On the generalized Hamming weights of certain Reed-Muller-type codes

There is a nice combinatorial formula of P. Beelen and M. Datta for the r-th generalized Hamming weight of an affine cartesian code. Using this combinatorial formula we give an easy to evaluate formula to compute the r-th generalized Hamming weight for a family of affine cartesian codes. If X is a set of projective points over a finite field we determine the basic parameters and the generalized Hamming weights of the Veronese type codes on X and their dual codes in terms of the basic parameters and the generalized Hamming weights of the corresponding projective Reed--Muller-type codes on X and their dual codes.

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1. Introduction

Let be a finite field and let be an -linear code of length and dimension , that is, is a linear subspace of with . The multiplicative group of is denoted by . The dual code of is given by

where , , and is the inner product of and .

Fix an integer . Given a subcode of (that is, is a linear subspace of ), the support of is the set of non-zero positions of , that is,

The -th generalized Hamming weight of , denoted , is the size of the smallest support of an -dimensional subcode [14, 16, 29]. Generalized Hamming weights have been extensively studied; see [2, 4, 9, 13, 15, 21, 25, 27, 30, 31] and the references therein. The study of these weights is related to trellis coding, –resilient functions, and was motivated by some applications from cryptography [29]. If , is the minimum distance of and is denoted .

In this note we give explicit formulas for the generalized Hamming weights of certain projective Reed-Muller-type codes and study the basic parameters (length, dimension, minimum distance) and the generalized Hamming weights of Veronese type codes and their dual codes.

These linear codes are constructed as follows. Let be a projective space over , let be a subset of where is the cardinality of the set , for all , and let be a polynomial ring with the standard grading, where is the

-vector space generated by the homogeneous polynomials in

of degree . Fix a degree . For each there is such that . Indeed suppose , there is at least one such that . Setting one has that and . Consider the evaluation map

This is a linear map between the -vector spaces and . The Reed–Muller–type-code of order associated to [5, 11], denoted , is the image of , that is

The -th generalized hamming weight of is sometimes denoted by . If , is the minimum distance of and is denoted by . The map is independent of the set of representatives that we choose for the points of , and the basic parameters of are independent of [19, Lemma 2.13] and so are the generalized Hamming weights of [8, Remark 1].

The basic parameters of are related to the algebraic invariants of the quotient ring , where is the vanishing ideal of (see for example [10, 20, 22]). Indeed, the dimension of is given by the Hilbert function of , that is,

the length of is the degree or the multiplicity of . Moreover, the regularity index of is the regularity of [28, pp. 226, 346] and is denoted . By the Singleton bound [27] one has for . Recall that the -invariant of , denoted , is the regularity index minus .

Let be subsets of and let be a projective cartesian set, where for all and . The Reed–Muller-type code is called an affine cartesian code [17].

There is a recent expression for the -th generalized Hamming weight of an affine cartesian code [1, Theorem 5.4], which depends on the -th monomial in ascending lexicographic order of a certain family of monomials (see [1] and the proof of Theorem 2.1). Using this result in Section 2 we give an easy to evaluate formula to compute the -th generalized Hamming weight for a family of affine cartesian codes (Theorem 2.1). Other formulas for the second generalized Hamming weight of an affine cartesian code are given in [7, Theorems 9.3 and 9.5].

Let be an integer and let be the set of all monomials in of degree , where . The map

is called the -th Veronese embedding. Given , the -th Veronese type code of degree is , the Reed–Muller-type code of degree on .

In Section 3 we are able to show that the Reed–Muller-type code over the set has the same basic parameters and the same generalized Hamming weights as the Veronese type code over the set for and (Theorem 3.2). As a consequence making we recover a result of Rentería and Tapia-Recillas [23, Proposition 1]. Also we show that the dual codes of and are equivalent (Theorem 3.5).

For all unexplained terminology and additional information we refer to [3, 28] (for the theory of Gröbner bases), and [18, 27] (for the theory of error-correcting codes and linear codes).

2. Generalized Hamming weights of some affine cartesian codes

In this section we present our main result on Hamming weights of certain cartesian codes. To avoid repetitions, we continue to employ the notations and definitions used in Section 1.

Let be a monomial order on and let be an ideal. If is a non-zero polynomial in , the leading monomial of is denoted by . The initial ideal of , denoted by , is the monomial ideal given by

A monomial is called a standard monomial of , with respect to , if is not in the ideal . The set of standard monomials, denoted , is called the footprint of . The footprint of is also called the Gröbner éscalier of . The image of the standard polynomials of degree , under the canonical map , , is equal to , and the image of is a basis of as a -vector space. This is a classical result of Macaulay [3, Chapter 5].

We come to our main result.

Theorem 2.1.

Let be a subset of , where and for . If and , then

where we set if or , and , are the unique integers such that and .

Proof. Setting , a polynomial ring with coefficients in , and , we order the set of all standard monomials of of degree at most with the lexicographic order (lex order for short), that is, if and only if the first non-zero entry of is positive. For , , the -th monomial of in decreasing lex order is

and the -th monomial of in ascending lex order, where , is

Case (I): . The case was proved in [17, Theorem 3.8]. Thus we may also assume . Therefore, applying [1, Theorem 5.4], we obtain that is given by

Case (II): . In this case the -th monomial of in ascending lex order is

Therefore, applying [1, Theorem 5.4], we obtain that is given by

Definition 2.2.

The set is called a projective torus.

Corollary 2.3.

Let be a projective torus in and let be the -th generalized Hamming weight of . Then

for , where , , .

Proof.

It follows readily from Theorem 2.1 making for . ∎

This corollary generalizes the case when is a projective torus in and :

Theorem 2.4.

[24, Theorem 3.5] Let be a projective torus in and let be the Reed–Muller-type code on of degree . Then its length is , its minimum distance is given by

where and are the unique integers such that , and , and the regularity of is .

The case when is a projective torus in and is treated in [6, Theorem 18].

3. Veronese type codes

Let be a polynomial ring over a field and let be the set of all monomials of of degree , where . The map

is called the -th Veronese embedding. Given , the -th Veronese type code of degree is , the Reed–Muller-type code of degree on . The next aim is to show that the Reed–Muller-type code has the same basic parameters and the same generalized Hamming weights as the Veronese type code for and .

Lemma 3.1.

is well-defined and injective.

Proof.

If , , , , then for some . Thus for all , that is, , here we are using as a short hand for . Thus is well-defined. To show that is injective assume that . Then for some one has for all . Pick such that and let . Note that for some . Then one has , that is, . For each , using the monomial , one has

Thus for all , that is, . ∎

We come to the main result of this section.

Theorem 3.2.

If , then the projective Reed–Muller-type codes and have the same basic parameters and the same generalized Hamming weights for and .

Proof.

Setting , let be a polynomial ring over the field with the standard grading. We can write , where , , and the ’s are in standard form, i.e., the first non-zero entry of is for all . By Lemma 3.1 the map is injective. Thus and have the same length. As are in standard form, for each there is such that . Therefore, by [19, Lemma 2.13], we may assume that the Reed–Muller-type code is the image of the evaluation map

(3.1)

and the Veronese type code is the image of the evaluation map

(3.2)

where for , and are polynomials in such that for . For any polynomial in , , one has

As is equal to , any in can be written as for some in . Therefore, using Eq. (3), we get

As a consequence, setting and , one has

(3.4)

where for in . This means that the linear codes and are equivalent [8, Remark 1]. Thus the dimension and minimum distance of and are the same, and so are the generalized Hamming weights. ∎

For convenience we recall the following classical result of Sørensen [26].

Theorem 3.3.

(Sørensen [26]) Let be a finite field and let be the classical projective Reed–Muller code of degree on the set . Then , the minimum distance of is given by

where and are the unique integers such that and , and the regularity of is .

Veronese codes are a natural generalization of the classical projective Reed–Muller codes.

Corollary 3.4.

[23, Proposition 1] If , then the projective Reed–Muller-type codes and have the same basic parameters for and .

Proof.

This follows at once from Theorem 3.2 making . ∎

As a byproduct we relate the dual codes of and .

Theorem 3.5.

If is a subset of , then and are equivalent codes and

where , with for all , is the vector that was given in the proof of Theorem 3.2.

Proof.

Let . Then , for all . By using Eq. (3.4) we conclude that

for all . Therefore

for all . Thus

(3.5)

Furthermore one has the equalities

(3.6)

and the equality follows from Eqs. (3.5) and (3.6). Thus and are equivalent codes [8, Remark 1]. ∎

Corollary 3.6.

If , , and , then the linear code is equivalent to

where is the subspace of generated by and .

Proof.

This result follows at once from Theorem 3.5 and [26, Theorem 2]. ∎

The rest of this section is devoted to show some explicit examples.

Example 3.7.

Let be the field . If , then by Theorem 3.3 the basic parameters of the classical projective Reed–Muller-type code of degree are given by

The dimension of is . The regularity of is and the -invariant is .

Example 3.8.

Let be the field . If , , and , then by Theorem 3.2 and Example 3.7 the parameters of the Veronese code of degree are given by

The regularity of is and the -invariant is .

Example 3.9.

Let be the field . If , is a projective torus in , and is the corresponding Veronese type code, then by Corollary 2.3, Theorem 2.4, [6, Theorem 18], and Macaulay [12], we obtain the following information for :