Generalized Hamming weight of Projective Toric Code over Hypersimplices

02/25/2020
by   Nupur Patanker, et al.
IISER Bhopal
0

The d-th hypersimplex of R^s is the convex hull in R^s of all integral points e_i_1+e_i_2+...+e_i_d such that 1 ≤ i_1 <... < i_d≤ s where e_i is the i-th unit vector in R^s. In [1], the authors have defined projective toric code of P of degree d denoted by C_P(d) and computed its dimension and minimum distance. In this note, we compute the second generalized Hamming weight of these codes. We also calculated the r-th generalized Hamming weight (1 ≤ r ≤ dim(C_P(d))) in certain cases.

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1. Introduction

Let be the finite field with elements of characteristic . The set of all vectors of length over , denote by , has the structure of vector space over . A subspace of is called a linear code. For any linear code over , the support of is defined by


For , the th generalized Hamming weight of code is defined by


In particular, the first generalized Hamming weight of is the usual minimum distance. The set of generalized Hamming weights is called the weight hierarchy of the code . The notions of generalized Hamming weights for linear codes were introduced by Tor Helleseth, Torleiv KlØve and Johannes Mykkeltveit in [2] and rediscovered by Wei in his paper [3]. The study of these weights was motivated from some applications in cryptography. The generalized Hamming weight of various linear codes have been studied for many years.

Recently, Delio Jaramillo, Maria Vaz Pinto and Rafael H. Villarreal in [1] have introduced projective toric code of th-hypersimplex of degree denoted by and computed its dimension and minimum distance. In this note, we prove the following theorem giving the second generalized Hamming weight of this code.

Theorem 1.1.

Let be the projective toric code of of degree and let be its second generalized Hamming weight. Then

The computation of -th generalized hamming weight for is also done in few cases.

2. Preliminaries

2.1. [1] Definition of the code

Let and be integers such that and . Let be the polynomial ring in variables over with standard grading. Let be the convex hull in of all integral points such that where is the -th unit vector in . The affine torus of the affine space is given by where is the multiplicative group of , and the projective torus of the projective space over the field is given by , where is the image of under the map , . The cardinality of is equal to . For polynomials and we denote the set of zeroes of in by and the set of zeroes of in by .

Let be the vector subspace of generated by all such that . be the set of all points of the projective torus of . We may assume that the first entry of each is . Thus, . The code is defined as the image of the evaluation map


The code has length . The minimum distance is given by

The following theorems give the dimension and minimum distance of .

Theorem 2.1.

[1], Theorem Let be the projective code of of degree . Then

Theorem 2.2.

[1], Theorem Let be the projective code of of degree and let be its minimum distance. Then

2.2. [6], Affine Hilbert function

Let be any field. Let denote the subset of consisting of polynomials of degree at most . For an ideal of , we denote by , the subset of consisting of polynomials of degree at most . The function

given by

is called the affine Hilbert function of . Note that if then .

Given a subset of , we define the affine Hilbert function of , denoted by , as , where is the vanishing ideal of in .

Proposition 2.3.

Fix an ordering on , then

  1. For any ideal of , we have .

  2. If is a monomial ideal of , then is the number of monomials of degree at most that does not lie in .

Proposition 2.4.

Let be a finite set. Then, for sufficiently large .

3. Second Generalized Hamming weight of

The second generalized Hamming weight of is given by

For , , and . Thus dimension of this code is and second generalized hamming weight doesn’t make sense. Also in case dimension of code is so we only consider and .

A polynomial is called square-free if all its monomials are square-free. For two distinct square-free polynomials in the following two lemmas give the upper bounds on the cardinality of the sets and . The first lemma has been proved in [1] Proposition . We give another proof of the proposition.

Lemma 3.1.

Let and . Let be a square-free polynomial of degree in . Let be the affine torus then

Proof.

We prove this lemma by induction on . For , we have either or .
For , we have to show that . By direct calculations, we obtain the following table, where ,

The first column contains the various choices of polynomial in two variables and of degree one and the second column specifies number of zeroes in of the corresponding polynomial. From the above table we clearly have that .

For , we have to show that By direct calculations, we have the following table, where ,

From the above table we clearly have that . Thus, the lemma is true for . Now, we assume that . We consider the following two cases:

  • If for some . Then where no term of is divisible by . Putting we get that is the zero polynomial. Thus, . If deg , then and

    Therefore, we assume that deg . By [1], Lemma we have that is square-free polynomial in and also we have deg . Let . Then, by induction hypothesis

  • If for any , then let . For , define . We have the following inclusion

    Therefore . For each , , we have the following cases

    1. If each term of contains then is a square-free polynomial in variables of degree .

    2. If there exists a term in not containing then is a square-free polynomial in variables of degree .

    Now, if each is of type , then

    as . But if there exists atleast one of type , then using the fact that

    we have

Lemma 3.2.

For and , let and be two distinct square-free polynomials of degree in . Let be the affine torus. Then

Proof.

We prove this lemma by applying induction on . For , we have and we have to show for any two distinct square-free polynomials and in two variables of degree one, we have . For , we obtain the following table by direct calculations.

From the above table we have . Thus, the lemma is true for . We assume that . Let . To prove the lemma we consider the following cases.

  • If and for some . Then, by induction hypothesis

  • If and for some and . Then, by inclusion-exclusion principle

    as .

  • If for any but for some . Let and for each , set and for some . Then

    Now, for we have the following inequalities

    and

    Using the inequalities, we have

    as .

  • If and for any . Then for , define and. Then

Now we come to the question of calculating the second generalized Hamming weight of . For the answer, we first consider the following two sets.

and


Clearly . Now we show that .

For some if then but if for some we have then consider the following polynomials

,

Then, and is a linearly independent set. Also and . Therefore, .

Lemma 3.3.

From Lemma , we conclude that

(3.1)

We have the following results from [4].

Lemma 3.4.

[4],Lemma Let be a finite subset of over a field . If is a set of homogeneous polynomials of , then if and only if .

Lemma 3.5.

[4],Lemma Let be a finite subset of over a field and be its vanishing ideal. If is a set of homogeneous polynomials of , then the number of points of is given by

Theorem 3.6.

[4], Corollary Let be a finite subset of over a field and be its vanishing ideal, and let be a monomial order. If is a finite set of homogeneous polynomials of and , then


and if .

Lemma 3.7.

Let be a monomial order, let be an ideal, let be a set of polynomials of of positive degree, and let be a set of initial terms of . If , then .

Let be the standard lexicographic order on with . Now, we have and . Let