Hyperplane Sections of Determinantal Varieties over Finite Fields and Linear Codes

09/12/2018
by   Peter Beelen, et al.
0

We determine the number of F_q-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify sections giving the maximum number of F_q-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determinantal varieties, and the determination of their weight distribution, minimum distance and generalized Hamming weights. The previously known results about these are generalized and expanded significantly

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1 Introduction

The classical determinantal variety defined by the vanishing all minors of a fixed size in a generic matrix is an object of considerable importance and ubiquity in algebra, combinatorics, algebraic geometry, invariant theory and representation theory. The defining equations clearly have integer coefficients and as such the variety can be defined over any finite field. The number of -rational points of this variety is classically known. We are mainly interested in a more challenging question of determining the number of -rational points of such a variety when intersected with a hyperplane in the ambient projective space, or more generally, with a linear subvariety of a fixed codimension in the ambient projective space. In particular, we wish to know which of these sections have the maximum number of -rational points. These questions are directly related to determining the complete weight distribution and the generalized Hamming weights of the associated linear codes, which are caledl determinantal codes. In this setting, the problem was considered in [2] and a beginning was made by showing that the determination of the weight distribution is related to the problem of computing the number of generic matrices of a given rank with a nonzero “partial trace”. More definitive results were obtained in the special case of varieties defined by the vanishing of all minors of a generic matrix. Here we settle the question of determination of the weight distribution and the minimum distance of determinantal codes in complete generality. Further, we determine some initial and terminal generalized Hamming weights of determinantal codes. We also show that the determinantal codes have a very low dual minimum distance (viz., 3), which makes them rather interesting from the point of view of coding theory. Analogous problems have been considered for other classical projective varieties such as Grassmannians, Schubert varieties, etc., leading to interesting classes of linear codes which have been of some current interest; see, for example, [17], [13], [14], [20], [12], and the survey [16].

As was mentioned in [2] and further explained in the next section and Remark 3.7, the results on the weight distribution of determinantal codes are also related to the work of Delsarte [9]

on eigenvalues of association schemes of bilinear forms using the rank metric as distance. We remark also that a special case of these results has been looked at by Buckhiester

[6].

A more detailed description of the contents of this paper is given in the next section, while the main results are proved in the two sections that follow the next section. An appendix contains self-contained and alternative proofs of some results that were deduced from the work of Delsarte and this might be of an independent interest.

2 Preliminaries

Fix throughout this paper a prime power , positive integers , and an matrix whose entries are independent indeterminates over . We will denote by the polynomial ring in the variables (, ) with coefficients in . As usual, by a minor of size or a minor of we mean the determinant of a submatrix of , where is a nonnegative integer . As per standard conventions, the only minor of is . We will be mostly interested in the class of minors of a fixed size, and this class is unchanged if is replaced by its transpose. With this in view, we shall always assume, without loss of generality, that . Given a field , we denote by the set of all matrices with entries in . Often and in this case we may simply write for . Note that can be viewed as an affine space over of dimension . For , the corresponding classical determinantal variety (over ) is denoted by and defined as the affine algebraic variety in given by the vanishing of all minors of ; in other words

Note that

only consists of the zero-matrix. For

, no minors of exist. This means that , which is in agreement with the above description of as the set of all matrices of rank at most .

It will also be convenient to define the sets

for as well as their cardinalities . The map that sends to its row-space is a surjection of onto the space of -dimensional subspaces of . Moreover for a given , the number of with row-space is equal to the number of matrices over of rank or equivalently, the number of

-tuples of linearly independent vectors in

. Since is the Gaussian binomial coefficient , we find

(1)

Using the Gaussian factorial , one can also give the following alternative expressions:

Note that . Next we define

Since is the disjoint union of , we have

(2)

The affine variety is, in fact, a cone; in other words, the vanishing ideal (which is precisely the ideal of generated by all minors of ) is a homogeneous ideal. Also it is a classical (and nontrivial) fact that is a prime ideal (see, e.g., [10]). Thus can also be viewed as a projective algebraic variety in , and viewed this way, we will denote it by . We remark that the dimension of is (cf. [10]). Moreover

We are now ready to define the codes we wish to study. Briefly put, the determinantal code is the linear code corresponding to the projective system . An essentially equivalent way to obtain this code is as follows: Denote and choose an ordering of the elements of . Further choose representatives for . Then consider the evaluation map

where denotes the space of homogeneous polynomials in of degree together with the zero polynomial. The image of this evaluation map can directly be identified with . A different choice of representatives or a different ordering of these representatives gives in general a different code, but basic quantities like minimum distance, weight distribution, and generalized Hamming weights are independent on these choices.

In [2], also another code was introduced. It can be obtained by evaluating functions in in all elements of . The parameters of determine those of and vice-versa, see [2, Prop. 1]. It is therefore sufficient to study either one of these two codes. In the remainder of this article, we will focus on and determine some of its basic parameters. It is in a sense also a more natural code to study, since the code is degenerate, whereas is nondegenerate [2]. We quote this and some other useful facts from [2, Prop.1, Lem. 1, Cor. 1]:

Fact 2.1

The code is a nondegenerate code of dimension and length For , denote by the coefficient matrix of . Then the Hamming weight of the corresponding codeword of depends only on . Consequently, if , then

As a result, the code has at most distinct weights, , given by for .

We call the function the partial trace. Note that , since . To determine the other weights , one would need to count the number of of rank at most with nonzero -th partial trace. Delsarte [9] used the theory of association schemes to solve an essentially equivalent problem of determining the number of with , and showed:

The case was already dealt with by Buckhiester in [6]. In the appendix of this paper, we obtain using different methods an alternative formula for , which may be of independent interest. For future use, we define for and . From Delsarte’s result it follows that

(3)

In fact, Delsarte considered codes obtained by evaluating elements from in all matrices of rank . Therefore his code can be seen as the projection of on the coordinates corresponding to matrices of rank .

Using Equation (3), we see that the nonzero weights of are given by

for . However, for a fixed , it is not obvious how are ordered or even which among them is the least. We will formulate a conjecture based (among others) on the following examples.

Example 2.2

If the code is trivial (containing only the zero word), while the code is not defined. Therefore the easiest nontrivial case occurs for . This case was considered in [2], where is was shown that

These formulae also follow fairly directly from Equations (3) and (2). It follows directly that and that is the minimum distance of .

Example 2.3

In this example we consider the determinantal code in case and . Using the formulae in Equations (3) and (2), we find the following table:

One sees that it is not true in general that whenever . However, in this example it is true that for a given , the weight is the smallest among all nonzero weights .

Example 2.4

In case in the previous example, all weights were the same. This holds in general: If , then and is a first order projective Reed-Muller code (cf. [15]). All nonzero codewords in this code therefore have weight . Note that combining this with Equations (3) and (2) we obtain for the following identity

Using Equation (2) with , we see that for apparently the following identity holds

This identity may readily be shown using for example [1, Thm 3.3] after interchanging the summation order. In any case, it is clear that Equations (3) and (2) may not always give the easiest possible expression for the weights.

While for and all weights are easy to compare with one another, the same cannot be said in case . We formulate the following conjecture.

Conjecture 2.5

Let be positive integers and an integer satisfying . The the following hold:

  1. All weights are mutually distinct.

  2. We have .

  3. For all , the weight lies between and .

3 Minimum distance of determinantal codes

Recall that in general for a linear code of length , i.e., for a linear subspace of , the Hamming weight of a codeword , denoted is defined by

The minimum distance of , denoted , is defined by

A consequence of Conjecture 2.5 would also be that is the minimum distance of . We will now show that this is indeed the case. We start by giving a rather compact expression for .

Proposition 3.1

Let , and be integers satisfying . Then

Proof.

First suppose that . In this case Example 2.2 implies that . On the other hand, using Equations (1) and (2), we see that , so the proposition holds for .

From now on we assume that (implying that also ). We will show that

(5)

Once we have shown this, the proposition follows using Equations (2) and (2). Let and suppose that . In that case, we may find uniquely determined square matrices

such that

(6)

The matrices and are indeed uniquely determined, since for and we have

and

These equations determine the values of given the matrix . The association of therefore is a well-defined map

The map is clearly surjective (one can for example choose as in the right hand side of Equation (6)), while the preimage of any matrix consist of the matrices of the form , with and as above and again chosen as in the right-hand-side of Equation (6). Equation (5) (and hence the proposition) then follows, since

Equation (5), and hence the proposition, follows directly from this. ∎

Note that the expression for from Equation (3) is considerably more involved that the expression obtained in the proof of Proposition 3.1. We now turn our attention to proving that actually is the minimum distance of the code . The proof involves several identities concerning and . The key is the following theorem in which the following quantity occurs:

Theorem 3.2

Let and , then

Proof.

Given a matrix , we denote by the matrix obtained from by deleting its -th row. Since either or , this defines a map . It is not hard to see that is surjective. In fact:

(7)

since if we obtain all elements of by adding a row from the rowspace of , while if we obtain all elements of by adding any row not from the rowspace of .

We will now prove the theorem by carefully counting the number of matrices such that , thus computing . The theorem then follows easily, since . We distinguish four cases:

  1. The -th column of is zero and has rank ,

  2. The -th column of is zero and has rank ,

  3. The -th column of is non zero and has rank ,

  4. The -th column of is non zero and has rank .

Case 1: The -th column of is zero and has rank . In this case , since otherwise . Therefore if and only if . By Equation (7), we find the following contribution to :

(8)

Case 2: The -th column of is zero and has rank . If , then by a similar reasoning as in case 1, we find a contribution to of magnitude

(9)

If , the situation is more complicated. If namely , then if and only if . Since and the -th column of is zero, all matrices with nonzero -th entry are in . This gives a contribution to of magnitude

(10)

If on the other hand , then if and only if . Since we already assumed that , we find a contribution to of magnitude

(11)

Case 3: The -th column of is non zero and has rank . Since the -th column of is non zero, the -th coordinates of elements from the row space of are distributed evenly over the elements of . This implies that regardless of the value of , a -th fraction of the matrices in contribute to . In total we find the contribution:

(12)

Case 4: The -th column of is non zero and has rank . Just as in case 3, since the -th column of is non zero, the -th coordinates of elements from the row space of are distributed evenly over the elements of . Therefore also the -th coordinates of elements not from the row space of are distributed evenly over the elements of . By a similar reasoning as in case 3, we find a contribution to of magnitude:

(13)

Adding all contributions to from Equations (8),(9),(10),(11),(12), and (13), the theorem follows. ∎

Corollary 3.3

Let and . Then

Proof.

By Equation (2) and Theorem 3.2 we have

The corollary now follows by the definition of and Equation (2). ∎

Corollary 3.4

Let and , then

In particular

Proof.

Using the previous corollary, we see that

This yields the first part of the corollary. The second part follows directly by choosing . ∎

We are now ready to prove our main theorem on the minimum distance.

Theorem 3.5

Let and . Then the minimum distance of the code is given by

Proof.

We already know that the only nonzero weights occurring in code are . Moreover, in case , we already know from Example 2.4 that the minimum distance is given by

Therefore we may assume . However, in this case the second part of Corollary 3.4 implies that cannot be larger than any of the other weights, since

The theorem then follows from Proposition 3.1. ∎

The above theorem gives a start to proving Conjecture 2.5. Exploring the above methods, we can do a little more as well as gain some information about codewords of minimum weight in .

Proposition 3.6

Let , then The code has exactly codewords of minimum weight and these codewords generate the entire code. More precisely, any codeword in is the sum of at most minimum weight codewords.

Proof.

Choosing in Corollary 3.4 and , we obtain that

so the first part of the proposition follows once we have shown that . In order to this, it is sufficient to produce one matrix of rank such that . However, this is easy to do: Let be a permutation matrix corresponding to a permutation on elements that fixes , but does not have other fixed points. Then , while any other diagonal element is zero. Now take to be the matrix such that if and , while otherwise. Then for any , we have , which is exactly what we wanted to show.

Now that we know that is strictly smaller than all other nonzero weights, the minimum weight codewords are exactly those such that has a coefficient matrix of rank . This gives exactly possibilities for and hence for . Now let be given. Assume that has coefficient matrix of rank . Since any matrix of rank can be written as the sum of matrices of rank , we can write for certain all having a coefficient matrix of rank . This implies that