# Second generalized Hamming weight of Projective Toric Code over Hypersimplices

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.

## Authors

• 5 publications
• 6 publications
02/25/2020

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### Determining the Generalized Hamming Weight Hierarchy of the Binary Projective Reed-Muller Code

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07/30/2019

### A note on projective toric codes

Let d≥ 1 be an integer, and let P be the convex hull in R^s of all integ...
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### Generalized Hamming weights of projective Reed--Muller-type codes over graphs

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## 1. Introduction

Let be the finite field with elements of characteristic . denotes the set of all vectors of length . Any non-empty subset of is a code and a subspace of is called a linear code. For any linear code, the generalized Hamming weights were introduced by Wei in [2]. 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, the authors in [1] have studied projective toric code of th-hypersimplex of degree denoted by and computed its dimension and minimum distance. In this note we compute the second generalized Hamming weight of this code as stated by the following theorem.

###### Theorem 1.1.

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

 d2(CP(d))=⎧⎪⎨⎪⎩q(q−2)d(q−1)s−d−2if d

The computation of -th generalized hamming weight for in under process.

## 2. Preliminaries

### 2.1. ([1]), Definition of the code CP(d)

Let . Let . Let and be an integer and 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 , denoted is equal to .

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

 evd:Ld→Fmq,  evd(f):=(f(P1),⋯,f(Pn))

.

In [1], Proposition and Theorem , the authors have calculated the dimension and minimum distance of which are stated in the following theorem.

###### Theorem 2.1.

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

 dimFq(CP(d))={(sd)if q≥3,1if q=2.
###### Theorem 2.2.

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

 δd=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩(q−2)d(q−1)s−d−1if d≤s/2, q≥3,(q−2)s−d(q−1)d−1if s/2

### 2.2. Generalized Hamming weight of Linear codes

The support of a linear code over is defined by

 supp(C):={1≤i≤n : xi≠0 for some x=(x1,⋯,xn)∈C}.

For , the th generalized Hamming weight of is defined by

 dr′(C):=min{ ∣supp(D)∣ : D is a linear subcode % of C with dim(D)=r′}

In particular, the first generalized Hamming weight of is the usual minimum distance. The weight hierarchy of the code is the set of generalized Hamming weights . These notions of generalized Hamming weights for linear codes were introduced by Wei in his paper [2].

An important property of generalized Hamming weight of have been listed in the following theorems.

###### Theorem 2.3.

[2]Monotonicity For an linear code with , we have

 1≤d1(C)

In the following section we give another proof for the minimum distance of in case .

### 2.3. Minimum distance for s<2d

Let denote the -th hypersimplex. Then we have a bijection between and given by the map

 ϕ:Δ(d,s)→Δ(s−d,s),  ϕ((a1,⋯,as))=(1−a1,⋯,1−as)

For , we define the polynomial

 f∗(t1,⋯,ts):=t1t2⋯tsf(t−11,⋯,t−1s).

Then, . Also, if and only if .

###### Proposition 2.4.

[1], Proposition 2.3 If , and , then

 ∣VT(f)∣=(q−1)s−1−(q−2)d(q−1)s−d−1.
###### Proposition 2.5.

If , and then,

 ∣VT(f)∣≤(q−1)s−1−(q−2)s−d(q−1)d−1.
###### Proof.

From the above we have and . Using the above proposition,

 ∣VT(f)∣=∣VT(f∗)∣≤(q−1)s−1−(q−2)s−d(q−1)d−1.

Conversely, let and consider the polynomial

 gk:=h1⋯hkt2k+1⋯ts,

where for . Let be the affine torus. Then . We have then by using the inclusion exclusion principle we get

Now,

 ∑1≤l1

therefore,

 ∣VT(gk)∣ =k∑i=1(−1)i−1(ki)(q−1)s−i =−(q−1)s−kk∑i=1(−1)i(ki)(q−1)k−i =−(q−1)s−k[−(q−1)s−k+k∑i=0(ki)(−1)i(q−1)k−i] =(q−1)s−(q−1)s−k(q−2)k =(q−1)s−(q−2)s−d(q−1)d. =(q−1)∣VT(gk)∣

## 3. Second Generalized Hamming weight of CP(d)

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 .

Before proceeding further we fix some notation. For polynomials and we denote the set of zeroes of in by and the set of zeroes of in by . Let be the standard lexicographic order on with .

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

 ∣VT(g)∣≤(q−1)s−(q−2)d(q−1)s−d.
###### Proof.

We prove the lemma by applying induction on . For , we have or . For , we have to check . We have the following table by direct calculations where

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

For , we have to check we have the following table by direct calculations where

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

 ∣VT(g)∣=(q−1)s−1≤(q−1)s−(q−2)d(q−1)s−d.

Therefore we assume that deg . We have by [1] Lemma that is square-free polynomial in of degree . Let . Therefore, by induction hypothesis

 ∣VT(g)∣ =∣VT(t1−α)∣+∣VT(h)∣−∣VT(t1−α)∩VT(h)∣ =(q−1)s−1+(q−2)∣VT1(h)∣ ≤(q−1)s−1+(q−2)[(q−1)s−1−(q−2)d−1(q−1)s−d] =(q−1)s−1+(q−2)(q−1)s−1−(q−2)d(q−1)s−d =(q−1)s−(q−2)d(q−1)s−d
• If for any , then let . For , define . Then there is an inclusion

 VT(g)↪∪q−1i=1({αi}×VT1(gi)), a↦a.

Then . 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

 ∣VT(g)∣ ≤q−1∑i=1∣VT1(gi)∣ ≤(q−1)[(q−1)s−1−(q−2)d−1(q−1)s−d] =(q−1)s−(q−2)d−1(q−1)s−d+1 ≤(q−1)s−(q−2)d(q−1)s−d,

since . But if there exists a of type then using the fact that

 (q−1)s−(q−2)d(q−1)s−d≥(q−1)s−(q−2)d′(q−1)s−d′,for d>d′,

we have

 ∣VT(g)∣ ≤q−1∑i=1∣VT1(gi)∣ ≤(q−1)[(q−1)s−1−(q−2)d(q−1)s−d−1] =(q−1)s−(q−2)d(q−1)s−d.

###### Lemma 3.2.

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

 ∣VT(f)∩VT(g)∣≤(q−1)s−q(q−2)d(q−1)s−d−1.
###### 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 have 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 have to consider the following cases.

• If and for some . Then

 ∣VT(f)∩VT(g)∣ =∣VT(t1−α)∣+∣VT(f1)∩VT(g1)∣−∣VT(t1−α)∩VT(f1)∩VT(g1)∣ =(q−1)s−1+(q−2)∣VT1(f1)∩VT1(g1)∣ ≤(q−1)s−1+(q−2)[(q−1)s−1−q(q−2)d−1(q−1)s−d−1] =(q−1)s−q(q−2)d(q−1)s−d−1,

the inequality follows from induction hypothesis.

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

 ∣VT(f)∩VT(g)∣ =∣(VT(t1−α)∪VT(f1))∩(VT(t1−β)∪VT(g1))∣ =∣VT(t1−α)∩VT(g1)∣+∣VT(t1−β)∩VT(f1)∣+∣VT(f1)∩VT(g1)∣ −∣VT(t1−α)∩VT(g1)∩VT(f1)∣−∣VT(t1−β)∩VT(f1)∩VT(g1)∣ =∣VT1(g1)∣+∣VT1(f1)∣+(q−3)∣VT1(f1)∩VT1(g1)∣ ≤2[(q−1)s−1−(q−2)d−1(q−1)s−d]+(q−3)[(q−1)s−1 −q(q−2)d−1(q−1)s−d−1] =2(q−1)s−1−2(q−2)d−1(q−1)s−d+(q−3)(q−1)s−1− −q(q−3)(q−2)d−1(q−1)s−d−1 =(q−1)s−(q−2)d−1(q−1)s−d−1(q2−q−2) ≤(q−1)s−q(q−2)d(q−1)s−d−1,

since .

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

 ∣VT(f)∩VT(g)∣ =∣(∪q−1i=1VT1(fi))∩VT((t1−β)g1)∣ =∣(∪q−1i=1VT(fi))∩VT(t1−β)∣+∣(∪q−1i=1VT1(fi))∩VT(g1)∣ −∣(∪q−1i=1VT1(fi))∩VT(t1−β)∩VT(g1)∣ =∣VT1(fj)∣+q−1∑i=1∣VT1(fi)∩VT1(g1)∣−∣VT1(fj)∩VT1(g1)∣.

For we have the following inequalities

 (q−1)s−q(q−2)d(q−1)s−d−1≥(q−1)s−q(q−2)d′(q−1)s−d′−1

and

 (q−1)s−(q−2)d(q−1)s−d≥(q−1)s−(q−2)d′(q−1)s−d′,

Using the inequalities, we have

 ∣VT(f)∩VT(g)∣ ≤(q−1)s−1−(q−2)d(q−1)s−d−1+(q−2)[(q−1)s−1−q(q−2)d(q−1)s−d−2] =(q−1)s−(q−2)d(q−1)s−d−1−q(q−2)d+1(q−1)s−d−2 =(q−1)s−(q−2)d(q−1)s−d−1[1+q(q−2)] =(q−1)s−(q−2)d(q−1)s−d+1 ≤(q−1)s−q(q−2)d(q−1)s−d−1,

since .

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

 ∣VT(f)∩VT(g)∣ ≤∣∪q−1i=1VT1(fi)∩∪q−1i=1VT1(gi)∣ ≤q−1∑i=1∣VT1(fi)∩VT1(gi)∣ ≤(q−1)[(q−1)s−1−q(q−2)d(q−1)s−d−2] =(q−1)s−q(q−2)d(q−1)s−d−1.

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

 A:={∣VT(F)∣: F={f1,f2}⊂Ld and F is linearly independent set }

and

 B:={∣VT(F)∣: F={f1,f2}⊂Ld,F is linearly independent set and LM(f1)≠LM(f2)}

Clearly . 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, .