    # Projective toric codes over hypersimplices

Let d≥ 1 be an integer, and let P be the convex hull in R^s of all integral points e_i_1+...+e_i_d such that 1≤ i_1<...< i_d≤ s, where e_i is the i-th unit vector in R^s. Given a finite field F_q, we determine the minimum distance of the projective toric code C_P(d) associated to the hypersimplex P using the projective footprint function of a graded ideal.

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## 1. Introduction

Let be a polynomial ring over a finite field with the standard grading, , let be an integer, and let be the convex hull in of all integral points such that , where is the -th unit vector in . The lattice polytope is called the -th hypersimplex of [15, p. 84]. 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 . The vanishing ideal of , denoted , is the graded ideal of generated by all homogeneous polynomials of that vanish at all points of . If and , we denote the set of zeros of in by , and the set of zeros of in by .

The projective toric code of of degree , denoted or simply , is the image of the evaluation map

 evd:Ld→Km,f↦(f(P1),…,f(Pm)),

where is the -vector subspace of generated by all , , such that and is the set of all points of the projective torus of . We may assume that the first entry of each is . A monomial of is in if and only if is squarefree and has degree , that is, an integral point of is in if and only if is in , where . The basic parameters of are the length , the dimension , and the minimum distance

 δ(CP(d)):=min{|T∖VT(g)|:g∈Ld∖I(T)}.

Toric codes were introduced by Hansen  and have been actively studied in the last decade, see  and the references therein. These codes are affine-variety codes in the sense of [3, p. 1]. If we replace by in the evaluation map, the image of the resulting map is the projective Reed–Muller-type code over the projective torus . The minimum distance of was determined in [12, Theorem 3.5]. Note that is the projective toric code of the convex hull in of all points in such that .

We are set out to solve part of the following problem using the projective footprint function of a graded ideal [9, 10].

###### Problem 1.1.

Find formulas for the minimum distance or more generally for the generalized Hamming weights of the projective toric code .

The length of is . In Proposition 2.6 we determine the dimension of . Our main result is a formula for the minimum distance of (Theorem 2.7).

For all unexplained terminology and additional information we refer to [4, 14, 17] (for the theory of Gröbner bases and Hilbert functions), and [8, 16] (for the theory of error-correcting codes and linear codes).

## 2. Minimum distance of certain projective toric codes

In this section we determine the basic parameters of . To avoid repetitions, we continue to employ the notations and definitions used in Section 1.

Let be a graded ideal of of Krull dimension . The Hilbert function of is:

 HI(d):=dimK(Sd/Id),   d=0,1,2,…,

where . By a theorem of Hilbert [14, p. 58], there is a unique polynomial of degree such that for . The degree of the zero polynomial is .

The degree or multiplicity of , denoted , is the positive integer given by

 deg(S/I):=(k−1)!limd→∞HI(d)/dk−1  if  k≥1,

and if . If , the ideal is referred to as a colon ideal. Note that is a zero-divisor of if and only if .

###### Lemma 2.1.

[9, Lemma 3.2] Let be a subset of the projective space over the finite field and let be its graded vanishing ideal. If is homogeneous, then

 |VX(f)|={deg(S/(I(X),f))% if (I(X):f)≠I(X),0if (I(X):f)=I(X).
###### Lemma 2.2.

[10, Lemma 3.3] Let be the ideal of generated by . If is a zero-divisor of and , then

 deg(S/(L,ta))=(q−1)s−1−(q−1−a1)⋯(q−1−as−1).

To prove the next result we use the footprint technique for projective Reed–Muller-type codes introduced in . A polynomial is called squarefree if all its monomials are squarefree.

###### Proposition 2.3.

If , and , then

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

We set . If , the inequality clearly holds because its right hand side is positive. Thus we may assume . Hence, by Lemma 2.1, one has and

 (2.1) |VT(f)|=deg(S/(I,f)).

Let be the lexicographical order on with . As the ideal is generated by the set  and is a Gröbner basis of , the initial ideal of is generated by the set of monomials . Let be the initial term of . Since is squarefree, so is . Note that is a zero-divisor of . Indeed if is regular on , then and because the only associated prime of is , a contradiction because . As , cannot be in . Therefore, by Lemma 2.2, we get

 deg(S/(L,ta))={(q−1)s−1−(q−2)d−1(q−1)s−d if as=1,(q−1)s−1−(q−2)d(q−1)s−d−1 if as=0.

Since , we obtain

 (2.2) deg(S/(L,ta))≤(q−1)s−1−(q−2)d(q−1)s−d−1.

According to [9, Lemma 4.1] the following inequality holds

 (2.3) deg(S/(I,f))≤deg(S/(in≺(I),in≺(f)))=deg(S/(L,ta)).

Therefore the inequality follows at once from Eqs. (2.1)–(2.3). ∎

###### Lemma 2.4.

Let be a squarefree polynomial of . If for some and , then is a polynomial in the variables and is squarefree.

###### Proof.

We can write , , for , and distinct monomials. Then

 (2.4) g=λ1t1f1+⋯+λrt1fr−βλ1f1−⋯−βλrfr.

We proceed by contradiction assuming that divides for some and choose and such that divides and does not divides for . As is squarefree, by Eq. (2.4), the monomial must be equal to for some , a contradiction because does not divides . This proves that is a polynomial in the variables . Hence are distinct monomials. As is squarefree, by Eq. (2.4), is squarefree for , that is, is squarefree, as required. ∎

###### Proposition 2.5.

Let be a squarefree polynomial in of total degree at most and let be the affine torus of . If and , then

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

We proceed by induction on . Note that because is squarefree. Assume that . Then because . If is a monomial, then and the inequality holds. Thus we may assume that is not a monomial. As and , there are essentially three cases to consider: (a) , where for , (b) , with , and (c) , with . To show the case we need to prove that . Take root of in . In case (a) one has

 x1x2+λ1x1+λ2x2=0  and  x2(x1+λ2)=−λ1x1.

Then and . Thus has at most roots in , as required. The cases (b) and (c) are also easy to show.

Assume . Then , and . By permuting variables we may assume that occurs in some monomial of of degree .

Case (I): for some . Setting and fixing a graded monomial order on with , by the division algorithm [1, Theorem 1.5.9, p. 30], we can write , for some in such that does not divide any monomial of , that is, is a polynomial in the variables . Making in this equality, we obtain that is the zero polynomial. Thus . If , then and

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

Thus we may assume that and . As is squarefree and , by Lemma 2.4, it follows that does not contain , is squarefree, and the total degree of is at most . Hence, by induction hypothesis, one has

 (2.5) |VT1(g1)|≤(q−1)s−1−(q−2)s−d(q−1)d−1,

where is the affine torus . Let be the elements of . Then

 VT(g1)=q−1⋃i=1({βi}×VT1(g1)),

and . Therefore, using the equalities and , together with Eq. (2.5), we get

 |VT(g)| = |VT(h)|+|VT(g1)|−|VT(h)∩VT(g1)| = (q−1)s−1+(q−2)|VT1(g1)| ≤ (q−1)s−1+(q−2)[(q−1)s−1−(q−2)s−d(q−1)d−1] = [(q−1)s−1+(q−2)(q−1)s−1]−(q−2)s−d+1(q−1)d−1 = (q−1)s−(q−2)s−d+1(q−1)d−1.

Case (II): for all . Let be the elements of , let be the polynomial for , and let be the affine torus . There is an inclusion

 VT(g)→q−1⋃i=1({βi}×VT1(gi)),    a↦a.

Hence, applying the induction hypothesis to each , one has

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

This completes the proof of the inequality. ∎

###### Proposition 2.6.

Let be the projective toric code of of degree . Then

 dimK(CP(d))={(sd) if q≥3, 1 if q=2.
###### Proof.

Assume that . The number of squarefree monomial of of degree is . Then one has . Hence it suffices to show that the evaluation map is injective. Take in , that is, is in . Using that is generated by the set , , and the monomials of are squarefree, it follows readily that .

Assume that . Then , , , and . ∎

We come to our main result.

###### Theorem 2.7.

Let be the projective toric code of of degree and let be its minimum distance. Then

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

Assume that and . We set and . Let and be the affine and projective torus in and , respectively. There is such that

 (2.6) δd=min{|T∖VT(g)|:0≠g∈Ld}=|T∖VT(f)|=(q−1)s−1−|VT(f)|.

Thus, by Proposition 2.3, one has . Consider the squarefree homogeneous polynomial of degree

 fd=h1⋯hd=(t1−t2)⋯(t2d−1−t2d),

where for . By Eq. (2.6), to prove the inequality it suffices to show that the polynomial has exactly roots in . If , one has . Thus we need only show the equality . As is equal to , using the inclusion-exclusion principle [2, p. 38, Formula 2.12], we get

 |VT(fd)| = ∑1≤ℓ1≤d|VT(hℓ1)|−∑1≤ℓ1<ℓ2≤d|VT(hℓ1)∩VT(hℓ2)| ±⋯+(−1)d−1|VT(h1)∩⋯∩VT(hd)|.

The variables occurring in and are disjoint for . Thus counting monomials in each of the intersections one obtains

 ∑1≤ℓ1<⋯<ℓi≤d|VT(hℓ1)∩⋯∩VT(hℓi)|=(di)(q−1)s−i,

and consequently the number of zeros of in is given by

 (2.7) |VT(fd)| = d∑i=1(−1)i−1(di)(q−1)s−i=(q−1)s−dd∑i=1(−1)i−1(di)(q−1)d−i = (q−1)s−d[(q−1)d−((q−1)−1)d] = (q−1)s−d[(q−1)d−(q−2)d] = (q−1)s−(q−2)d(q−1)s−d=(q−1)ϕ(d).

Assume that , and . There is such that . We set and . To prove the inequality pick such that

 δd=min{|T∖VT(g)|:0≠g∈Ld}=|T∖VT(f)|=(q−1)s−1−|VT(f)|.

We can write in standard form . Then is a zero of in if and only if is a zero of in . Setting and noticing , by Proposition 2.5, one has

 |VT(f)|=|VT1(g)|≤(q−1)s−1−(q−2)s−d(q−1)d−1=(q−1)s−1−η1(d).

Thus , as required. Consider the squarefree homogeneous polynomial of degree

 gk:=h1⋯hkt2k+1⋯tk+d=(t1−t2)⋯(t2k−1−t2k)t2k+1⋯tk+d,

where for . To prove the inequality it suffices to show that the polynomial has exactly roots in . Thus we need only show the equality . Since is equal to , , and , using Eq. (2.7) with and noticing , we get

 |VT(gk)| = |VT(h1⋯hk)|=|VT(fk)| = (q−1)s−(q−2)k(q−1)s−k = (q−1)s−(q−2)s−d(q−1)d=(q−1)ϕ1(d).

Assume that . Then is generated by and the toric code is generated by , . Since for all , one has .

Assume that . Then , , and . Thus . ∎

The minimum distance of the projective Reed–Muller-type code is non-increasing as a function of [11, Proposition 5.2]. This is no longer the case for the minimum distance of the projective toric code as the next simple example shows.

###### Example 2.8.

For and , the list of values of the length, dimension and minimum distance of are given by the following table.

 d1234m8888dimK(CP(d))4641δ(CP(d))4248

## Acknowledgments

Computations with Macaulay  were important to study some examples to have a better understanding of toric codes over hypersimplices.

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