Generalized Hamming weights of projective Reed--Muller-type codes over graphs

12/10/2018
by   Jose Martinez-Bernal, et al.
Cinvestav
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Let G be a connected graph and let X be the set of projective points defined by the column vectors of the incidence matrix of G over a field K of any characteristic. We determine the generalized Hamming weights of the Reed--Muller-type code over the set X in terms of graph theoretic invariants. As an application to coding theory we show that if G is non-bipartite and K is a finite field of char(K)≠ 2, then the r-th generalized Hamming weight of the linear code generated by the rows of the incidence matrix of G is the r-th weak edge biparticity of G. If char(K)=2 or G is bipartite, we prove that the r-th generalized Hamming weight of that code is the r-th edge connectivity of G.

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1. Introduction

In this work we study basic parameters of projective Reed–Muller-type codes over graphs using an algebraic geometric approach via graph theory and commutative algebra, and give some applications to linear codes whose generator matrices are incidence matrices of graphs.

Let be a field of characteristic , let be a connected graph with vertex set and edge set , and let and be the vertices and edges of , respectively. The incidence matrix of , over the field , is the matrix given by if and otherwise. The edge biparticity of , denoted , is the minimum number of edges whose removal makes the graph bipartite. The -th weak edge biparticity of , denoted , is the minimum number of edges whose removal results in a graph with bipartite components. If , is the weak edge biparticity of and is denoted by . The -th edge connectivity of , denoted , is the minimum number of edges whose removal results in a graph with connected components. If , is the edge connectivity of and is denoted by . We will use these invariants to study the minimum distance and the Hamming weights of Reed-Muller-type codes over graphs.

The edge biparticity and the edge connectivity are well studied invariants of a graph [16, 33]. In Section 2 we give an algebraic method for computing the edge biparticity (Proposition 2.3). For a discussion of computational and algorithmic aspects of edge bipartization problems we refer to [25]. One has the following relationships [7, 16]:

where is the vertex connectivity and is the minimum degree of the vertices of .

The set of columns of can be regarded as a set of points in a projective space over the field . Consider a polynomial ring over the field with the standard grading. The vanishing ideal of is the graded ideal of generated by the homogeneous polynomials of that vanish at all points of . Fix integers and . The aim of this work is to determine the following number in terms of the combinatorics of the graph :

where is the set of zeros or projective variety of in , and is the class of modulo . This is equivalent to determine

because .

A projective Reed–Muller-type code of degree on [8, 13], denoted , is the image of the following evaluation linear map

The motivation to study comes from algebraic coding theory because—over a finite field—the -th generalized Hamming weight of the Reed–Muller-type code of degree is equal to [11, Lemma 4.3(iii)].

Generalized Hamming weights were introduced by Wei [18, 21, 30]. For convenience we recall this notion. Let be a finite field and let be a linear code of length and dimension , that is, is a linear subspace of with . Let be an integer. Given a linear subspace of , the support of is the set

The -th generalized Hamming weight of , denoted , is given by

The set is called the weight hierarchy of the code . The following duality of Wei [30, Theorem 3] is a classical result in this area that shows a strong relationship between the weight hierarchies of and its dual :

These numbers are a natural generalization of the notion of minimum distance and they have several applications from cryptography (codes for wire–tap channels of type II), –resilient functions, trellis or branch complexity of linear codes, and shortening or puncturing structure of codes; see [1, 3, 6, 9, 11, 12, 17, 19, 24, 27, 28, 30, 31, 32] and the references therein. If , we obtain the minimum distance of which is the most important parameter of a linear code. In this paper we give combinatorial formulas for the weight hierarchy of for .

Our main results are:

Theorems 2.10, 2.11, 2.12 Let be a connected graph with vertices, edges, -th weak edge biparticity , -th edge connectivity , and let be the incidence matrix of over a field of characteristic . If is the set of column vectors of , then

Thus computing and is equivalent to computing the -th generalized Hamming weight of for or . These are the only cases that matter.

The incidence matrix code of a graph over a finite field of characteristic , denoted , is the linear code generated by the rows of the incidence matrix of . As an application to coding theory we obtain the following combinatorial formulas for the generalized Hamming weights of when is connected (Corollary 2.13).

The minimum distance of the incidence matrix code of the graph is defined as

where is the Hamming weight of the vector , that is, the number of non-zero entries of . The minimum distance of is , the st Hamming weight of this code. Then we can recover the combinatorial formulas of Dankelmann, Key and Rodrigues [4, Theorems 1–3] for the minimum distance of in terms of the weak edge biparticity and the edge connectivity of (Corollary 2.14).

Using Macaulay 2 [14], SageMath [26], and Wei’s duality [30, Theorem 3], we can compute the weight hierarchy of . In Sections 3 and 4, we illustrate this with some examples and procedures. There are algebraic methods that can be used to obtain lower bounds for or equivalently for and [11, Theorem 4.9].

2. Reed–Muller-type codes over connected graphs

In this section we present our main results. To avoid repetitions, we continue to employ the notations and definitions used in Section 1.

Lemma 2.1.

Let be a connected graph and let be a minimum set of edges whose removal makes the graph bipartite. Then there is such that the edges of whose vertices have positive sign or negative sign are precisely .

Proof.

If is bipartite, there is nothing to prove. If is non-bipartite, pick a bipartition , of the graph . Setting if and if , note that the vertices of each have the same sign. Indeed if the vertices of have different sign, then is bipartite, a contradiction. ∎

The edge biparticity of a graph can be easily expressed by considering all possible ways of making a vertex-signed graph.

Lemma 2.2.

Let be a connected graph, let be the set of surjective maps , and let be the set of edges of whose vertices have the same sign. Then

Proof.

If is bipartite, and there is nothing to prove. Assume that is non-bipartite. Then for . By Lemma 2.1, there is such that . Thus, one has the inequality “”. To show the reverse inequality take in . It suffices to show that . The vertex set of can be partitioned as , where (resp. ) is the set of vertices of with positive (resp. negative) sign. Then is bipartite with bipartition , . Thus . ∎

This lemma can be used to compute . Let be a field of . Each in defines a linear polynomial

The number of points of where does not vanish is equal to . As a consequence one obtains the following algebraic formula for the edge biparticity.

Proposition 2.3.

Let be a connected non-bipartite graph over a field of . Then

Proof.

As for , the result follows from Lemma 2.2. ∎

This result can be used in practice to compute using Macaulay [14] (see the examples and procedures of Sections 3 and 4).

Remark 2.4.

If we allow to be in such that not all of them are zero, we obtain the minimum distance of . This follows from [11, Lemma 4.3(iii)].

The following result is well known.

Proposition 2.5.

[2, 15, 20] Let be a connected graph with vertices and let be its incidence matrix over a field . Then

Corollary 2.6.

Let be a connected graph with vertices and edges and let resp. be the code resp. dual code of . Then

  • resp. is an resp. code if and is non-bipartite.

  • resp. is an resp. code if or is bipartite.

Proof.

This follow from Proposition 2.5 noticing that . ∎

Lemma 2.7.

Let be a connected graph and let be a field. The following hold.

(a) If , is non-bipartite and is a linear form in , then .

(b) If and is a linear form in in variables, then .

(c) If and is a linear form in , then , for some .

Proof.

Let be the linear map , , where is the transpose of the incidence matrix of . Fix a linear from of and set . Then is in if and only if . For use below notice that .

(a): By Proposition 2.5, . Thus , that is, .

(b): Again by Proposition 2.5, has dimension . Hence is generated by the vector . As is in the kernel of , it is a scalar multiple of , and since at least one of the is zero, we get , that is, .

(c): Since is generated by and , the result follows. ∎

Lemma 2.8.

Let be a connected bipartite graph with bipartition , . The following hold.

  • If is a field and is a linear form of that vanishes at all points of , then for some .

  • If and are in , then

    contains an odd cycle.

Proof.

(a): It follows adapting the proof of Lemma 2.7.

(b): As is connected and bipartite, there is a path in of even length joining and . Then, adding the new edge to the path , gives an odd cycle of . ∎

Lemma 2.9.

Let be a connected non-bipartite graph. If and are edges of , then the graph has at most bipartite connected components.

Proof.

Let be the connected components of . We proceed by contradiction assuming that are bipartite. Consider the graph . If for some , then has bipartite components, a contradiction. Thus for . Hence, joins and for some with . If , the graph is bipartite and connected, and has bipartite components, a contradiction. Thus and in this case has bipartite components, a contradiction. ∎

We come to one of our main results.

Theorem 2.10.

Let be a connected non-bipartite graph with vertices and edges, let be a field of , and let be the incidence matrix of . If is the set of column vectors of and is the -th weak edge biparticity of , then

Proof.

Assume . First we show the inequality . We proceed by contradiction assuming that . Then for some set consisting of linear forms which are linearly independent modulo . Let be the points in and let be the edges of corresponding to these points. Consider the graph . Let be the bipartite connected components of . Since , is at most . Let be the set of points corresponding to the columns of the incidence matrix of . Note that vanishes at all points of for . Then, by Lemma 2.7, are linear forms in the variables . For each , let , be the bipartition of and set . Then, by Lemma 2.8, is in the -linear space generated by , a contradiction because is linearly independent over and .

Now we show the inequality . Note that, by Lemma 2.7, it suffices to find a set of linearly independent forms of degree such that . We set . There are edges of such that the graph

has exactly connected bipartite components (see Lemma 2.9). We denote the connected components of by , where are bipartite. Consider a bipartition , of for and set

Let be the point in that corresponds to for . To complete the proof of the case we need only show the equality . To show the inclusion “” fix an edge with and set

Note that for , otherwise has bipartite components. As a consequence intersects , otherwise , joins and for some , and the graph has a bipartite components, a contradiction.

Case (1): for some . As , either or , otherwise the graph has bipartite components, a contradiction. Hence, as , we get that . Thus .

Case (2): and for some . Then using the bipartitions of and we get and . Thus .

Case (3): for some and . Then using the bipartition of we get . Thus .

To show the inclusion “” take and denote by its corresponding edge in . Then there is such that . We proceed by contradiction assuming , that is, for . Then is an edge of . Thus is an edge of for some . If , then for by construction of the ’s, a contradiction. Thus . If or , then would not be bipartite, a contradiction. Hence joins with , and consequently for by construction of the ’s, a contradiction. Thus for some , as required.

Assume . We claim that is equal to , the number of edges of . The set of all squarefree monomials such that is an edge of is -linearly independent modulo . This follows using that the vanishing ideal of is the intersection of the vanishing ideals of the points of and using a well known formula for the vanishing ideal of a projective point [22, p. 398, Corollary 6.3.19]. Therefore . As is a non-decreasing function of and it is bounded from above by the number of points of (see [10]), the claim follows. Therefore, since , one has . Thus for . ∎

We come to another of our main results.

Theorem 2.11.

Let be a connected graph with vertices and edges, let be a field of , and let be the incidence matrix of . If is the set of column vectors of and is the -th edge connectivity of , then

Proof.

Assume . First we show the inequality . We proceed by contradiction assuming that . Then for some set consisting of linear forms which are linearly independent modulo . We set . Let be the points in and let be the edges of corresponding to these points. Consider the graph and denote by its connected components. Since , cannot have components, that is, . Let be the set of points corresponding to the columns of the incidence matrix of . Note that vanishes at all points of for . Indeed, take a point in , then its corresponding edge is in for some , then for and , that is, . We set for . As , by Lemma 2.7, is a linear combination of for . Therefore

and consequently . Thus and the inclusion above is an equality. Therefore taking classes modulo , we get

As are linearly independent, so are because , a contradiction because by construction of the ’s and since , one has .

Next we show the inequality . Note that by Lemma 2.7. It suffices to find a set of forms of degree whose image in is linearly independent over and . We set . There are edges of such that the graph

has exactly connected components . For , we set

Note that and have no common variables for and any sum of the polynomials is a linear form in variables. Hence, by Lemma 2.8, is linearly independent.

Let be the point in that corresponds to for . To complete the proof of the case we need only show the equality . To show the inclusion “” fix an edge with and set

Note that for , otherwise has components, a contradiction. As a consequence joins and for some . Thus and .

To show the inclusion “” take and denote by its corresponding edge in . Then there is such that . We proceed by contradiction assuming , that is, for . Then is an edge of . As , we get for by construction of , a contradiction.

If , the equality for follows from the proof of Theorem 2.10. ∎

The next result is a hybrid of Theorems 2.10 and 2.11 and is characteristic free.

Theorem 2.12.

Let be a connected bipartite graph with vertices and edges, let be a field of any characteristic, and let be the incidence matrix of . If is the set of column vectors of and is the -th edge connectivity of , then

Proof.

Let , be the bipartition of . Consider the set of all points in such that is an edge of with and , where is the -th unit vector in . Noticing that the polynomial vanishes at all points of and the equality , the result follows adapting Lemma 2.7 and the proof of Theorem 2.11 with playing the role of . ∎

Corollary 2.13.

Let be the code of a connected graph with vertices, edges, -th weak edge biparticity , -th edge connectivity , over a finite field of . Then the -th generalized Hamming weight of is given by

Proof.

Note that the linear code is the image of —the vector space of linear forms of —under the evaluation map , . Note that the image of the linear function , under the map , gives the -th row of the incidence matrix of . This means that is the Reed–Muller-type code . Hence, the result follows using the equality [11, Lemma 4.3(iii)] and Theorems 2.10, 2.11, and 2.12. ∎

Corollary 2.14.

[4, Theorems 1–3] Let be the code of a connected graph with vertices, edges, weak edge biparticity , edge connectivity , over a finite field of . Then the minimum distance of is given by

Proof.

It follows from Corollary 2.13 making . ∎

3. Examples

Let be a connected graph and let be the incidence matrix code of over a finite field of characteristic . Using Macaulay 2 [14], SageMath [26], and Wei’s duality [30, Theorem 3], we can compute the weight hierarchy of . We illustrate this with some examples.

Note that, by Theorem 2.13, we can compute the corresponding higher weak biparticity and edge connectivity numbers of the graph. Conversely any algorithm that computes these graph invariants can be used to compute the weight hierarchy of .

Example 3.1.

Let be the graph of Figure 1. Recall that the dimension of is if and is if (Corollary 2.6). For use below we denote the dual code by .