then is sparse in . If has density in , then is dense in .
To simplify the notation, throughout the paper the variable in and is omitted when it is clear from the context. We remark that notions of lower density (the ) and upper density (the ) are also used and appear in the literature, but are not required here.
3.1. Asymptotic Estimation
Since in several instances we will obtain estimates on density functions, we recall the standard notation used to describe the asymptotic growth of functions (see [11] for example).
Definition 3.2.
Let be realvalued a function defined on an infinite domain . We denote by , and the following sets.
If are functions of more than one variable, we will put in evidence the variable with respect to which the asymptotic estimate is made by writing expressions such as
where all the other variables are treated as constants. We will also need the following fact.
Lemma 3.3.
Let be a polynomial of degree . Then as , when is viewed as a function on an infinite subset .
The next simple consequences of the previous lemma will be particularly useful in the sequel.
Lemma 3.4.
Let be nonnegative integers. The following hold.

The binomial coefficient of and is a polynomial in of degree . In particular,

For all we have as .
3.2. Some Preliminary Formulæ
We conclude this section on density functions and asymptotic estimation by establishing some technical results that will be needed later.
Proposition 3.5.
Let be a fixed linear code of dimension . For all we have
Proof.
Denote by the lattice of linear subspaces of . For any subspace define . Then for all we have
where . We now use Möbius inversion [37, Proposition 3.7.1] in the lattice and obtain, for all ,
where denotes the Möbius function of . Therefore for all of dimension we have
(3.1) 
The formula in the statement now follows from (3.1) and the fact that
Since the quantity will arise a number of times in our results, we introduce the following notation.
Notation 3.6.
For nonnegative integers , , and a prime power , let
We can now give a precise asymptotic estimate of as grows.
Proposition 3.7.
Let be nonnegative integers satisfying . Then
In particular,
Proof.
Observe that
For a polynomial in indeterminate , write to denote its leading term. We have
hence
The inner sum can be expressed as:
which is in as for each value of satisfying , and contributes in the product of terms yielding the leading term of . Moreover,
It is easy to check that for and , the quantity attains its maximum value at , and hence
The proposition follows. ∎
4. DistanceRegular Spaces from Coding Theory
In this section we briefly describe three distanceregular spaces in coding theory, namely, the Hamming space, the matrix rank metric space, and the vector rank metric space. We also provide asymptotic estimations for some of the parameters associated with these spaces.
Notation 4.1.
Throughout the paper, denotes a prime power, and , are integers that satisfy . The results that we will obtain for the general ary space of dimension will be applied substituting , , , , or depending on the context.
4.1. Hamming Space
Denote by the Hamming distance on . Then is a ary distanceregular space, called the Hamming space. The weight induced by is denoted by . We use the symbol for the Hammingmetric covering radius. The linear codes in are the block codes. We write that is an code to say that is an linear code in of dimension . For all , the size of the ball of radius in the Hamming space is estimated to be
(4.1) 
The following upper bounds for the covering radius and minimum distance of a linear code with the Hamming metric are well known. They are called the redundancy bound and the Singleton bound, respectively. See [24, Corollary 11.1.3] and [24, Theorem 2.4.1] respectively.
Proposition 4.2.
Let be an code. Then and if , then .
An code with and minimum distance is called MDS. It is known that a dimensional MDS code exists for all whenever . Moreover, the weight distribution of an MDS code is uniquely determined. See for example [28, Chapter 11].
Remark 4.3.
For all , the family of dimensional MDS codes is balanced. This is easy to see. If have the same Hamming weight, then there is a monomial transformation taking to . This induces a bijection on the linear codes containing and those containing . In particular, , being a Hamming distance isometry, maps an MDS code containing to the equivalent MDS code containing .
We conclude this subsection on the Hamming space by giving explicit formulæ for its intersection numbers. These expressions are wellknown (see [28, Chapter 21] for ). We also include asymptotic estimations for such numbers.
Let be integers satisfying . The intersection numbers of the Hamming space are given by
where
(4.3) 
Lemma 4.4.
Let be integers. Then the expression in (4.3) is positive if and only if the following three inequalities hold:
In particular if or or .
Proof.
The fact that is positive if and only if satisfies the inequalities shown can be seen by inspection of its binomial factors. The value is zero if and only if for all satisfying . It can be checked that this occurs if or or . ∎
We now derive asymptotic estimates of the intersection numbers of the Hamming space as the field size grows.
Proposition 4.5.
Let be integers such that intersection number for the Hamming space is positive. Let . Then
In particular, as ,
Proof.
If there exists some nonnegative such that , then , and the leading term in of
is . Let and suppose that is nonzero. Then from Lemma 4.4 it holds that . We claim that . Clearly since otherwise at least one of inequalities in the statement of Lemma 4.4 does not hold. If then and only if , so and . If then and only if , so and . If then and only if , so and . ∎
4.2. Matrix RankMetric Space
The rank distance between matrices is defined to be . Then is a ary distanceregular space, called the (matrix) rank metric space. We use the symbol for the rank metric covering radius. A code in is called a (matrix) rank metric code. Clearly, the weight induced by is matrix rank. The size of the ball of radius in is given by
(4.4) 
There exists a Singletontype bound also for rank metric codes. See [14, Theorem 5.4] for a proof using association schemes, or [22, Section 3] for a linear algebra proof.
Proposition 4.6.
The dimension of a matrix code in with minimum rank distance is at most .
A rank metric code of minimum rank distance and dimension is called a maximum rank distance code and we say that is an MRD code. It is known (see [14, Section 6]) that for every there exists an MRD code of minimum distance . Notice that we always assume . See Notation 4.1.
Remark 4.7.
For all , the family of MRD codes of minimum rank distance is invariant. This can be seen by a very similar argument as given in Remark 4.3. If have the same rank over , then there exist invertible matrices satisfying . Then multiplication by and induces a bijection between the linear codes containing and those containing , which moreover preserves the MRD property, being an linear isometry of .
4.3. Vector RankMetric Space
Define the rank weight of a vector as the dimension over of the subspace generated by its components. The rank distance between vectors is . Then the pair is a ary distance regular space, called the (vector) rank metric space (see [17] for further details). The codes in are the vector rank metric codes. We write that is an code to say that is an linear code in of dimension over . The rank metric covering radius of such a code is denoted by .
Clearly, can be also viewed as a ary distanceregular space, which in fact is isomorphic (as linear space) and isometric to the matrix space . Let us make these isomorphisms more explicit. Given an basis of and given a vector , denote by the matrix over defined by
Then the following hold (see e.g. [22, Section 1]).
Proposition 4.8.
For every basis of , the map is an linear bijective isometry from to . In particular, if is an