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
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
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 .
Find formulas for the minimum distance or more generally for the generalized Hamming weights of the projective toric code .
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:
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
and if . If , the ideal is referred to as a colon ideal. Note that is a zero-divisor of if and only if .
[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
[10, Lemma 3.3] Let be the ideal of generated by . If is a zero-divisor of and , then
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.
If , and , then
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
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
Since , we obtain
According to [9, Lemma 4.1] the following inequality holds
Let be a squarefree polynomial of . If for some and , then is a polynomial in the variables and is squarefree.
We can write , , for , and distinct monomials. Then
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. ∎
Let be a squarefree polynomial in of total degree at most and let be the affine torus of . If and , then
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
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
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
where is the affine torus . Let be the elements of . Then
and . Therefore, using the equalities and , together with Eq. (2.5), we get
Case (II): for all . Let be the elements of , let be the polynomial for , and let be the affine torus . There is an inclusion
Hence, applying the induction hypothesis to each , one has
This completes the proof of the inequality. ∎
Let be the projective toric code of of degree . Then
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.
Let be the projective toric code of of degree and let be its minimum distance. Then
Assume that and . We set and . Let and be the affine and projective torus in and , respectively. There is such that
Thus, by Proposition 2.3, one has . Consider the squarefree homogeneous polynomial of degree
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
The variables occurring in and are disjoint for . Thus counting monomials in each of the intersections one obtains
and consequently the number of zeros of in is given by
Assume that , and . There is such that . We set and . To prove the inequality pick such that
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
Thus , as required. Consider the squarefree homogeneous polynomial of degree
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
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.
For and , the list of values of the length, dimension and minimum distance of are given by the following table.
Computations with Macaulay  were important to study some examples to have a better understanding of toric codes over hypersimplices.
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