Upper Bounds on Matching Families in
Matching families are one of the major ingredients in the construction of locally decodable codes (LDCs) and the best known constructions of LDCs with a constant number of queries are based on matching families. The determination of the largest size of any matching family in , where is the ring of integers modulo , is an interesting problem. In this paper, we show an upper bound of for the size of any matching family in , where and are two distinct primes. Our bound is valid when is a constant, and . Our result improves an upper bound of Dvir et al.
upper bound, matching families, locally decodable codes.
Locally Decodable Codes. A classical error-correcting code allows one to encode any message of symbols as a codeword of symbols such that the message can be recovered even if gets corrupted in a number of coordinates. However, to recover even a small fraction of the message, one has to consider all or most of the coordinates of the codeword. In such a scenario, more efficient schemes are possible. They are known as locally decodable codes (LDCs). Such codes allow the reconstruction of any symbol of the message by looking at a small number of coordinates of the codeword, even if a constant fraction of the codeword has been corrupted.
Let be positive integers and let be a finite field. For any , we denote by the Hamming distance between and .
(Locally Decodable Code) A code is said to be -locally decodable if there is a randomized decoding algorithm such that
for every and such that , , where the probability is taken over the random coins of ; and
makes at most queries to .
The efficiency of is measured by its query complexity and length (as a function of ). Ideally, one would like both and to be as small as possible.
Implicit discussion of the notion of LDCs dates back to [3, 34, 30]. Katz and Trevisan  were the first to formally define LDCs and prove (superlinear) lower bounds on their length. Kerenidis and de Wolf  showed a tight (exponential) lower bound for the length of 2-query LDCs. Woodruff  obtained superlinear lower bounds for the length of -query LDCs, where . More lower bounds for specific LDCs can be found in [19, 13, 29, 16, 36, 33]. On the other hand, many constructions of LDCs have been proposed in the past decade. These constructions can be classified into three generations based on their technical ideas. The first-generation LDCs [3, 24, 6, 12] are based on (low-degree) multivariate polynomial interpolation. In such a code, each codeword consists of evaluations of a low-degree polynomial in at all points of , for some finite field . The decoder recovers the value of the unknown polynomial at a point by shooting a line in a random direction and decoding along it using noisy polynomial interpolation [5, 28, 35]. The second-generation LDCs [7, 38] are also based on low-degree multivariate polynomial interpolation but with a clever use of recursion. The third-generation LDCs, known as matching vector codes (MV codes), were initiated by Yekhanin  and developed further in [31, 25, 17, 20, 22, 23, 10, 8, 15]. The constructions involve novel combinatorial and algebraic ideas, where the key ingredient is the design of large matching families in . The interested reader may refer to Yekhanin  for a good survey of LDCs.
Matching Families. Let and be positive integers. For any vectors , we denote by their dot product.
(Matching Family) Let . Two families of vectors form an -matching family in if
for every ; and
for every such that .
The matching family defined above is of size . Dvir et al.  showed that, if there is an -matching family of size in , then there is an -query LDC encoding messages of length as codewords of length . Hence, large matching families are interesting because they result in short LDCs. For any , it is interesting to determine the largest size of any -matching family in . When , this largest size is often denoted by , which is clearly a universal upper bound for the size of any matching family in .
(Set System) Let and be two disjoint subsets of . A collection of subsets of is said to be a -set system over if
for every ; and
for every such that .
The set system defined above is of size . When and , it is easy to show that the -set system yields an -matching family of size in . To see this, let be the characteristic vector of for every , where for every and 0 otherwise. Clearly, and form an -matching family of size in .
When is a prime power and , Deza et al.  and Babai et al.  showed that the largest size of any -set systems over cannot be greater than . For any integer , Sgall  showed that the largest size of any -set system over is bounded by . On the other hand, Grolmusz  constructed a -set system of (superpolynomial) size over when has distinct prime divisors. Grolmusz’s set systems result in superpolynomial-sized matching families in , which have been the key ingredient for Efremenko’s LDCs .
Bounds. Due to the difficulty of determining precisely, it is interesting to give both lower and upper bounds for . When , a simple lower bound for is . To see this, let be the set of all 0-1 vectors of Hamming weight (i.e., the number of nonzero components) in . Let for every , where 1 is the all-one vector. Then and form a matching family of size . When is a composite number with distinct prime factors, the -set systems of [21, 27, 15] result in superpolynomial-sized matching families in . In particular, we have that when for two distinct primes and . On the other hand, Dvir et al.  obtained upper bounds for for various settings of the integers and . More precisely, they showed that
for any integers and , where is a term that tends to 0 as approaches infinity;
for any prime and integer ;
for any integers and such that and .
In particular, the latter two bounds imply that when for two distinct primes and such that .
Our Results. Dvir et al.  conjectured that for any integers and . A special case where the conjecture is open is when is a constant, and for two distinct primes such that and . In this paper, we show that for this special case, which improves the best known upper bound that can be derived from results of Dvir et al. in , i.e., .
Our Techniques. Let be a matching family of size , where for two distinct primes and . We say that are equivalent (and write ) if there is a such that for every , where is the set of units of . Clearly, no two elements of (resp. ) can be equivalent to each other. Let . We say that is of type if and . We can partition the set of pairs according to their types. Let be the number of pairs of type . Then we have the following observations:
These observations in turn imply that and enable us to reduce the problem of upper-bounding to that of establishing an upper bound for .
As in , we establish an upper bound for by using an interesting relation between matching families and the expanding properties of the projective graphs (which will be explained shortly). Let
We define the projective -space over to be the pair . We call the elements of points and the elements of hyperplanes. We say that a point lies on a hyperplane if . The projective graph is defined to be a bipartite graph with classes of vertices , where a point and a hyperplane are adjacent if and only if lies on . Vertices that are adjacent to each other are called neighbors. A set has the unique neighbor property if, for every , there is a hyperplane such that is adjacent to but to no other points in (see also Definition 3.1). Without loss of generality, let be the set of pairs of type , where . Let . It is straightforward to see that satisfies the unique neighbor property (Lemma 3.8). For any , we denote by the neighborhood of , i.e., the collection of vertices in that are adjacent to some vertex in . Since every point in must have a unique neighbor in , we have that
We show that has some kind of expanding property (see Theorem 3.1), meaning that is large for certain choices of , which allows us to obtain the expected upper bound for (see Theorems 3.2 and 3.3). When is a prime, such an expanding property of was proved by Alon  using the spectral method and it says that
where is arbitrary.
Let be the adjacency matrix of , where the rows are labeled by the points, the columns by the hyperplanes, and if and only if and are adjacent. Note that the matrix may take many different forms because the sets and are not ordered. However, from now on, we always assume that and are identical to each other as ordered sets. Hence, is symmetric. Let be the characteristic vector of , where the components of are labeled by the elements and if and only if . Alon  obtained both an upper bound and a lower bound for that jointly result in (3), where with the superscript denoting the transpose of a matrix. More precisely, Alon  determined the eigenvalues of and represented as a linear combination of the eigenvectors of . In this paper, we develop their spectral method further and show a tensor lemma on (see Lemma 2.1), which says that is a tensor product of and when , where and are two distinct primes. As in , we determine the eigenvalues of and represent as a linear combination of the eigenvectors of . We obtain both an upper bound and a lower bound for , which are then used to show that has some kind of expanding property (see Theorem 3.1).
Subsequent Work. Recently, in a follow-up work, Bhowmick et al.  obtained new upper bounds for . They used different techniques and showed that for any integers and . In particular, their upper bound translates into for the special case we consider in this paper.
2 Projective Graphs and Associated Matrices
Let be a positive integer. We denote by and the all-zero (either row or column) vector of dimension , all-one (either row or column) vector of dimension , identity matrix of order and all-one matrix of order , respectively. We denote by an all-zero matrix whose size is clear from the context. We also define
Let and be two matrices. We define their tensor product to be the block matrix . We say that if can be obtained from by simultaneously permuting the rows and columns (i.e., apply the same permutation to both the rows and columns). Clearly, and have the same eigenvalues if .
In this section, we study the projective graph defined in Section 1. We also follow the notation there. Let be the number of points (or hyperplanes) in the projective -space over . Chee and Ling  showed that
and for every point and hyperplane . When is prime, Alon  showed that is an eigenvalue of of multiplicity 1 and is an eigenvalue of of multiplicity . Furthermore, an eigenvector of with eigenvalue is 1 and linearly independent eigenvectors of with eigenvalue can be chosen to be the columns of , where . However, the eigenvalues of have not been studied when is composite. Here, we determine the eigenvalues of when for two distinct primes and .
(Tensor Lemma) Let be an integer and let for two distinct primes and . Then .
Proof: Let be the mapping defined by , where
for every . Then is well-defined. To see this, let and for and . If and , then there are integers and such that
for every . Let be an integer such that
Let and , where and . It is clear that is injective and (this is clear from (5)). It follows that is bijective and
Let and be as above. Then if and only if and . Hence, the -entry of is equal to 1 if and only if the -entry of and the -entry of are both equal to 1. Hence, . It follows that
In fact, we could have concluded that and therefore in Lemma 2.1. The sole reason that we did not do so is that those matrices may take different forms, as noted in Section 1. To facilitate further analysis, we make the matrices unique such that . This can be achieved by making the sets and unique. To do so, we first make and unique as ordered sets, where and .
Figure 1: Ordered Point Sets
|(0, 0, 1)||(0, 0, 1)||(0, 0, 1)||(0, 3, 4)||(0, 3, 1)||(3, 0, 4)||(3, 0, 1)||(3, 3, 4)||(3, 3, 1)|
|(0, 1, 0)||(0, 1, 0)||(0, 4, 3)||(0, 1, 0)||(0, 1, 3)||(3, 4, 0)||(3, 4, 3)||(3, 1, 0)||(3, 1, 3)|
|(0, 1, 1)||(0, 1, 1)||(0, 4, 1)||(0, 1, 4)||(0, 1, 1)||(3, 4, 4)||(3, 4, 1)||(3, 1, 4)||(3, 1, 1)|
|(1, 0, 0)||(0, 1, 2)||(0, 4, 5)||(0, 1, 2)||(0, 1, 5)||(3, 4, 2)||(3, 4, 5)||(3, 1, 2)||(3, 1, 5)|
|(1, 0, 1)||(1, 0, 0)||(4, 0, 3)||(4, 3, 0)||(4, 3, 3)||(1, 0, 0)||(1, 0, 3)||(1, 3, 0)||(1, 3, 3)|
|(1, 1, 0)||(1, 0, 1)||(4, 0, 1)||(4, 3, 4)||(4, 3, 1)||(1, 0, 4)||(1, 0, 1)||(1, 3, 4)||(1, 3, 1)|
|(1, 1, 1)||(1, 0, 2)||(4, 0, 5)||(4, 3, 2)||(4, 3, 5)||(1, 0, 2)||(1, 0, 5)||(1, 3, 2)||(1, 3, 5)|
|(1, 1, 0)||(4, 4, 3)||(4, 1, 0)||(4, 1, 3)||(1, 4, 0)||(1, 4, 3)||(1, 1, 0)||(1, 1, 3)|
|(1, 1, 1)||(4, 4, 1)||(4, 1, 4)||(4, 1, 1)||(1, 4, 4)||(1, 4, 1)||(1, 1, 4)||(1, 1, 1)|
|(1, 1, 2)||(4, 4, 5)||(4, 1, 2)||(4, 1, 5)||(1, 4, 2)||(1, 4, 5)||(1, 1, 2)||(1, 1, 5)|
|(1, 2, 0)||(4, 2, 3)||(4, 5, 0)||(4, 5, 3)||(1, 2, 0)||(1, 2, 3)||(1, 5, 0)||(1, 5, 3)|
|(1, 2, 1)||(4, 2, 1)||(4, 5, 4)||(4, 5, 1)||(1, 2, 4)||(1, 2, 1)||(1, 5, 4)||(1, 5, 1)|
|(1, 2, 2)||(4, 2, 5)||(4, 5, 2)||(4, 5, 5)||(1, 2, 2)||(1, 2, 5)||(1, 5, 2)||(1, 5, 5)|
For example, as shown in Figure 1, we may set and . Then both and have been made unique as ordered sets. (Here, each equivalence class in is represented by the first element when its elements are arranged in lexicographical order, and these representatives are subsequently also arranged in lexicographical order.) Once and have been made unique as ordered sets, we can simply set , where and . For example, as shown in Figure 1, consists of columns and the th column corresponds to for every . From now on, we suppose that the point sets and have always been made unique, such as in the way illustrated above. Then we have
Let and . We define an matrix
For every , the columns of are linearly independent eigenvectors of with eigenvalue , where and .
Proof: The proof consists of simple verification. For example, when , we have that , where the first equality is due to (10). Similarly, we can verify for . The linear independence of the columns of can be checked using the linear independence of the columns of and .
We have that
Proof: Note that and and for every integer . Both equalities follow from simple calculations.
3 Main Result
In this section, we present our main result, i.e., a new upper bound for , where is the product of two distinct primes and . As noted in the Section 1, our arguments consist of a series of reductions. First of all, we reduce the problem of finding an upper bound for to one of establishing an upper bound for , the number of pairs of type . The latter problem is in turn reduced to the study of the projective graph . More precisely, we follow the techniques of  and use the unique neighbor property of . However, the validity of the technique depends on some expanding property of .
3.1 An Expanding Property
We follow the notations of Section 2. In this section, we show that the projective graph has some kind of expanding property (see Theorem 3.1), in the sense that is large for certain choices of . Expanding properties of the projective graph , where is a prime, have been studied by Alon  using the well-known spectral method. In Section 2, we made the observation that the graph is a tensor product of the graphs and . This observation enables us to obtain interesting properties (see Lemmas 2.2 and 2.3) which in turn facilitate our proof that has some kind of expanding property.
Let be the set of nonnegative integers and let be the field of real numbers. For any vectors , we let and . Furthermore, we define the weight of to be . For a set , we denote by its characteristic vector whose components are labeled by the elements and if and 0 otherwise. Due to Lemmas 2.2 and 2.3, the column vectors of form a basis of the vector space . Therefore, there is a real vector
such that can be written as a linear combination of the columns of , say
Let . The main idea of Alon’s spectral method in  is to establish both a lower bound and an upper bound for the following number:
The quantities , and can be written as:
Proof: Lemma 2.3 shows that . Then we have
which is the second equality. The first and third equalities can be proved similarly.
Lemma 3.1 allows us to represent as an explicit function of , and . Let
We have that .
for every , by Lemma 2.3. As an immediate consequence, we then have that
On the other hand, recall that , and have been made unique as ordered sets in Section 2. For every , there exists such that . Let be the mapping defined by
and let be the mapping defined by
We have that for every .
Proof: Suppose that for . Then the representation of in Section 2 shows that . It is easy to see that and .
For every and , let be the preimage of under and let be the preimage of under . Let and be two real vectors defined by