Evaluation codes and their basic parameters

TL;DR

This paper provides formulas and bounds for generalized Hamming weights of evaluation codes, including Reed--Muller, toric, and squarefree codes, advancing understanding of their parameters and minimum distances.

Contribution

It introduces degree formulas for generalized Hamming weights and establishes lower bounds, specifically analyzing Reed--Muller, toric, and squarefree evaluation codes.

Findings

Degree formulas for generalized Hamming weights of Reed--Muller codes

Minimum distance of toric codes over hypersimplices determined

First and second generalized Hamming weights of squarefree evaluation codes identified

Abstract

The aim of this work is to give degree formulas for the generalized Hamming weights of evaluation codes and to show lower bounds for these weights. In particular, we give degree formulas for the generalized Hamming weights of Reed--Muller-type codes, and we determine the minimum distance of toric codes over hypersimplices, and the 1st and 2nd generalized Hamming weights of squarefree evaluation codes.

Equations

$$\mathrm{ev}:S\to K^m,f\mapsto \left(f(P_1),\dots ,f(P_m)\right).$$

$$(I:(F)):=\{g\in S|g(F)\in I\}$$

$$\chi (D):=\{i|\exists (a_1,\dots ,a_m)\in D,a_i\ne 0\}.$$

$${\delta }_r({\mathcal{L}}_X)=\mathrm{deg}(S/I)-\max \{\mathrm{deg}(S/(I,F))|F\in {\mathcal{L}}_{\prec ,r}\}\text{mbox}for1\le r\le {\dim }_K({\mathcal{L}}_X).$$

$${\mathrm{fp}}_r({\mathcal{L}}_X):=\mathrm{deg}(S/I)-\max \{\mathrm{deg}(S/({\mathrm{in}}_{\prec }(I),N))|N\in {\mathcal{N}}_{\prec ,r}\}.$$

$${\mathrm{fp}}_r({\mathcal{L}}_X)\le {\delta }_r({\mathcal{L}}_X)\text{mbox}for1\le r\le {\dim }_K({\mathcal{L}}_X).$$

$${\mathrm{ev}}_d:K{V}_d\to K^m,f\mapsto \left(f(P_1),\dots ,f(P_m)\right),$$

$$\delta ({\mathcal{C}}_{\mathcal{P}}(d))=\{\begin{array}{ll}(q-2{)}^d(q-1{)}^{s-d}& \text{mbox}ifd\le s/2,q\ge 3,\\ (q-2{)}^{s-d}(q-1{)}^d& \text{mbox}ifs/2<d<s,q\ge 3,\\ (q-1{)}^s& \text{mbox}ifd=s,\\ 1& \text{mbox}ifq=2.\end{array}$$

$${\delta }_2({\mathcal{C}}_{\le d})=\{\begin{array}{ll}(q-2{)}^{s-1}(q-1)& \text{mbox}ifd=s,\\ (q-2{)}^d(q-1{)}^{s-d-1}q& \text{mbox}ifd<s.\end{array}$$

$$H_I^a(d):={\dim }_K(S_{\le d}/I_{\le d}),d=0,1,2,\dots ,$$

$$\dim (S[u]/I^h)=\dim (S/I)+1\text{mbox}and{H}_I^a(d)=H_{I^h}(d)\text{mbox}ford\ge 0,$$

$$\mathrm{deg}(S/I)=\sum _{\mathrm{ht}({\mathfrak{q}}_i)=\mathrm{ht}(I)}\mathrm{deg}(S/{\mathfrak{q}}_i).$$

$$I_P=(t_1-p_1,\dots ,t_s-p_s),\mathrm{deg}(S/I_P)=1,$$

$$\bigcap _{i=1}^m{\mathfrak{p}}_i=I(X)=(I(X):(F))=\bigcap _{i=1}^m({\mathfrak{p}}_i:(F))=\bigcap _{i\ne k}({\mathfrak{p}}_i:(F))\subset {\mathfrak{p}}_k.$$

$$D_j:=\{p_{i,j}:i=1,\dots ,m\}=\{a_{1,j},\dots ,a_{d_j,j}\},$$

$$f_j:=(t_j-a_{1,j})\cdots (t_j-a_{d_j,j}),j=1,\dots ,s,$$

$$|V_X(F)|=\{\begin{array}{ll}\mathrm{deg}(S/(I(X),F))& \text{mbox}if(I(X):(F))\ne I(X),\\ 0& \text{mbox}if(I(X):(F))=I(X).\end{array}$$

$$I_P=(t_1-p_1,\dots ,t_s-p_s),\mathrm{deg}(S/I_P)=1,$$

$$\mathrm{deg}(S/(I(X),F))=\mathrm{deg}(S/I(X))=\sum _{i=1}^m\mathrm{deg}(S/I_{P_i})=|X|=|V_X(F)|.$$

$$(I(X),F)=I_{P_1}\bigcap \cdots \bigcap I_{P_n},$$

$$|V_X(F)|=\sum _{P_i\in V_X(F)}\mathrm{deg}(S/I_{P_i})=\sum _{i=1}^n\mathrm{deg}(S/I_{P_i})=\mathrm{deg}(S/(I(X),F)).$$

$${\Delta }_{\prec }(I)\subset {\Delta }_{\prec }({\mathrm{in}}_{\prec }(g_1),\dots ,{\mathrm{in}}_{\prec }(g_n)),$$

$$|V_X(F)|=\mathrm{deg}(S/(I(X),F))={\dim }_K(S/(I(X),F))=|{\Delta }_{\prec }(I(X),F)|.$$

$$|V_X(F)|=\mathrm{deg}(S/(I,F))\le \mathrm{deg}(S/({\mathrm{in}}_{\prec }(I),{\mathrm{in}}_{\prec }(F)))\le \mathrm{deg}(S/I)=|X|,$$

$${\Delta }_{\prec }(J)={\Delta }_{\prec }(I,F)\subset {\Delta }_{\prec }({\mathrm{in}}_{\prec }(g_1),\dots ,{\mathrm{in}}_{\prec }(g_n),{\mathrm{in}}_{\prec }(F))=$$

$${\Delta }_{\prec }({\mathrm{in}}_{\prec }(I),{\mathrm{in}}_{\prec }(F))={\Delta }_{\prec }(L)\subset {\Delta }_{\prec }({\mathrm{in}}_{\prec }(g_1),\dots ,{\mathrm{in}}_{\prec }(g_n))={\Delta }_{\prec }(I).$$

$$H_J^a(d)=\mathrm{deg}(S/J)\le H_L^a(d)=\mathrm{deg}(S/L)\le H_I^a(d)=\mathrm{deg}(S/I)$$

$$\mathrm{deg}(S/J)=\sum _{\mathfrak{p}\in \mathrm{Ass}(J)}\mathrm{deg}(S/\mathfrak{p})<\sum _{\mathfrak{p}\in \mathrm{Ass}(I)}\mathrm{deg}(S/\mathfrak{p})=\mathrm{deg}(S/I),$$

$$\chi (D):=\{i|\exists (a_1,\dots ,a_m)\in D,a_i\ne 0\}.$$

$${\delta }_r(C):=\min \{|\chi (D)|:D\text{mbox}isasubcodeofC\text{mbox}with{\dim }_K(D)=r\}.$$

$$1\le {\delta }_1(C)<\cdots <{\delta }_k(C)\le m,$$

Full text

The first author was supported by a scholarship from CONACYT, Mexico. The second author was partially supported by the Center for Mathematical Analysis, Geometry and Dynamical Systems of Instituto Superior Técnico, Universidade de Lisboa. The third author was supported by SNI, Mexico.

1. Introduction

Let $S=K[t_{1},\ldots,t_{s}]=\bigoplus_{d=0}^{\infty}S_{d}$ be a polynomial ring over a finite field $K=\mathbb{F}_{q}$ with the standard grading and let $X=\{P_{1},\ldots,P_{m}\}$ be a set of distinct points in the affine space $\mathbb{A}^{s}:=K^{s}$. The evaluation map, denoted ${\rm ev}$, is the $K$-linear map given by

[TABLE]

The kernel of ${\rm ev}$, denoted $I(X)$, is the vanishing ideal of $X$ consisting of the polynomials of $S$ that vanish at all points of $X$. This map induces an isomorphism of $K$-linear spaces between $S/I(X)$ and $K^{m}$. Let $\mathcal{L}$ be a linear subspace of $S$ of finite dimension. The image of $\mathcal{L}$ under the evaluation map, denoted $\mathcal{L}_{X}$, is called an evaluation code on $X$ [50, 52].

Let $\prec$ be a monomial order on $S$ [10, p. 54] and let $I=I(X)$ be the vanishing ideal of $X$. The monomials of $S$ are denoted $t^{a}:=t_{1}^{a_{1}}\cdots t_{s}^{a_{s}}$, $a=(a_{1},\dots,a_{s})$ in $\mathbb{N}^{s}$, where $\mathbb{N}=\{0,1,\ldots\}$. We denote the initial monomial of a non-zero polynomial $f\in S$ by ${\rm in}_{\prec}(f)$ and the initial ideal of $I$ by ${\rm in}_{\prec}(I)$. A monomial $t^{a}$ is called a standard monomial of $S/I$, with respect to $\prec$, if $t^{a}\notin{\rm in}_{\prec}(I)$. The footprint of $S/I$, denoted $\Delta_{\prec}(I)$, is the set of all standard monomials of $S/I$. The footprint has been used in connection with many kinds of codes [12, 13, 14, 24].

The linear code $\mathcal{L}_{X}$ is called a standard evaluation code on $X$ relative to $\prec$ if $\mathcal{L}$ is a linear subspace of $K\Delta_{\prec}(I)$, the $K$-linear space spanned by $\Delta_{\prec}(I)$. A polynomial $f$ is called a standard polynomial of $S/I$ if $f\neq 0$ and $f$ is in $K\Delta_{\prec}(I)$. As the field $K$ is finite, there are only a finite number of standard polynomials.

The aim of this work is to introduce general methods, similar to those of [9, 17, 37], to study the basic parameters of the family of evaluation codes and those of certain interesting subfamilies, such as, affine and projective Reed–Muller-type codes, generalized toric codes, toric codes over hypersimplices and squarefree evaluation codes. This will also allow us to gain insight of the geometry of affine varieties and systems of polynomial equations over finite fields [3, 22, 30, 31]. We study standard evaluation codes first and then, using a monomial order, we show how to transform an evaluation code into a standard evaluation code.

If $F$ is a finite subset of $S$, the affine variety of $F$ in $X$, denoted $V_{X}(F)$, is the set of all $\alpha\in X$ such that $f(\alpha)=0$ for all $f\in F$. The ideal

[TABLE]

is referred to as a colon ideal. This ideal is useful to determine whether or not the affine variety $V_{X}(F)$ is non-empty (Lemma 2.5).

The degree of the quotient ring $S/I$, denoted $\deg(S/I)$, is defined using Hilbert functions at the beginning of Section 2. This invariant plays a unifying role in the theory of affine and projective varieties over finite fields. For instance, using the degree, one has similar formulas for $|V_{X}(F)|$ when $X$ is a set of affine or projective points (cf. Lemma 2.8 and [17, Lemma 3.4]). The footprint of $S/I$ combined with the degree (Theorems 2.11 and 2.12) will be used to compute and to find lower bounds for the generalized Hamming weights of evaluation codes over $X$.

We now turn our attention to standard evaluation codes and present degree formulas, and degree-footprint lower bounds formulas, for their generalized Hamming weights. The parameters of the linear code $\mathcal{L}_{X}$ that we consider are:

• (a)

length: $|X|$,

• (b)

dimension: $\dim_{K}(\mathcal{L}_{X})$, and

• (c)

$r$-th generalized Hamming weight: $\delta_{r}(\mathcal{L}_{X})$.

For convenience we recall the notion of generalized Hamming weight of a linear code [25, 54]. Let $C$ be a $[m,k]$ linear code of length $m$ and dimension $k$, that is, $C$ is a linear subspace of $K^{m}$ with $k=\dim_{K}(C)$. Let $1\leq r\leq k$ be an integer. Given a subcode $D$ of $C$ (that is, $D$ is a linear subspace of $C$), the support $\chi(D)$ of $D$ is the set

[TABLE]

The $r$-th generalized Hamming weight of $C$, denoted $\delta_{r}(C)$, is the size of the smallest support of an $r$-dimensional subcode. If $r=1$, $\delta_{r}(C)$ is the minimum distance of $C$ and is denoted simply by $\delta(C)$. Generalized Hamming weights have received a lot of attention; see [7, 26, 28, 52, 54] and the references therein. The study of these weights is related to trellis coding, $t$–resilient functions, and was motivated by some applications from cryptography [54].

There are combinatorial formulas for the generalized Hamming weights of some interesting families of evaluation codes [4, 8, 17, 24, 39]. The work done by Heijnen and Pellikaan [24] relates footprints and generalized Hamming weights and introduce methods to study certain evaluation codes (cf. [24, Section 7]). These methods were used in [5] to determine the generalized Hamming weights of affine Cartesian codes.

If $F\subset S$, the $K$-linear subspace of $S$ spanned by $F$ is denoted by $KF$. Let $\prec$ be a monomial order on $S$. Given a linear subspace $\mathcal{L}$ of $K\Delta_{\prec}(I)$ and an integer $1\leq r\leq\dim_{K}(\mathcal{L})$, let $\mathcal{L}^{*}$ be the set of nonzero elements of $\mathcal{L}$, and let $\mathcal{L}_{\prec,r}$ be the set of all subsets $F=\{f_{1},\ldots,f_{r}\}$ of $\mathcal{L}^{*}$ such that ${\rm in}_{\prec}(f_{1}),\ldots,{\rm in}_{\prec}(f_{r})$ are distinct monomials and $f_{i}$ is monic for $i=1,\ldots,r$.

One of the main result of Section 3 is the following degree formula for the $r$-th generalized Hamming weight $\delta_{r}(\mathcal{L}_{X})$ of a standard evaluation code $\mathcal{L}_{X}$.

Theorem 3.4.* Let $X$ be a subset of $\mathbb{A}^{s}$, let $I$ be the vanishing ideal of $X$, let $\mathcal{L}$ be a linear subspace of $K\Delta_{\prec}(I)$, and let $\mathcal{L}_{X}$ be the standard evaluation code on $X$ relative to $\prec$. Then*

[TABLE]

This theorem can be applied to any evaluation code $\mathcal{L}_{X}$ by constructing a standard evaluation code $\widetilde{\mathcal{L}}_{X}$ on $X$, relative to a monomial order $\prec$, such that $\widetilde{\mathcal{L}}_{X}=\mathcal{L}_{X}$ (Proposition 3.5, Example 6.1). We will apply this theorem to interesting subfamilies of evaluation codes.

Fix a degree $d\geq 1$ and let $S_{\leq d}=\bigoplus_{i=0}^{d}S_{i}$ be the $K$-linear subspace of $S$ of all polynomials of degree at most $d$. If $\mathcal{L}$ is equal to $S_{\leq d}$, then the resulting evaluation code $\mathcal{L}_{X}$ is called a Reed-Muller-type code of degree $d$ on $X$ [11, 20] and is denoted by $C_{X}(d)$. If $X=\mathbb{A}^{s}$, we obtain the generalized Reed–Muller code $\mathcal{R}_{q}(d,s)$ [26, p. 524] or $q$-ary Reed–Muller code [24]. As an application of Theorem 3.4 we obtain a formula to compute the $r$-th generalized Hamming weight of $C_{X}(d)$ using the degree and a graded monomial order (Corollary 3.6, Example 6.2).

The minimum distance of $C_{X}(d)$ can be computed using the standard polynomials of $S/I$ of degree at most $d$ (Corollary 3.6). Using this together with Proposition 2.2, we show that the minimum distance of $C_{X}(d)$ can be computed recursively using only the standard polynomials of $S/I$ of degree $d$ (Corollary 3.7, Example 6.4).

To compute $\delta_{r}(\mathcal{L}_{X})$ is a very difficult problem even for $r=1$ because $|\mathcal{L}^{*}|=q^{k}-1$, where $k$ is the dimension of $\mathcal{L}_{X}$. In practice the formula of Theorem 3.4 can only be used to compute examples for small values of $r,q,s$, and $k$. Using Theorem 3.4, we show lower bounds for $\delta_{r}(\mathcal{L}_{X})$, in terms of the degree and the footprint of $S/I$, which are easier to compute.

Let $\mathcal{N}_{\prec,r}$ be the family of all subsets $N$ of ${\rm in}_{\prec}(\mathcal{L}^{*}):=\{{\rm in}_{\prec}(f)|\,f\in\mathcal{L}^{*}\}$ with $r$ distinct elements. The $r$-th footprint of the evaluation code $\mathcal{L}_{X}$, denoted ${\rm fp}_{r}(\mathcal{L}_{X})$, is given by

[TABLE]

The $r$-th footprint of $\mathcal{L}_{X}$ is easier to compute than $\delta_{r}(\mathcal{L}_{X})$ because $|{\rm in}_{\prec}(\mathcal{L}^{*})|$ is $k$. The other main result of Section 3 is the following degree-footprint lower bound for $\delta_{r}(\mathcal{L}_{X})$.

Theorem 3.9.* Let $X$ be a subset of $\mathbb{A}^{s}$, let $I=I(X)$ be the vanishing ideal of $X$, let $\mathcal{L}$ be a linear subspace of $K\Delta_{\prec}(I)$, and let $\mathcal{L}_{X}$ be the standard evaluation code on $X$. Then*

[TABLE]

As an application of Theorem 3.9 we obtain a lower bound for the $r$-generalized Hamming weight of $C_{X}(d)$ (Corollary 3.10, Example 6.5).

The scope of our results include another interesting family of evaluation codes that we now introduce. Let $\mathbb{X}=\{P_{1},\ldots,P_{m}\}$ be a set of non-zero distinct points in $(\mathbb{F}_{q})^{s}$ such that the first non-zero entry of each $P_{i}$ is $1$. If $\mathcal{L}=S_{d}$, the evaluation code $\mathcal{L}_{\mathbb{X}}$ on $\mathbb{X}$ is called a projective Reed–Muller-type code on $\mathbb{X}$ [20]. In particular, by making $\mathcal{L}=S_{d}$ and $\mathbb{X}$ equal to the set of all non-zero points of $(\mathbb{F}_{q})^{s}$ whose first non-zero entry is $1$, we obtain the classical projective Reed–Muller code studied by Lachaud and Sørensen[32, 47].

As we just saw, one can apply our results on evaluation codes to non-classical projective Reed–Muller-type codes (Example 6.6, Procedure A.8). One can study Reed-Muller-type codes over a set $X\subset\mathbb{A}^{s}$ by using projective Reed–Muller-type codes over the set $\mathbb{X}=\{(x,1)|\,x\in X\}$, because the corresponding codes on $X$ and $\mathbb{X}$ have the same basic parameters [35].

We show examples of how to use Hilbert’s Nullstellensatz over $\mathbb{F}_{q}$ [16] (Proposition 3.12) to estimate the parameters of evaluation codes over affine varieties that are in a broad sense algebraic geometry codes [52, p. 192] (Examples 6.7 and 6.8). Then, we give a projective version over $\mathbb{F}_{q}$ of Hilbert’s Nullstellensatz (Theorem 3.13) that can be used in the case of evaluation codes over projective varieties defined by a given set of homogeneous polynomials (Example 6.9).

The rest of this paper is devoted to show applications of the results of Sections 2 and 3 to two other families of evaluation codes that we now introduce.

First we introduce toric codes over hypersimplices. Let $1\leq d\leq s$ be an integer, $s\geq 2$, and let $\mathcal{P}$ be the convex hull in $\mathbb{R}^{s}$ of all integral points $\mathbf{e}_{i_{1}}+\cdots+\mathbf{e}_{i_{d}}$ such that $1\leq i_{1}<\cdots<i_{d}\leq s$, where $\mathbf{e}_{i}$ is the $i$-th unit vector in $\mathbb{R}^{s}$. The lattice polytope $\mathcal{P}$ is called the $d$-th hypersimplex of $\mathbb{R}^{s}$ [51, p. 84]. The affine torus of the affine space $\mathbb{A}^{s}:=K^{s}$ is given by $T:=(K^{*})^{s}$, where $K^{*}$ is the multiplicative group of the field $K$. The toric code of $\mathcal{P}$ of degree $d$, denoted $\mathcal{C}_{\mathcal{P}}(d)$ or simply $\mathcal{C}_{d}$, is the image of the evaluation map

[TABLE]

where $KV_{d}$ is the $K$-linear subspace of $S_{d}$ spanned by the set $V_{d}$ of all $t^{a}$ such that $a\in\mathcal{P}\cap\mathbb{Z}^{s}$, and $\{P_{1},\ldots,P_{m}\}$ is the set of all points of the affine torus $T$. A monomial $t^{a}$ of $S$ is in $KV_{d}$ if and only if $t^{a}$ is squarefree and has degree $d$. The set $V_{d}$ is precisely the set of squarefree monomials of $S$ of degree $d$.

Toric codes were introduced by Hansen [23] and have been actively studied in the last decade, see [46] and the references therein. These codes are affine-variety codes in the sense of [2, p. 1]. For $q\geq 3$, the toric code $\mathcal{C}_{\mathcal{P}}(d)$ is a standard evaluation code on $T$, relative to any monomial order $\prec$, because the linear space $\mathcal{L}=KV_{d}$ is spanned by $V_{d}$, all elements of $V_{d}$ are standard monomials of $S/I(T)$, and $\mathcal{L}_{T}=\mathcal{C}_{\mathcal{P}}(d)$ (Lemma 4.2).

We solve part of the following problem by determining the minimum distance of $\mathcal{C}_{\mathcal{P}}(d)$.

Problem 1.1**.**

Find formulas for the minimum distance or more generally for the generalized Hamming weights of the toric code $\mathcal{C}_{\mathcal{P}}(d)$.

We come to our main result on toric codes.

Theorem 4.5.* Let $\mathcal{C}_{\mathcal{P}}(d)$ be the toric code of $\mathcal{P}$ of degree $d$ and let $\delta(\mathcal{C}_{\mathcal{P}}(d))$ be its minimum distance. Then*

[TABLE]

The 2nd generalized Hamming weight of the toric code $\mathcal{C}_{\mathcal{P}}(d)$ has been recently determined by Patanker and Singh [42].

We now introduce the family of squarefree evaluation codes. Elements of this family are generalized toric codes in the sense of [33, 43, 46]. Let $V_{\leq d}$ be the set of all squarefree monomials of $S$ of degree at most $d$. If we replace $V_{d}$ by $V_{\leq d}$ in the evaluation map of Eq. (1.1), the image of the resulting map, denoted $\mathcal{C}_{\leq d}$, is called a squarefree evaluation code of degree $d$ on $T$. If we replace $KV_{d}$ by $S_{\leq d}$ in the evaluation map of Eq. (1.1), the image of the resulting map is the Reed–Muller-type code $C_{T}(d)$ over the affine torus $T$. The parameters of $C_{T}(d)$ have been determined in [5, 17, 18, 34, 44].

We solve part of the following problem by determining easy to evaluate formulas for the minimum distance and the 2nd generalized Hamming weight of $\mathcal{C}_{\leq d}$.

Problem 1.2**.**

Find formulas for the minimum distance or more generally for the generalized Hamming weights of the squarefree evaluation code $\mathcal{C}_{\leq d}$.

The first main result about squarefree evaluation codes is:

Theorem 5.5.* If $q\geq 3$, then the minimum distance of $\mathcal{C}_{\leq d}$ is $(q-2)^{d}(q-1)^{s-d}$. *

We obtain formulas to compute the generalized Hamming weights of squarefree evaluation codes, and also obtain the corresponding footprint lower bounds (Corollaries 5.3 and 5.4).

The second main result about squarefree evaluation codes is the following formula.

Theorem 5.6.* If $q\geq 3$, then the second generalized Hamming weight of $\mathcal{C}_{\leq d}$ is*

[TABLE]

We include one section with some examples illustrating some of our results (Section 6) and an appendix with implementations in Macaulay$2$ [21] that show how some of our results can be used in practice (Appendix A).

For all unexplained terminology and additional information we refer to [10, 48, 53] (for the theory of Gröbner bases and Hilbert functions), and [26, 36, 52] (for the theory of error-correcting codes and linear codes).

2. Preliminaries: Affine varieties over finite

sets

In this section we study affine varieties defined over finite sets. The results of this section do not need the hypothesis that the field is finite.

Let $S=K[t_{1},\ldots,t_{s}]$ be a polynomial ring over a field $K$ and let $I$ be an ideal of $S$. The Krull dimension of $S/I$ is denoted by $\dim(S/I)$. We say that $I$ has dimension $k$ if $\dim(S/I)$ is equal to $k$. The $K$-linear space of polynomials in $S$ (resp. $I$) of degree at most $d$ is denoted by $S_{\leq d}$ (resp. $I_{\leq d}$). The function

[TABLE]

is called the affine Hilbert function of $S/I$. Let $u=t_{s+1}$ be a new variable and let $I^{h}\subset S[u]$ be the homogenization of $I$, where $S[u]$ is given the standard grading. One has the following two well-known facts

[TABLE]

where $H_{I^{h}}(d)=\dim_{K}(S[u]/I^{h})_{d}$, see for instance [53, Lemma 8.5.4]. If $k=\dim(S/I)$, by Hilbert theorem [48, p. 58], there is a unique polynomial $h^{a}_{I}(z)=\sum_{i=0}^{k}a_{i}z^{i}\in\mathbb{Q}[z]$ of degree $k$ such that $h^{a}_{I}(d)=H_{I}^{a}(d)$ for $d\gg 0$. By convention the degree of the zero polynomial is $-1$. The integer $k!\,a_{k}$, denoted ${\rm deg}(S/I)$, is called the degree of $S/I$. The degree of $S/I$ is equal to $\deg(S[u]/I^{h})$. If $k=0$, then $H_{I}^{a}(d)=\deg(S/I)=\dim_{K}(S/I)$ for $d\gg 0$. Note that the degree of $S/I$ is positive if $I\subsetneq S$ and is [math] otherwise.

An element $f\in S$ is called a zero-divisor of $S/I$—as an $S$-module—if there is $\overline{0}\neq\overline{a}\in S/I$ such that $f\overline{a}=\overline{0}$, and $f$ is called regular on $S/I$ otherwise. Note that $f$ is a zero-divisor of $S/I$ if and only if $(I\colon f)\neq I$. An associated prime of $I$ is a prime ideal $\mathfrak{p}$ of $S$ of the form $\mathfrak{p}=(I\colon f)$ for some $f$ in $S$. The radical of $I$ is denoted by ${\rm rad}(I)$. The ideal $I$ is radical if $I={\rm rad}(I)$.

Theorem 2.1**.**

[53, Lemma 2.1.19, Corollary 2.1.30]** If $I$ is an ideal of $S$ and $I=\mathfrak{q}_{1}\cap\cdots\cap\mathfrak{q}_{m}$ is an irredundant primary decomposition with ${\rm rad}(\mathfrak{q}_{i})=\mathfrak{p}_{i}$, then the set of zero-divisors $\mathcal{Z}_{S}(S/I)$ of $S/I$ is equal to $\bigcup_{i=1}^{m}\mathfrak{p}_{i}$, and $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{m}$ are the associated primes of $I$.

Proposition 2.2**.**

([15], [35, p. 411])* Let $I=I(X)$ be the vanishing ideal of a set $X$ of affine points over a field $K$. Then, $H_{I}^{a}$ is increasing until it reaches constant value $|X|$, and $\delta(C_{X}(d))$ is decreasing, as a function of $d$, until it reaches constant value $1$. In particular, $\deg(S/I)=|X|$.*

The least integer $r_{0}\geq 0$ such that $H_{I}^{a}(d)=|X|$ for $d\geq r_{0}$, denoted ${\rm reg}(H_{I}^{a})$, is the index of regularity of the affine Hilbert function.

Proposition 2.3**.**

(Additivity of the degree [41, Proposition 2.5])* If $I$ is an ideal of $S$ and $I=\mathfrak{q}_{1}\bigcap\cdots\bigcap\mathfrak{q}_{m}$ is an irredundant primary decomposition, then*

[TABLE]

Lemma 2.4**.**

[30, p. 389]** Let $X$ be a finite subset of $\mathbb{A}^{s}$, let $P$ be a point in $X$, $P=(p_{1},\ldots,p_{s})$, and let $I_{P}$ be the vanishing ideal of $P$. Then, $I_{P}$ is a prime ideal of height $s$,

[TABLE]

and $I(X)=\bigcap_{P\in X}I_{P}$ is the primary decomposition of $I(X)$.

Lemma 2.5**.**

Let $X$ be a finite subset of $\mathbb{A}^{s}$ over a field $K$ and let $F=\{f_{1},\ldots,f_{r}\}$ be a set of polynomials of $S$. Then, the following conditions are equivalent.

1. (a)

$(I(X)\colon(F))=I(X)$. 2. (b)

$V_{X}(F)=\emptyset$. 3. (c)

$(I(X),F)=S$.

Proof.

(a) $\Rightarrow$ (b): We can write $X=\{P_{1},\ldots,P_{m}\}$ and $I(X)=\bigcap_{i=1}^{m}\mathfrak{p}_{i}$, where $\mathfrak{p}_{i}$ is equal to $I_{P_{i}}$, the vanishing ideal of $P_{i}$. We proceed by contradiction. Assume that $V_{X}(F)\neq\emptyset$. Pick $P_{k}$ in $V_{X}(F)$. Then, $f_{i}(P_{k})=0$ and $f_{i}\in\mathfrak{p}_{k}$ for all $i$. Note that $(\mathfrak{p}_{k}\colon(F))=(1)$. Therefore

[TABLE]

Hence $\mathfrak{p}_{i}\subset(\mathfrak{p}_{i}\colon(F))\subset\mathfrak{p}_{k}$ for some $i\neq k$, see [53, p. 74]. Thus, $\mathfrak{p}_{i}=\mathfrak{p}_{k}$, a contradiction.

(b) $\Rightarrow$ (a): We proceed by contradiction. Assume that $I(X)\subsetneq(I(X)\colon(F))$. Pick a polynomial $g$ such that $gf_{i}\in I(X)$ for all $i$ and $g\notin I(X)$. Then, there is $P$ in $X$ such that $g(P)\neq 0$. Thus, $f_{i}(P)=0$ for all $i$, that is, $P\in V_{X}(F)$, a contradiction.

(c) $\Rightarrow$ (b): We can write $1=f+\sum_{i=1}^{r}g_{i}f_{i}$, where $f\in I(X)$ and $g_{i}\in S$ for all $i$. If $V_{X}(F)\neq\emptyset$, picking $\alpha\in V_{X}(F)$ and evaluating the last equality at $\alpha$, we get $1=0$, a contradiction.

(a) $\Rightarrow$ (c): If $(I(X),F)\subsetneq S$, pick a maximal ideal $\mathfrak{m}$ of $S$ that contains $(I(X),F)$. Then, by Lemma 2.4 and [53, 2.1.48, p. 74], $\mathfrak{m}=I_{P_{k}}$ for some $P_{k}$ in $X$. Thus, $F\subset I_{P_{k}}$ and $P_{k}\in V_{X}(F)$, a contradiction because conditions (a) and (b) are equivalent. ∎

The next result gives a sufficient conditions for an ideal of dimension zero to be radical. As usual we denote the derivative of a univariate polynomial $f$ by $f^{\prime}$.

Lemma 2.6**.**

(Seidenberg’s lemma [45])* Let $I\subsetneq S$ be an ideal of dimension zero. If $I$ contains a univariate polynomial $f_{i}\in K[t_{i}]$ with $\gcd(f_{i},f_{i}^{\prime})=1$ for $i=1,\ldots,s$, then $I$ is an intersection of finitely many maximal ideals, and any proper ideal of $S$ containing $I$ is a radical ideal.*

If $K=\mathbb{F}_{q}$ is a finite field, by taking $f_{j}=t_{j}^{q}-t_{j}$ for $j=1,\ldots,s$, the next result follows directly from Seidenberg’s lemma.

Proposition 2.7**.**

Let $X$ be a finite subset of an affine space $\mathbb{A}^{s}$ over a field $K$. For each $1\leq j\leq s$, there is a univariate polynomial $f_{j}$ in $K[t_{j}]$ that vanishes at all points of $X$ such that $\gcd(f_{j},f_{j}^{\prime})=1$, and any proper ideal of $S$ containing $I(X)$ is a radical ideal.

Proof.

Let $P_{1},\ldots,P_{m}$ be the points of $X$. We can write $P_{i}=(p_{i,1},\ldots,p_{i,s})$ with $p_{i,j}\in K$ for all $1\leq i\leq m$ and $1\leq j\leq s$. For each $1\leq j\leq s$ consider the set

[TABLE]

where $a_{1,j},\ldots,a_{d_{j},j}$ are distinct elements of $K$ and $d_{j}=|D_{j}|$ for $j=1,\ldots,s$. The univariate polynomials given by

[TABLE]

vanish at all points of $X$. Each $f_{j}$ is a separable polynomial of $K[t_{j}]$. Hence, $f_{j}^{\prime}$ is relatively prime to $f_{j}$ [27, Theorem 4.5, p. 231]. Thus, the result follows from Lemma 2.6. ∎

The next result is an analog of [17, Lemma 3.4] for affine spaces.

Lemma 2.8**.**

(cf. [31, Proposition 6.2.12, p. 262])*

Let $X$ be a finite subset of $\mathbb{A}^{s}$ and let $I(X)$ be its vanishing ideal. If $F$ is a finite subset of $S$, then*

[TABLE]

Proof.

Let $P_{1},\ldots,P_{m}$ be the points of $X$, $m=|X|$, and let $P=P_{k}$ be a point in $X$, with $P=(p_{1},\ldots,p_{s})$. Then, the vanishing ideal $I_{P}$ of $P$ is a maximal ideal of $S$ of height $s$,

[TABLE]

and $I(X)=\bigcap_{P\in X}I_{P}$ is an irredundant primary decomposition of $I(X)$ (Lemma 2.4). In particular the ideal $I(X)$ is an unmixed radical ideal of dimension [math].

Assume that $(I(X)\colon(F))\neq I(X)$. If $(I(X)\colon(F))=S$, then $(F)\subset I(X)$, $X=V_{X}(F)$, and

[TABLE]

Hence, we may assume the strict inclusions $I(X)\subsetneq(I(X)\colon(F))\subsetneq{S}$. By Lemma 2.5, one has $(I(X),F)\subsetneq S$, and by Proposition 2.7, $(I(X),F)$ is a radical ideal. Any prime ideal $\mathfrak{p}$ containing $(I(X),F)$ is equal to $I_{P_{i}}$ for some $i$. Therefore, we may assume that

[TABLE]

for some $n\geq 1$, $(F)\subset I_{P_{i}}$ for $i\leq n$, and $(F)\not\subset I_{P_{i}}$ for $i>n$. As a consequence, noticing that $P_{i}\in V_{X}(F)$ if and only if $(F)\subset I_{P_{i}}$ and by additivity of the degree of Proposition 2.3, we get

[TABLE]

Now assume $(I(X)\colon(F))=I(X)$. Then, by Lemma 2.5, $V_{X}(F)=\emptyset$ and $|V_{X}(F)|=0$. ∎

Let $I\subset S$ be an ideal, let $\prec$ be a monomial order, and let $\Delta_{\prec}(I)$ be the set of standard monomials of $S/I$. The image of $\Delta_{\prec}(I)$, under the canonical map $S\mapsto S/I$, $x\mapsto\overline{x}$, is a basis of $S/I$ as a $K$-vector space. This is a classical result of Macaulay [10, Chapter 5]. In particular, $H_{I}^{a}(d)$ is the number of standard monomials of $S/I$ of degree at most $d$.

Definition 2.9**.**

Let $I$ be an ideal of $S$ and let $\prec$ be a monomial order. A subset $\mathcal{G}=\{g_{1},\ldots,g_{n}\}$ of $I$ is called a Gröbner basis of $I$ if ${\rm in}_{\prec}(I)=({\rm in}_{\prec}(g_{1}),\ldots,{\rm in}_{\prec}(g_{n}))$.

Lemma 2.10**.**

[7, p. 2]** Let $I\subset S$ be an ideal generated by $\mathcal{G}=\{g_{1},\ldots,g_{n}\}$, then

[TABLE]

with equality if $\mathcal{G}$ is a Gröbner basis of $I$.

Proof.

Take $t^{a}$ in $\Delta_{\prec}(I)$. We set $L:=({\rm in}_{\prec}(g_{1}),\ldots,{\rm in}_{\prec}(g_{n}))$. Note that $\Delta_{\prec}(L)$ is the set of all monomials that are not in $L$. If $t^{a}\notin\Delta_{\prec}(L)$, then $t^{a}\in L$, that is, we can write $t^{a}=t^{c}\,{\rm in}_{\prec}(g_{i})$ for some $i$ and some $t^{c}$. Then, $t^{a}={\rm in}_{\prec}(t^{c}g_{i})$, with $t^{c}g_{i}$ in $I$, a contradiction. Thus, $t^{a}\in\Delta_{\prec}(L)$. Assume that $\mathcal{G}$ is a Gröbner basis of $I$, that is, ${\rm in}_{\prec}(I)=L$. Then, $\Delta_{\prec}(I)=\Delta_{\prec}(L)$. ∎

Theorem 2.11**.**

(cf. [3, Theorem 8.32])* Let $X$ be a finite subset of an affine space $\mathbb{A}^{s}$ over a field $K$. If $F=\{f_{1},\ldots,f_{r}\}$ is a set of polynomials of $S$, then*

[TABLE]

Proof.

First assume that $(I(X)\colon(F))\neq I(X)$. Then, by Lemma 2.5, $(I(X),F)\subsetneq S$. The first equality follows from Lemma 2.8, the second equality follows from the definition of the degree since $S/(I(X),F)$ has Krull dimension [math], and the third equality is Macaulay’s theorem that $\Delta_{\prec}(I(X),F)$ is a basis for $S/(I(X),F)$ as a $K$-vector space [10, Chapter 5].

Now assume that $(I(X)\colon(F))=I(X)$. Then, by Lemma 2.5, $(I(X),F)=S$ and $V_{X}(F)=\emptyset$. Thus, $\deg(S/(I(X),F))=0$, $\Delta_{\prec}(I(X),F)=\emptyset$, and all numbers in the equality above are [math]. ∎

For vanishing ideals, the next result is the affine analog of [17, Lemma 4.1].

Theorem 2.12**.**

Let $X$ be a finite subset of $\mathbb{A}^{s}$, let $I=I(X)$ be the vanishing ideal of $X$, and let $\prec$ be a monomial order. If $F$ is a finite set of polynomials of $S$ and $(I\colon(F))\neq I$, then

[TABLE]

and $\deg(S/(I,F))<\deg(S/I)$ if $(F)\not\subset I$.

Proof.

The equality on the left follows from Lemma 2.8 and the equality on the right follows from Lemma 2.4 and the additivity of the degree of Proposition 2.3. We set $J=(I,F)$ and $L=({\rm in}_{\prec}(I),{\rm in}_{\prec}(F))$, where ${\rm in}_{\prec}(F)=\{{\rm in}_{\prec}(f)|\,f\in F\}$. As $I\subsetneq(I\colon(F))$, we can pick $a$ in $(I\colon(F))$ and $a$ not in $I$. Then, $f\overline{a}=\overline{0}$ for all $f\in(F)$. Thus, all elements of $(F)$ are zero divisors of $S/I$. Hence, as $I$ is a finite intersection of maximal ideals of $S$, by Theorem 2.1 and [53, 2.1.49, p. 74], there is an associated prime ideal $\mathfrak{p}$ of $S/I$ such that $(F)\subset\mathfrak{p}$. Thus, $J\subset\mathfrak{p}\subsetneq S$. The rings $S/J$, $S/L$, and $S/I$ have Krull dimension [math] since $\dim(S/I)=\dim(S/{\rm in}_{\prec}(I))=0$. Pick a Gröbner basis $\mathcal{G}=\{g_{1},\ldots,g_{n}\}$ of $I$. Then, $J$ is generated by $\mathcal{G}\cup F$ and by Lemma 2.10 one has the inclusions

[TABLE]

Thus, $\Delta_{\prec}(J)\subset\Delta_{\prec}(L)\subset\Delta_{\prec}(I)$. Recall that $H_{I}^{a}(d)$, the affine Hilbert function of $I$ at $d$, is the number of standard monomials of degree at most $d$. Hence, $H_{J}^{a}(d)\leq H_{L}^{a}(d)\leq H_{I}^{a}(d)$ for $d\geq 0$. Then, by Hilbert theorem [48, p. 58], $H_{J}^{a}$, $H_{L}^{a}$, $H_{I}^{a}$ are polynomial functions of degree equal to $\dim(S/I)=0$, and so they become constant for $d\gg 0$. Thus,

[TABLE]

for $d\gg 0$, that is, $\deg(S/J)\leq\deg(S/L)\leq\deg(S/I)$. Now, assume that $(F)\not\subset I$. As $I$ is a radical ideal, there is at least one minimal prime of $I$ that does not contain $(F)$. By Proposition 2.7, $J$ is a radical ideal. Hence, ${\rm Ass}(J)\subsetneq{\rm Ass}(I)$, and using the additivity of the degree of Proposition 2.3, we get

[TABLE]

and consequently $\deg(S/J)<\deg(S/I)$. ∎

3. Generalized Hamming weights of evaluation

codes

In this section we give formulas, in terms of the degree and a graded monomial order, for the generalized Hamming weights of standard evaluation codes, and show degree-footprint lower bounds for these weights which are much easier to compute. To avoid repetitions, we continue to employ the notations and definitions used in Sections 1 and 2. Throughout this section we assume that $K$ is a finite field $\mathbb{F}_{q}$.

Lemma 3.1**.**

Let $\mathcal{L}_{X}$ be a standard evaluation code on $X$ relative to a monomial order $\prec$. Then, $\mathcal{L}\cap I(X)=(0)$ and $\mathcal{L}\simeq\mathcal{L}_{X}$.

Proof.

We set $I=I(X)$. Take $f\in\mathcal{L}\cap I$. If $f\neq 0$, then ${\rm in}_{\prec}(f)\in{\rm in}_{\prec}(I)$, a contradiction since all monomials of $f$ are standard monomial of $S/I$. Hence, the evaluation maps gives an isomorphism between $\mathcal{L}$ and $\mathcal{L}_{X}$. ∎

Let $I$ be an ideal of $S$, let $\prec$ be a monomial order on $S$, let $\mathcal{L}$ be a linear subspace of $K\Delta_{\prec}(I)$, let $\mathcal{L}_{r}$ be the set of all linearly independent subsets $F=\{f_{1},\ldots,f_{r}\}$ of $\mathcal{L}$ with $r$ elements, let $\mathcal{L}_{\prec,r}$ be the set of all subsets $F=\{f_{1},\ldots,f_{r}\}$ of $\mathcal{L}^{*}=\mathcal{L}\setminus\{0\}$ such that ${\rm in}_{\prec}(f_{1}),\ldots,{\rm in}_{\prec}(f_{r})$ are distinct monomials and $f_{i}$ is monic for $i=1,\ldots,r$. The set of all standard polynomials of $S/I$ is denoted by $\Delta_{\prec}^{p}(I)$, that is, $\Delta_{\prec}^{p}(I)$ is equal to $(K\Delta_{\prec}(I))\setminus\{0\}$.

Lemma 3.2**.**

Let $\mathcal{L}$ be a $K$-linear subspace of $S$ spanned by a finite subset of $\Delta_{\prec}^{p}(I)$, and let $F=\{f_{1},\ldots,f_{r}\}$ be a subset of $\mathcal{L}^{*}=\mathcal{L}\setminus\{0\}$. The following hold.

1. (a)

If $f_{1},\ldots,f_{r}$ are linearly independent over $K$, then there is a set $G=\{g_{1},\ldots,g_{r}\}\subset\mathcal{L}^{*}$ such that $KF=KG$, ${\rm in}_{\prec}(g_{1}),\ldots,{\rm in}_{\prec}(g_{r})$ distinct, and ${\rm in}_{\prec}(f_{i})\succeq{\rm in}_{\prec}(g_{i})$ for all $i$. 2. (b)

If ${\rm in}_{\prec}(f_{1}),\ldots,{\rm in}_{\prec}(f_{r})$ are distinct, then $f_{1},\ldots,f_{r}$ are linearly independent over $K$. 3. (c)

$\mathcal{L}_{\prec,r}\subset\mathcal{L}_{r}$, and if $F$ is in $\mathcal{L}_{r}$, then there is $G$ in $\mathcal{L}_{\prec,r}$ such that $KF=KG$.

Proof.

(a): Note that all elements of $\mathcal{L}^{*}$ are standard polynomials of $S/I$. We proceed by induction on $r$. The case $r=1$ is clear. Assume that $r>1$. Permuting the $f_{i}$’s if necessary we may assume that ${\rm in}_{\prec}(f_{1})\succeq\cdots\succeq{\rm in}_{\prec}(f_{r})$.

Case ($\mathrm{a}_{1}$): Assume that ${\rm in}_{\prec}(f_{1})\succ{\rm in}_{\prec}(f_{2})$. By applying the induction hypothesis to the set $F^{\prime}=\{f_{2},\ldots,f_{r}\}$, we obtain a set $G^{\prime}=\{g_{2},\ldots,g_{r}\}\subset\mathcal{L}^{*}$ such that $KF^{\prime}$ is equal to $KG^{\prime}$, the monomials ${\rm in}_{\prec}(g_{2}),\ldots,{\rm in}_{\prec}(g_{r})$ are distinct, and ${\rm in}_{\prec}(f_{i})\succeq{\rm in}_{\prec}(g_{i})$ for $i=2,\ldots,r$. Setting $g_{1}=f_{1}$ and $G=G^{\prime}\cup\{g_{1}\}$, we get $KF=KG$, and the monomial ${\rm in}_{\prec}(g_{1})$ is distinct from ${\rm in}_{\prec}(g_{2}),\ldots,{\rm in}_{\prec}(g_{r})$ because ${\rm in}_{\prec}(f_{1})\succ{\rm in}_{\prec}(f_{i})\succeq{\rm in}_{\prec}(g_{i})$ for $i=2,\ldots,r$.

Case ($\mathrm{a}_{2}$): Assume there is $k\geq 2$ such that ${\rm in}_{\prec}(f_{1})={\rm in}_{\prec}(f_{i})$ for $i\leq k$ and ${\rm in}_{\prec}(f_{1})\succ{\rm in}_{\prec}(f_{i})$ for $i>k$. We set $h_{i}=f_{1}-f_{i}$ for $i=2,\ldots,k$ and $h_{i}=f_{i}$ for $i=k+1,\ldots,r$. Note that ${\rm in}_{\prec}(f_{1})\succ{\rm in}_{\prec}(h_{i})$ for $i\geq 2$, $h_{2},\ldots,h_{r}$ are in $\mathcal{L}^{*}$, and $h_{2},\ldots,h_{r}$ are linearly independent over $K$. By applying the induction hypothesis to the set $H=\{h_{2},\ldots,h_{r}\}$, we obtain a set $G^{\prime}=\{g_{2},\ldots,g_{r}\}\subset\mathcal{L}^{*}$ such that $KH=KG^{\prime}$, ${\rm in}_{\prec}(g_{2}),\ldots,{\rm in}_{\prec}(g_{r})$ distinct, and ${\rm in}_{\prec}(h_{i})\succeq{\rm in}_{\prec}(g_{i})$ for $i=2,\ldots,r$. Setting $g_{1}=f_{1}$ and $G=G^{\prime}\cup\{g_{1}\}$, we get $KF=KG$, and the monomial ${\rm in}_{\prec}(g_{1})$ is distinct from ${\rm in}_{\prec}(g_{2}),\ldots,{\rm in}_{\prec}(g_{r})$ because ${\rm in}_{\prec}(f_{1})\succ{\rm in}_{\prec}(h_{i})\succeq{\rm in}_{\prec}(g_{i})$ for $i\geq 2$.

(b): By hypothesis, ${\rm in}_{\prec}(f_{1})\succ\cdots\succ{\rm in}_{\prec}(f_{r})$. Assume that $\sum_{i=1}^{r}\lambda_{i}f_{i}=0$, $\lambda_{i}\in K$ for all $i$. We proceed by contradiction assuming $\lambda_{1}=\cdots=\lambda_{k-1}=0$ and $\lambda_{k}\neq 0$ for some $k$. Setting $f:=\sum_{i=k}^{r}\lambda_{i}f_{i}$, we get ${\rm in}_{\prec}(f)={\rm in}_{\prec}(f_{k})$ and $f\neq 0$, a contradiction.

(c): This follows at once from parts (a) and (b). ∎

Let $C$ be a $[m,k]$ linear code of length $m$ and dimension $k$ over a finite field $K=\mathbb{F}_{q}$, and let $1\leq r\leq k$ be an integer. Given a subcode $D$ of $C$, the support $\chi(D)$ of $D$ is the set

[TABLE]

The support $\chi(\beta)$ of a vector $\beta\in K^{m}$ is $\chi(K\beta)$, that is, $\chi(\beta)$ is the set of all non-zero entries of $\beta$. The $r$-th generalized Hamming weight of $C$, denoted $\delta_{r}(C)$, is the size of the smallest support of an $r$-dimensional subcode:

[TABLE]

The weight hierarchy of $C$ is the sequence $(\delta_{1}(C),\ldots,\delta_{k}(C))$. The integer $\delta_{1}(C)$ is the minimum distance of $C$ and is denoted by $\delta(C)$. According to [54, Theorem 1, Corollary 1] the weight hierarchy is an increasing sequence

[TABLE]

and $\delta_{r}(C)\leq m-k+r$ for $r=1,\ldots,k$. For $r=1$ this is the Singleton bound for the minimum distance. Notice that $\delta_{r}(C)\geq r$.

Lemma 3.3**.**

[17, Lemma 2.1]** Let $D$ be a subcode of $C$ of dimension $r\geq 1$. If $\beta_{1},\ldots,\beta_{r}$ is a $K$-basis for $D$ with $\beta_{i}=(\beta_{i,1},\ldots,\beta_{i,m})$ for $i=1,\ldots,r$, then $\chi(D)=\bigcup_{i=1}^{r}\chi(\beta_{i})$ and the number of elements of $\chi(D)$ is the number of non-zero columns of the matrix:

[TABLE]

One of the main result of this section is the following degree formula for the $r$-th generalized Hamming weight $\delta_{r}(\mathcal{L}_{X})$ of a standard evaluation code $\mathcal{L}_{X}$.

Theorem 3.4**.**

Let $X$ be a subset of $\mathbb{A}^{s}$, let $I$ be the vanishing ideal of $X$, let $\mathcal{L}$ be a linear subspace of $K\Delta_{\prec}(I)$, and let $\mathcal{L}_{X}$ be the standard evaluation code on $X$ relative to $\prec$. Then

[TABLE]

*Proof. *Let $P_{1},\ldots,P_{m}$ be the points of $X$ and let $D$ be a subcode of $\mathcal{L}_{X}$ of dimension $r$. The evaluation map ${\rm ev}$ induces an isomorphism of $K$-vector spaces between $\mathcal{L}$ and $\mathcal{L}_{X}$ (Lemma 3.1). Hence, by Lemma 3.3, there are $f_{1},\ldots,f_{r}$ linearly independent elements of $\mathcal{L}$, that is, the set $F:=\{f_{1},\ldots,f_{r}\}$ is in $\mathcal{L}_{r}$, such that $D=\bigoplus_{i=1}^{r}K\beta_{i}$, where $\beta_{i}$ is $(f_{i}(P_{1}),\ldots,f_{i}(P_{m}))$, and the support $\chi(D)$ of $D$ is equal to $\bigcup_{i=1}^{r}\chi(\beta_{i})$. Consider the matrix $A$ with rows $\beta_{1},\ldots,\beta_{r}$. The $j$-th column of $A$ is not zero if and only if $P_{j}$ is in $X\setminus V_{X}(F)$. Therefore, since the number of non-zero columns of $A$ is $|\chi(D)|$ (Lemma 3.3), we get:

[TABLE]

Conversely let $F=\{f_{1},\ldots,f_{r}\}$ be a set in $\mathcal{L}_{r}$, then there is a subcode $D$ of $\mathcal{L}_{X}$ of dimension $r$ with $|\chi(D)|=|X\setminus V_{X}(F)|$. Indeed, setting

[TABLE]

and using Lemma 3.3, we obtain that $|\chi(D)|=|X\setminus V_{X}(F)|$. The affine varieties defined by the elements of $\mathcal{L}_{r}$ and $\mathcal{L}_{\prec,r}$ are the same:

[TABLE]

Indeed, the inclusion “$\supset$” is clear since $\mathcal{L}_{\prec,r}\subset\mathcal{L}_{r}$ (Lemma 3.2). To show the inclusion “$\subset$” take $F\in\mathcal{L}_{r}$. Then, by Lemma 3.2, there is $G\in\mathcal{L}_{\prec,r}$ such that $KF=KG$. Thus, $V_{X}(F)=V_{X}(G)$ with $G\in\mathcal{L}_{\prec,r}$, that is, $V_{X}(F)$ is in the right hand side of Eq. (3.2). By Proposition 2.2, $|X|=\deg(S/I)$. Hence, by Eqs. (3.1)–(3.2) and Lemma 2.8, we obtain

[TABLE]

The next proposition allows us to transform any evaluation code into a standard evaluation code relative to a monomial order, that is, one can apply Theorem 3.4 to any evaluation code after picking a monomial order and making a suitable transformation. We illustrate the case of generalized toric codes in Example 6.1 (Procedure A.2) and the case of projective Reed–Muller-type codes in Example 6.6 (Procedure A.8).

Proposition 3.5**.**

Let $\mathcal{L}_{X}$ be an evaluation code on $X$ and let $\prec$ be a monomial order. Then, there is a standard evaluation code $\widetilde{\mathcal{L}}_{X}$ on $X$ relative to $\prec$ such that $\widetilde{\mathcal{L}}_{X}=\mathcal{L}_{X}$.

Proof.

Let $X$ be the set of points $\{P_{1},\ldots,P_{m}\}$ of the affine space $\mathbb{A}^{s}$and let $\mathcal{G}$ be a Gröbner basis of $I=I(X)$. Pick a $K$-basis $\{f_{1},\ldots,f_{k}\}$ of $\mathcal{L}$. For each $i$, let $r_{i}$ be the remainder on division of $f_{i}$ by $\mathcal{G}$, that is, by the division algorithm [10, Theorem 3, p. 63], for each $i$ we can write $f_{i}=h_{i}+r_{i}$, where $h_{i}\in I$, and $r_{i}=0$ or $r_{i}$ is a standard polynomial of $S/I$. We set

[TABLE]

The evaluation code $\widetilde{\mathcal{L}}_{X}$ is a standard evaluation code on $X$ relative to $\prec$ since $\widetilde{\mathcal{L}}$ is a linear subspace of $K\Delta_{\prec}(I)$. To show the inclusion ${\mathcal{L}}_{X}\subset\widetilde{\mathcal{L}}_{X}$ take a point $P$ in ${\mathcal{L}}_{X}$. Then, $P$ is equal to $(f(P_{1}),\ldots,f(P_{m}))$ for some $f\in\mathcal{L}$. Using the equations $f_{i}=h_{i}+r_{i}$, $i=1,\ldots,k$, we can write $f=h+r$, where $h\in I$ and $r\in\widetilde{\mathcal{L}}$. Hence, $P$ is equal to $(r(P_{1}),\ldots,r(P_{m}))$, that is, $P\in\widetilde{\mathcal{L}}_{X}$. To show the inclusion $\widetilde{\mathcal{L}}_{X}\subset{\mathcal{L}}_{X}$ take a point $Q$ in $\widetilde{\mathcal{L}}_{X}$. Then, $Q$ is equal to $(g(P_{1}),\ldots,g(P_{m}))$ for some $g\in\widetilde{\mathcal{L}}$. Using the equations $f_{i}=h_{i}+r_{i}$, $i=1,\ldots,k$, we can write $g=g_{1}+g_{2}$, where $g_{1}\in I$ and $g_{2}\in{\mathcal{L}}$. Hence, $Q$ is equal to $(g_{2}(P_{1}),\ldots,g_{2}(P_{m}))$, that is, $Q\in{\mathcal{L}}_{X}$. ∎

Let $\mathcal{L}_{X}$ be a standard evaluation code on $X$ relative to $\prec$ of dimension $k$. Fix an integer $1\leq r\leq k$. The generalized Hamming weights of $\mathcal{L}_{X}$ are hard to compute because the size of $\mathcal{L}_{\prec,r}$ could be very large as we now explain. The Grassmannian of $\mathcal{L}$, denoted $G(r,k)$, is the set of $r$-dimensional subspaces of $\mathcal{L}$. Consider the equivalence relation on $\mathcal{L}_{\prec,r}$ given by: $F\sim G$ if and only if $KF=KG$. The map

[TABLE]

is bijective. This follows from Lemma 3.2. If $r=1$, then $|\mathcal{L}_{\prec,r}|=|G(1,k)|=(q^{k}-1)/(q-1)$. The following formula for $G(r,k)$ can be found in [49, Proposition 1.7.2, p. 57]:

[TABLE]

Let $F$ and $G$ be in $\mathcal{L}_{\prec,r}$. If $KF=KG$, then $\{{\rm in}_{\prec}(f)|\,f\in F\}$ is equal to $\{{\rm in}_{\prec}(g)|\,g\in G\}$. The converse does not hold in general but it may hold for some special families of standard evaluation codes.

One of the applications of Theorem 3.4 is a formula for the $r$-th generalized Hamming weight $\delta_{r}(C_{X}(d))$ of an affine Reed–Muller-type code $C_{X}(d)$ that can be used to compute $\delta_{r}(C_{X}(d))$ for small values of $q,r,d,s$ using the software system Macaulay$2$ [21] (Example 6.2). In the next result we assume that $\prec$ is a graded monomial order, that is, monomials are first compared by their total degrees (Example 6.3). For use below, let $\mathcal{F}_{\prec,d,r}$ be the set of all $F=\{f_{1},\ldots,f_{r}\}$ contained in $S_{\leq d}$ such that $f_{i}$ is a standard monic polynomial of $S/I$ for all $i$ and ${\rm in}_{\prec}(f_{1}),\ldots,{\rm in}_{\prec}(f_{r})$ are distinct monomials.

Corollary 3.6**.**

Let $K=\mathbb{F}_{q}$ be a finite field, let $X$ be a subset of $\mathbb{A}^{s}$, and let $I=I(X)$ be the vanishing ideal of $X$. If $\prec$ is a graded monomial order, then

[TABLE]

Proof.

Let $P_{1},\ldots,P_{m}$ be the points of $X$ and let $\mathcal{L}=K\Delta_{\prec}(I)_{\leq d}$ be the linear space of standard polynomials of $S/I$ of degree at most $d$ together with the zero vector. Note that $\mathcal{L}_{\prec,r}=\mathcal{F}_{\prec,d,r}$. Hence, by Theorem 3.4, we need only show that $\mathcal{L}_{X}=C_{X}(d)$. Clearly, $\mathcal{L}_{X}\subset C_{X}(d)$. To show the other inclusion take a point $Q$ in $C_{X}(d)$, that is, $Q=(f(P_{1}),\ldots,f(P_{m}))$ for some $f\in S_{\leq d}$. As $\prec$ is a graded monomial order, by the division algorithm [10, Theorem 3, p. 63], $f$ can be written as $f=p+h$, where $p$ is in $I_{\leq d}$ and $h$ is a $K$-linear combination of standard monomials of degree at most $d$. Hence, $Q=(h(P_{1}),\ldots,h(P_{m}))$ and $Q$ is in $\mathcal{L}_{X}$. ∎

The minimum distance of the linear code $C_{X}(d)$ is $\delta_{1}(C_{X}(d))$ and can be computed using the standard monic polynomials of $S/I$ of degree at most $d$ (Corollary 3.6). The next result can be used to compute the minimum distance of $C_{X}(d)$ recursively using only the standard monic polynomials of $S/I$ of degree $d$ (Example 6.4).

Corollary 3.7**.**

Let $K=\mathbb{F}_{q}$ be a finite field, let $X$ be a subset of $\mathbb{A}^{s}$, let $I=I(X)$ be the vanishing ideal of $X$, let $\prec$ be a graded monomial order, and let $\Delta_{\prec}^{p}(I)_{d}$ be the set of standard monic polynomials of $S/I$ of degree $d$. If $d\geq 2$ and $\delta(C_{X}(d-1))>1$, then

[TABLE]

Proof.

Let $\Delta_{\prec}^{p}(I)_{\leq d}$ be the set of standard monic polynomials of $S/I$ of degree at most $d$. As $\mathcal{F}_{\prec,d,1}$ is equal to $\Delta_{\prec}^{p}(I)_{\leq d}$, by Corollary 3.6, one has

[TABLE]

This proves the inequality “$\leq$”. To show the inequality “$\geq$”pick $f$ in $\Delta_{\prec}^{p}(I)_{\leq d}$ such that $\delta(C_{X}(d))=\deg(S/I)-\deg(S/(I,f))$. It suffices to show that $f$ has degree $d$, because then Eq. (3.4) becomes an equality. If $\deg(f)<d$, then by Proposition 2.2, we get

[TABLE]

a contradiction. ∎

For non-graded orders we obtain the following upper bound for $\delta_{r}(C_{X}(d))$.

Corollary 3.8**.**

Let $K=\mathbb{F}_{q}$ be a finite field, let $X$ be a subset of $\mathbb{A}^{s}$, and let $I=I(X)$ be the vanishing ideal of $X$. If $\prec$ is a monomial order, then

[TABLE]

Proof.

Let $\mathcal{L}=K\Delta_{\prec}(I)_{\leq d}$ be the linear space of standard polynomials of $S/I$ of degree at most $d$ together with the zero vector. Note that $\mathcal{L}_{\prec,r}=\mathcal{F}_{\prec,d,r}$. As $\mathcal{L}_{X}\subset C_{X}(d)$, one has $\delta_{r}(C_{X}(d))\leq\delta_{r}(\mathcal{L}_{X})$. Therefore, the inequality follows from Theorem 3.4. ∎

Fix a monomial order $\prec$ on $S$, let $X$ be a subset of $\mathbb{A}^{s}$, and let $I=I(X)$ be its vanishing ideal. Given a $K$-linear subspace $\mathcal{L}$ of $K\Delta_{\prec}(I)$, let $\mathcal{N}_{\prec,r}$ be the family of all subsets $N$ of ${\rm in}_{\prec}(\mathcal{L}^{*}):=\{{\rm in}_{\prec}(f)|\,f\in\mathcal{L}^{*}\}$ with $r$ distinct elements. The $r$-th footprint of the standard evaluation code $\mathcal{L}_{X}$, denoted ${\rm fp}_{r}(\mathcal{L}_{X})$, is given by

[TABLE]

The other main result of Section 3 is the following lower bound for $\delta_{r}(\mathcal{L}_{X})$.

Theorem 3.9**.**

Let $X$ be a subset of $\mathbb{A}^{s}$, let $I=I(X)$ be the vanishing ideal of $X$, let $\mathcal{L}$ be a linear subspace of $K\Delta_{\prec}(I)$, and let $\mathcal{L}_{X}$ be the standard evaluation code on $X$. Then

[TABLE]

Proof.

By Theorem 3.4, there is $F\in\mathcal{L}_{\prec,r}$ such that $\delta_{r}(\mathcal{L}_{X})=\deg(S/I)-\deg(S/(I,F))$. Hence, by Theorem 2.12, and noticing that ${\rm in}_{\prec}(F)\in\mathcal{N}_{\prec,r}$ because $F\in\mathcal{L}_{\prec,r}$, we get

[TABLE]

Thus, ${\rm fp}_{r}(\mathcal{L}_{X})\leq\delta_{r}(\mathcal{L}_{X})$. ∎

Given integers $d,r\geq 1$, let $\mathcal{M}_{\prec,d,r}$ be the set of all subsets $M$ of $\Delta_{\prec}(I)\cap S_{\leq d}$ with $r$ distinct elements. The $r$-th footprint of the Reed–Muller-type code $C_{X}(d)$, denoted ${\rm fp}_{I}(d,r)$, is given by

[TABLE]

We come to one of the main applications of Theorem 3.9.

Corollary 3.10**.**

Let $K=\mathbb{F}_{q}$ be a finite field, let $X$ be a subset of $\mathbb{A}^{s}$, let $I=I(X)$ be the vanishing ideal of $X$, and let $\prec$ be a graded monomial order. Then

[TABLE]

Proof.

Let $\mathcal{L}=K\Delta_{\prec}(I)_{\leq d}$ be the $K$-vector space generated by the set $\Delta_{\prec}(I)_{\leq d}$ of all standard monomials of $S/I$ of degree at most $d$. As $\prec$ is a graded monomial order, by the division algorithm [10, Theorem 3, p. 63], one has:

[TABLE]

that is, $C_{X}(d)$ is the standard evaluation code $\mathcal{L}_{X}$ on $X$. Hence, the inequality follows directly from Theorem 3.9 by noticing the following. The set ${\rm in}_{\prec}(\mathcal{L}^{*})$ of initial terms of $\mathcal{L}^{*}$ is equal to $\Delta_{\prec}(I)_{\leq d}$, $\mathcal{N}_{\prec,r}$ is equal to $\mathcal{M}_{\prec,d,r}$, and $\dim_{K}(\mathcal{L}_{X})$ is equal to $|\Delta_{\prec}(I)_{\leq d}|=H_{I}^{a}(d)$. ∎

Remark 3.11**.**

Let $I=I(X)$ be the vanishing ideal of a subset $X$ of $\mathbb{A}^{s}$. The following hold.

• (a)

$r\leq\delta_{r}(C_{X}(d))\leq|X|$ for $d\geq 1$ and $1\leq r\leq H_{I}^{a}(d)$. This follows from the fact that the weight hierarchy is an increasing sequence (see [54, Theorem 1]).

• (b)

If $d\geq{\rm reg}(H_{I}^{a})$, then $C_{X}(d)=K^{|X|}$ and $\delta_{r}(C_{X}(d))=r$ for $1\leq r\leq|X|$.

• (c)

If $C_{X}(d)$ is non-degenerate, i.e., for each $1\leq i\leq|X|$ there is $\alpha\in C_{X}(d)$ whose $i$-th entry is non-zero, then $\delta_{r}(C_{X}(d))=|X|$ when $r=H_{I}^{a}(d)$. This follows from Lemma 2.5 and Corollary 3.6 noticing that $V_{X}(F)=\emptyset$ for $F\in\mathcal{F}_{\prec,d,r}$.

Hilbert’s Nullstellensatz over finite fields

The next result is well-known, see [16] for an expository account of Nullstellensatz-type results like this. For convenience we give a short proof of this result using our degree driven approach. We show examples of how to use this result to estimate the basic parameters of evaluation codes over affine varieties defined by a given set of polynomials of $S$ (Examples 6.7 and 6.8).

Proposition 3.12**.**

Let $K$ be a finite field $\mathbb{F}_{q}$, let $\mathbb{A}^{s}=K^{s}$ be the affine space over $K$, and let $G$ be a finite set of polynomials of $S$. If $X=V_{\mathbb{A}^{s}}(G)$ and $(I(\mathbb{A}^{s})\colon(G))\neq I(\mathbb{A}^{s})$, then

[TABLE]

Proof.

As $I(\mathbb{A}^{s})=(t_{1}^{q}-t_{1},\ldots,t_{s}^{q}-t_{s})$ [27, p. 137], we need only show $(I(\mathbb{A}^{s}),G)=I(X)$. The ideal $J:=(I(\mathbb{A}^{s}),G)$ is contained in $I(X)$ because $G\subset I(V_{\mathbb{A}^{s}}(G))$. By Lemma 2.8 and Proposition 2.2, we get $\deg(S/J)=|V_{\mathbb{A}^{s}}(G)|$ and $\deg(S/I(X)=|X|$, respectively. Thus, $S/J$ and $S/I(X)$ have the same degree. Therefore, using the inclusion $J\subset I(X)$, by additivity of the degree and Seidenberg’s lemma the equality follows. ∎

As a byproduct we obtain the next projective version over finite fields of Hilbert’s projective Nullstellensatz over algebraically closed fields [22, Theorem A.4.6, p. 476]. This result can be used to estimate the parameters of evaluation codes over projective varieties defined by a given set of homogeneous polynomials of $S$ [9, 17] (Example 6.9).

Theorem 3.13**.**

Let $\mathbb{P}^{s-1}$ be a projective space over a finite field $K=\mathbb{F}_{q}$, let $\mathbb{X}=V_{\mathbb{P}^{s-1}}(G)$ be the projective variety defined by a finite set $G$ of homogeneous polynomials of $S\setminus\{0\}$, and let $I=I(\mathbb{X})$ be its homogeneous vanishing ideal. If and $(I(\mathbb{P}^{s-1})\colon(G))\neq I(\mathbb{P}^{s-1})$, then

[TABLE]

Proof.

As $I(\mathbb{P}^{s-1})=\{t_{i}t_{j}^{q}-t_{i}^{q}t_{j}|\,1\leq i<j\leq s\})$ [40, Theorem 2.1], we need only show that ${\rm rad}(I(\mathbb{P}^{s-1}),G)$ is equal to $I(\mathbb{X})$. The ideal $J:=(I(\mathbb{P}^{s-1}),G)$ is a subset of $I(\mathbb{X})$ because $G\subset I(V_{\mathbb{P}^{s-1}}(G))$. Thus, ${\rm rad}(J)$ is a subset of $I(\mathbb{X})$ because $I(\mathbb{X})$ is a radical ideal. Using [17, Lemma 3.4] and Proposition 2.2, we get

[TABLE]

and $\deg(S/I(\mathbb{X})=|\mathbb{X}|$, respectively. Thus, $S/J$ and $S/I(\mathbb{X})$ have the same degree. Let $[P_{1}],\ldots,[P_{m}]$ be the points of $\mathbb{X}$ and for each $i$ let $\mathfrak{p}_{i}$ be the homogeneous vanishing ideal of the point $[P_{i}]$. Then, $I(\mathbb{X})=\bigcap_{i=1}^{m}\mathfrak{p}_{i}$. As $J$ and $I(\mathbb{X})$ have height $s-1$ and $J\subset I(\mathbb{X})$, the ideal $J$ has an irredundant primary decomposition of the form $J=\mathfrak{q}_{1}\bigcap\cdots\bigcap\mathfrak{q}_{n}\cap\mathfrak{q}$ such that $n\geq m$, $\mathfrak{q}_{i}$ is $\mathfrak{p}_{i}$-primary of height $s-1$ for $i=1,\ldots,n$, and ${\rm rad}(\mathfrak{q})=(t_{1},\ldots,t_{s})$. Hence, by additivity of the degree Proposition 2.3, one has

[TABLE]

Therefore, $n=m$ and consequently ${\rm rad}(J)=I(\mathbb{X})$. ∎

4. Minimum distance of toric

codes

To avoid repetitions, we continue to employ the notations and definitions used in Section 1. In this section we determine the minimum distance of the toric code $\mathcal{C}_{\mathcal{P}}(d)$. Throughout this section we assume that $K$ is a finite field $\mathbb{F}_{q}$.

Lemma 4.1**.**

(cf. [38, Lemma 3.3])* Let $d_{1},\ldots,d_{s}$ be positive integers and let $L$ be the ideal of $S$ generated by $t_{1}^{d_{1}},\ldots,t_{s}^{d_{s}}$. If $t^{a}=t_{1}^{a_{1}}\cdots t_{s}^{a_{s}}$ is not in $L$, then*

[TABLE]

Proof.

The colon ideal $(L\colon t^{a})$ is equal to $(t_{1}^{d_{1}-a_{1}},\ldots,t_{s}^{d_{s}-a_{s}})$. Since $L$ and $(L\colon t^{a})$ are complete intersections, one has $\deg(S/L)=\prod_{i=1}^{s}d_{i}$ and $\deg(S/(L\colon t^{a}))=\prod_{i=1}^{s}(d_{i}-a_{i})$. Taking affine Hilbert functions in the exact sequence

[TABLE]

we obtain $\deg(S/(L,t^{a}))=\deg(S/L)-\deg(S/(L\colon t^{a}))$. ∎

A polynomial is called squarefree if all its monomials are squarefree. The set of all squarefree monomials of $S$ of degree $d$ (resp. degree at most $d$) is denoted by $V_{d}$ (resp. $V_{\leq d}$).

Lemma 4.2**.**

Let $\prec$ be a monomial order on $S$ and let $I=I(T)$ be the vanishing ideal of an affine torus $T=(\mathbb{F}_{q}^{*})^{s}$ over a finite field with $q\geq 3$ elements. The following hold.

1. (a)

The initial ideal ${\rm in}_{\prec}(I)$ of $I$ is generated by $\{t_{1}^{q-1},\ldots,t_{s}^{q-1}\}$. In particular if $f$ is a squarefree polynomial of $S$, then $f$ is a standard polynomial of $S/I$. 2. (b)

If $\mathcal{L}=KV_{d}$ or $\mathcal{L}=KV_{\leq d}$, then the evaluation code $\mathcal{L}_{T}$ on $T$ is a standard evaluation code on $T$ relative to $\prec$. 3. (c)

The toric code $\mathcal{C}_{\mathcal{P}}(d)$ over the hypersimplex $\mathcal{P}$ and the squarefree evaluation code $\mathcal{C}_{\leq d}$ are standard evaluation codes on $T$ relative to $\prec$.

Proof.

(a): The ideal $I$ is generated by the set $\mathcal{B}=\{t_{i}^{q-1}-1\}_{i=1}^{s}$ and this set is a Gröbner basis of $I$ (see [19] and [34, Lemma 2.3] ). Then, the initial ideal $L:={\rm in}_{\prec}(I)$ of $I$ is generated by the set of monomials $\{t_{i}^{q-1}\}_{i=1}^{s}$. As $q\geq 3$ and $f$ is squarefree, none of the terms of $f$ can be in $L$. Thus, $f$ is a standard polynomial.

(b): By part (a), $V_{d}$ and $V_{\leq d}$ are subset of $K\Delta_{\prec}(I)$. Thus, $\mathcal{L}$ is a linear subspace of $K\Delta_{\prec}(I)$ and $\mathcal{L}_{T}$ is a standard evaluation code.

(c): This follows from part (b) by recalling that $\mathcal{C}_{\mathcal{P}}(d)$ and $\mathcal{C}_{\leq d}$ are the images of $KV_{d}$ and $KV_{\leq d}$ under the evaluation map, respectively. ∎

To prove the next proposition we use the results of Section 2.

Proposition 4.3**.**

If $0\neq f\in KV_{d}$, $q\geq 3$ and $1\leq d<s$, then

[TABLE]

Proof.

Let $\prec$ be a monomial order on $S$ and let $I=I(T)$ be the vanishing ideal of the affine torus $T=(\mathbb{F}_{q}^{*})^{s}$. By Theorem 2.11, one has

[TABLE]

The initial ideal $L:={\rm in}_{\prec}(I)$ of $I$ is generated by the set $\{t_{i}^{q-1}\}_{i=1}^{s}$ (Lemma 4.2). Let $t^{a}={\rm in}_{\prec}(f)=t_{1}^{a_{1}}\cdots t_{s}^{a_{s}}$ be the initial monomial of $f$. Since $f$ is squarefree, so is $t^{a}$. As $q\geq 3$, $t^{a}$ cannot be in $L$. Therefore, by Theorem 2.12 and Lemma 4.1, we get

[TABLE]

where $d=\deg(f)$. Thus, the inequality follows at once from Eq. (4.1). ∎

Proposition 4.4**.**

Let $\mathcal{C}_{\mathcal{P}}(d)$ be the toric code of $\mathcal{P}$ of degree $d$. Then, the length of $\mathcal{C}_{\mathcal{P}}(d)$ is equal to $(q-1)^{s}$ and its dimension is given by

[TABLE]

Proof.

The length $m$ of $\mathcal{C}_{\mathcal{P}}(d)$ is the number of points of $T$, that is, $m=(q-1)^{s}$. Assume that $q\geq 3$. The number of squarefree monomials of $S$ of degree $d$ is $\binom{s}{d}$. Then, one has $\dim_{K}(KV_{d})=\binom{s}{d}$. Hence, it suffices to note that, by Lemmas 3.1 and 4.2, the evaluation map gives an isomorphism between $KV_{d}$ and $\mathcal{C}_{\mathcal{P}}(d)$.

Assume that $q=2$. Then, $T=\{(1,\ldots,1)\}$, $m=1$, $\mathcal{C}_{\mathcal{P}}(d)=\mathbb{F}_{2}$, and $\dim_{K}(\mathcal{C}_{\mathcal{P}}(d))=1$. ∎

We come to the main result of this section.

Theorem 4.5**.**

Let $\mathcal{C}_{\mathcal{P}}(d)$ be the toric code of $\mathcal{P}$ of degree $d$ and let $\delta(\mathcal{C}_{\mathcal{P}}(d))$ be its minimum distance. Then

[TABLE]

Proof.

Assume that $s\geq 2d$ and $q\geq 3$. We set $\eta(d)=(q-2)^{d}(q-1)^{s-d}$ and $\phi(d)=(q-1)^{s}-\eta(d)$. Let $T=(K^{*})^{s}$ be the affine torus in $\mathbb{A}^{s}$, let $\prec$ be a monomial order on $S$, and let $\mathcal{L}:=KV_{d}$ be the linear space generated by $V_{d}$. By Lemma 4.2, $\mathcal{L}_{X}=\mathcal{C}_{\mathcal{P}}(d)$ is a standard evaluation code. Then, by Theorems 2.11 and 3.4, there is $0\neq f\in\mathcal{L}$ such that

[TABLE]

Thus, by Proposition 4.3, one has $|V_{T}(f)|\leq\phi(d)$ and $\delta(\mathcal{C}_{\mathcal{P}}(d))\geq\eta(d)$. Consider the squarefree homogeneous polynomial of degree $d$

[TABLE]

where $h_{i}=t_{2i-1}-t_{2i}$ for $i=1,\ldots,d$. By Eq. (4.3), to prove the inequality $\delta(\mathcal{C}_{\mathcal{P}}(d))\leq\eta(d)$ it suffices to show that the polynomial $f_{d}$ has exactly $\phi(d)$ roots in $T$. As $V_{T}(f_{d})$ is equal to $\bigcup_{i=1}^{d}V_{T}(h_{i})$, using the inclusion-exclusion principle [1, p. 38, Formula 2.12], we get

[TABLE]

The variables occurring in $h_{i}$ and $h_{j}$ are disjoint for $i\neq j$. Thus, counting monomials in each of the intersections, one obtains

[TABLE]

and consequently the number of zeros of $f_{d}$ in $T$ is given by

[TABLE]

Assume that $s<2d$, $d<s$ and $q\geq 3$. The affine torus $T$ in $\mathbb{A}^{s}$ is a group under componentwise multiplication and the map $\sigma\colon{T}\rightarrow{T}$, $[(x_{1},\ldots,x_{s})]\mapsto[(x_{1}^{-1},\ldots,x_{s}^{-1})]$ is a group isomorphism. Setting $Q_{i}:=\sigma(P_{i})$ for $i=1,\ldots,m$, we can write

[TABLE]

We denote the toric code of $\mathcal{P}$ of degree $d$ with respect to $\{P_{1},\ldots,P_{m}\}$ by $\mathcal{C}_{d}$ and denote the toric code of

[TABLE]

of degree $s-d$ with respect to $\{Q_{1},\ldots,Q_{m}\}$ by $\mathcal{C}_{s-d}$. Thus,

[TABLE]

The basic parameters of the projective toric codes $\mathcal{C}_{d}$ and $\mathcal{C}_{s-d}$ do not depend on how we order the elements of $T$. For use below, let $\{M_{1},\ldots,M_{n}\}$ be the set of all squarefree monomials of $S$ of degree $d$. Setting $M_{i}^{c}:=t_{1}\cdots t_{s}/M_{i}$ for $i=1,\ldots,n$, note that $\{M_{1}^{c},\ldots,M_{n}^{c}\}$ is the set of all squarefree monomials of $S$ of degree $s-d$. If $f\in KV_{d}$, writing $f=\sum_{i=1}^{n}\lambda_{i}M_{i}$, $\lambda_{i}\in\mathbb{F}_{q}$ for all $i$, we set $f^{c}:=\sum_{i=1}^{n}\lambda_{i}M_{i}^{c}$. From the equalities

[TABLE]

we get $t_{1}\cdots t_{s}f^{c}\left({t_{1}^{-1}},\ldots,{t_{s}^{-1}}\right)=f(t_{1},\ldots,t_{s})$. Hence, setting $P_{k}=(p_{k,1},\ldots,p_{k,s})$, one has

[TABLE]

for $k=1,\ldots,m$. There is a commutative diagram

[TABLE]

where all the maps are isomorphisms of linear spaces. By Eq. (4.5), $f(P_{k})=0$ if and only if $f^{c}(Q_{k})=0$. Hence, the linear codes $\mathcal{C}_{d}$ and $\mathcal{C}_{s-d}$ have the same minimum distance. Since $s-d\leq s/2$, by the previous part, it follows that

[TABLE]

Thus, the minimum distance of $\mathcal{C}_{d}$ is equal to $(q-2)^{s-d}(q-1)^{d}$, as required.

Assume that $d=s$. Then, $KV_{d}$ is generated by $f:=t_{1}\cdots t_{d}$ and the toric code $\mathcal{C}_{\mathcal{P}}(d)$ is generated by $(f(P_{1}),\ldots,f(P_{m}))$. Since $f(P_{i})\neq 0$ for all $i$, one has $\delta(\mathcal{C}_{\mathcal{P}}(d))=m=(q-1)^{s}$.

Assume that $q=2$. Then, ${T}=\{(1,\ldots,1)\}$, $m=1$, and $\mathcal{C}_{\mathcal{P}}(d)=\mathbb{F}_{2}$. Thus, $\delta(\mathcal{C}_{\mathcal{P}}(d))=1$. ∎

Corollary 4.6**.**

Let $f$ be a squarefree homogeneous polynomial in $S_{d}\setminus\mathbb{F}_{q}$ and let $T$ be the affine torus $(\mathbb{F}_{q}^{*})^{s}$ of $\mathbb{A}^{s}$. If $q\geq 3$ and $s/2<d<s$, then

[TABLE]

and this upper bound is sharp.

*Proof. *Let $\delta(\mathcal{C}_{\mathcal{P}}(d))$ be the minimum distance of $\mathcal{C}_{\mathcal{P}}(d)$. By Theorem 4.5, one has

[TABLE]

Therefore, $|V_{T}(f)|\leq(q-1)^{s}-(q-2)^{s-d}(q-1)^{d}$ and this upper bound is sharp. For convenience we construct a polynomial in $KV_{d}$ where equality occurs. There is $1\leq k\leq d-1$ such that $s=d+k$. Consider the following squarefree homogeneous polynomial of degree $d$

[TABLE]

where $h_{i}=t_{2i-1}-t_{2i}$ for $i=1,\ldots,k$. Since $V_{T}(g_{k})$ is equal to $V_{T}(f_{k})=\bigcup_{i=1}^{k}V_{T}(h_{i})$, where $f_{k}:=h_{1}\cdots h_{k}$, and $k=s-d$, using Eq. (4.4) with $d\rightarrow k=s-d$ and noticing $s>2k$, we get

[TABLE]

The minimum distance $\delta(C_{T}(d))$ of the affine Reed–Muller-type code $C_{T}(d)$ is non-increasing as a function of $d$ (Proposition 2.2). This is no longer the case for the minimum distance of the toric code $\mathcal{C}_{\mathcal{P}}(d)$ (Example 6.10).

5. Squarefree affine evaluation

codes

In this section we determine the minimum distance and the 2nd generalized Hamming weight of a squarefree evaluation code. To avoid repetitions, we continue to employ the notations and definitions used in Sections 1 and 3.

Proposition 5.1**.**

Let $f$ be a squarefree polynomial in $S\setminus\mathbb{F}_{q}$ of degree at most $d$ and let $T$ be the affine torus $(\mathbb{F}_{q}^{*})^{s}$ of $\mathbb{A}^{s}$. If $q\geq 3$ and $d\leq s$, then

[TABLE]

with equality if $d\geq 1$ and $f=(t_{1}-1)\cdots(t_{d}-1)$.

Proof.

Let $I=I(T)$ be the vanishing ideal of $T$. If $f$ is not a zero divisor of $S/I$, then $(I\colon f)=I$ and, by Lemma 2.8, the set $V_{T}(f)$ is empty and the required inequality is clear. Thus, we may assume that $f$ is a zero-divisor of $S/I$. In particular $d\geq 1$. If $e=\deg(f)$, then

[TABLE]

Thus, we may also assume that $d=\deg(f)$. Let $\prec$ be a graded monomial order and let $t^{a}={\rm in}_{\prec}(f)$ be the initial monomial of $f$. Note that $t^{a}$ is squarefree and $d=\deg(t^{a})$ since $\prec$ is graded. The initial ideal $L:={\rm in}_{\prec}(I)$ of $I$ is generated by the set of monomials $\{t_{i}^{q-1}\}_{i=1}^{s}$ (Lemma 4.2). As $q\geq 3$, $t^{a}$ cannot be in $L$. Hence, by Theorem 2.12 and Lemma 4.1, we get

[TABLE]

Now, assume that $f=h_{1}\cdots h_{d}$, where $h_{i}=t_{i}-1$ for $i=1,\ldots,d$. As in the proof of Theorem 4.5, using the inclusion-exclusion principle [1, p. 38, Formula 2.12], the formula for $|V_{T}(f)|$ follows readily. ∎

Proposition 5.2**.**

Let $\mathcal{C}_{\leq d}$ be the squarefree evaluation code of degree $d$ on the affine torus $T=(\mathbb{F}_{q}^{*})^{s}$. Then, the length of $\mathcal{C}_{\leq d}$ is $(q-1)^{s}$, and the dimension of $\mathcal{C}_{\leq d}$ is given by

[TABLE]

Proof.

The length $m$ of the code $\mathcal{C}_{\leq d}$ is the number of points of $T$, that is, $m=(q-1)^{s}$. Assume that $q\geq 3$. The number of squarefree monomial of $S$ of degree at most $d$ is equal to $k:=\sum_{i=0}^{d}\binom{s}{i}$. Then, one has $k=\dim_{K}(KV_{\leq d})$. Setting $\mathcal{L}=KV_{\leq d}$, by Lemmas 3.1 and 4.2, one has $\mathcal{L}\simeq\mathcal{L}_{T}=\mathcal{C}_{\leq d}$. Thus, $\dim_{K}(\mathcal{C}_{\leq d})=k$.

Assume that $q=2$. Then $T=\{(1,\ldots,1)\}$, $m=1$, $\mathcal{C}_{\leq d}=\mathbb{F}_{2}$, and $\dim_{K}(\mathcal{C}_{\leq d})=1$. ∎

Let $\prec$ be a monomial order and let $\mathcal{H}_{\prec,d,r}$ be the set of all $F=\{f_{1},\ldots,f_{r}\}\subset(KV_{\leq d})^{*}$ such that ${\rm in}_{\prec}(f_{1}),\ldots,{\rm in}_{\prec}(f_{r})$ are distinct monomials and $f_{i}$ is monic for all $i$.

Corollary 5.3**.**

Let $K=\mathbb{F}_{q}$ be a finite field and let $I=I(T)$ be the vanishing ideal of the affine torus $T=(\mathbb{F}_{q}^{*})^{s}$. If $q\geq 3$ and $\prec$ is a graded monomial order, then

[TABLE]

Proof.

We set $\mathcal{L}=KV_{\leq d}$. By Lemma 4.2, $\mathcal{L}_{T}$ is a standard evaluation code on $T$ relative to $\prec$. Then, $\mathcal{C}_{\leq d}=\mathcal{L}_{X}$, and the formula for $\delta_{r}(\mathcal{C}_{\leq d})$ is a direct consequence of Theorem 3.4. ∎

To show a lower bound for $\delta_{r}(\mathcal{C}_{\leq d})$, let $\mathcal{J}_{\prec,d,r}$ be the family of all sets $M=\{t^{a_{1}},\ldots,t^{a_{r}}\}$ such that $t^{a_{1}},\ldots,t^{a_{r}}$ are distinct squarefree monomials of $S_{\leq d}$. The $r$-th squarefree footprint of $\mathcal{C}_{\leq d}$ of degree $d$, denoted $\rho_{I}(d,r)$, is given by

[TABLE]

Corollary 5.4**.**

Let $K=\mathbb{F}_{q}$ be a finite field and let $I=I(T)$ be the vanishing ideal of the affine torus $T=(\mathbb{F}_{q}^{*})^{s}$. If $q\geq 3$ and $\prec$ is a graded monomial order, then

[TABLE]

Proof.

We set $\mathcal{L}=KV_{\leq d}$. By Lemma 4.2, $\mathcal{L}_{T}$ is a standard evaluation code on $T$ relative to $\prec$. Then, $\mathcal{C}_{\leq d}=\mathcal{L}_{X}$, and the lower bound for $\delta_{r}(\mathcal{C}_{\leq d})$ is a direct consequence of Theorem 3.9. ∎

We come to one of the main results of this section.

Theorem 5.5**.**

Let $\mathcal{C}_{\leq d}$ be the squarefree evaluation code of degree $d$ on the affine torus $T=(\mathbb{F}_{q}^{*})^{s}$. If $q\geq 3$, then the minimum distance $\delta(\mathcal{C}_{\leq d})$ of $\mathcal{C}_{\leq d}$ is $(q-2)^{d}(q-1)^{s-d}$.

Proof.

Let $\prec$ be a monomial order on $S$. We set $\mathcal{L}=KV_{\leq d}$. As $q\geq 3$, by Lemma 4.2, $\mathcal{L}_{T}$ is a standard evaluation code on $T$ relative to $\prec$. Then, by Corollary 5.3, we get

[TABLE]

and the result follows from Proposition 5.1. ∎

We come to another of the main results of this section.

Theorem 5.6**.**

If $q\geq 3$ and $d\geq 1$, then the second generalized Hamming weight of $\mathcal{C}_{\leq d}$ is

[TABLE]

*Proof. *Let $\prec$ be a graded monomial order and let $I=I(T)$ be the vanishing ideal of $T$. The initial ideal of $I$ is $L=(\{t_{i}^{q-1}\}_{i=1}^{s})$ (Lemma 4.2). If $s=1$, then $d=s=1$ and $\delta_{2}(\mathcal{C}_{\leq 1})=q-1$. Indeed, take $\{f_{1},f_{2}\}$ in $\mathcal{H}_{\prec,1,2}$ and notice that $(f_{1},f_{2})=S$, then $\deg(S/(t_{1}^{q-1}-1,f_{1},f_{2}))=0$ and consequently, by Corollary 5.3, $\delta_{2}(\mathcal{C}_{\leq 1})=q-1$. Thus, we may assume $s\geq 2$.

The support of a monomial $t^{a}$, denoted ${\rm supp}(t^{a})$, is the set of all $t_{i}$ that occur in $t^{a}$. Take $\{t^{a},t^{b}\}\in\mathcal{J}_{\prec,d,2}$, $t^{a}=\prod_{i=1}^{s}t_{i}^{a_{i}}$ and $t^{b}=\prod_{i=1}^{s}t_{i}^{b_{i}}$. We set $e=\deg(t^{a})$, $n=\deg(t^{b})$, $A={\rm supp}(t^{a})$, and $B={\rm supp}(t^{b})$. We may assume $e\leq n\leq d$. As $t^{a}$ and $t^{b}$ are squarefree monomials, one has

[TABLE]

Therefore, picking a new variable $u$ and setting $J=(L,t^{a},t^{b})$, from [17, p. 343] and using that $t^{a},t^{b}$ are squarefree monomials, we obtain

[TABLE]

Case (a): Assume $d=s$. As $t^{a},t^{b}$ are distinct squarefree monomials, one has $e<s$. Indeed, if $e=s$, then $t^{a}=t_{1}\cdots t_{s}$, $s=e\leq n\leq s$, and $t^{a}=t^{b}$, a contradiction. First we show the inequality $\delta_{2}(\mathcal{C}_{\leq d})\geq(q-2)^{s-1}(q-1)$. By Corollary 5.4, $\delta_{2}(\mathcal{C}_{\leq d})\geq\rho_{I}(d,2)$. Thus, it suffices to show the inequality $\rho_{I}(d,2)\geq(q-2)^{s-1}(q-1)$. Hence, we need only show

[TABLE]

for any $\{t^{a},t^{b}\}$ in $\mathcal{J}_{\prec,d,2}$. Note that, by Eq. (5.1), this inequality is equivalent to

[TABLE]

That this inequality holds follows recalling that $e<s$ and noticing the next two inequalities

[TABLE]

Now, we show the inequality $\delta_{2}(\mathcal{C}_{\leq d})\leq(q-2)^{s-1}(q-1)$. By Corollary 5.3 and Theorem 2.11, it suffices to find $\{f_{1},f_{2}\}$ in $\mathcal{H}_{\prec,s,2}$ such that

[TABLE]

Setting $f_{1}=(t_{1}-1)\cdots(t_{s}-1)$ and $f_{2}=(t_{2}-1)\cdots(t_{s}-1)$, one has $|V_{T}(f_{1},f_{2})|=|V_{T}(f_{2})|$. As $\deg(f_{2})=d-1$ and $d=s$, by Proposition 5.1, we get

[TABLE]

Case (b): Assume $d<s$. First we show the inequality $\delta_{2}(\mathcal{C}_{\leq d})\geq(q-2)^{d}(q-1)^{s-d-1}q$. By Corollary 5.4, $\delta_{2}(\mathcal{C}_{\leq d})\geq\rho_{I}(d,2)$. Thus, it suffices to show $\rho_{I}(d,2)\geq(q-2)^{d}(q-1)^{s-d-1}q$. Hence, we need only show that the inequality

[TABLE]

holds for any $\{t^{a},t^{b}\}$ in $\mathcal{J}_{\prec,d,2}$. Note that, by Eq. (5.1), this inequality is equivalent to

[TABLE]

To show this inequality we consider two subcases.

($\mathrm{b}_{1}$): Assume $e=d$. Then, $e=n=d$, and Eq. (5.2) is equivalent to

[TABLE]

This inequality follows by noticing that $|A\cup B|\geq d+1$. Indeed, if $|A\cup B|=d$, then $A=B$ and $t^{a}=t^{b}$, a contradiction.

($\mathrm{b}_{2}$): Assume $e<d$. Setting $k=d-e$, one has $(q-1)^{k+1}\geq(q-2)^{k}q$ because $k\geq 1$. This inequality is easy to show using induction on $k\geq 1$. That Eq. (5.2) holds true now follows from the following two inequalities

[TABLE]

Finally, we show the inequality $\delta_{2}(\mathcal{C}_{\leq d})\leq(q-2)^{d}(q-1)^{s-d-1}q$. By Corollary 5.3 and Theorem 2.11, it suffices to find $\{f_{1},f_{2}\}$ in $\mathcal{H}_{\prec,d,2}$ such that

[TABLE]

Setting $f_{1}=(t_{1}-1)\cdots(t_{d}-1)$, $f_{2}=(t_{2}-1)\cdots(t_{d+1}-1)$, and $g=(t_{1}-1)\cdots(t_{d+1}-1)$, one has $|V_{T}(f_{1})\cup V_{T}(f_{2}))|=|V_{T}(g)|$ and, by Proposition 5.1, we get

[TABLE]

6. Examples

This section includes examples illustrating some of our results. In Appendix A we give the implementations in Macaulay$2$ [21] and Magma [6] that are used in the examples.

Example 6.1**.**

Let $K$ be the field $\mathbb{F}_{5}$, let $S=K[t_{1},t_{2}]$ be a polynomial ring in two variables, and let $\mathcal{L}$ be the $K$-linear space generated by all monomials $t_{1}^{a_{1}}t_{2}^{a_{2}}$ such that $(a_{1},a_{2})$ is one of the solid points of the point configuration depicted on the left of Figure 1. The evaluation code $\mathcal{L}_{T}$, over the torus $T=(\mathbb{F}_{5}^{*})^{2}$, is a generalized toric code in the sense of [33, 43]. Let $\prec$ be a monomial order. The vanishing ideal $I$ of $T$ is generated by the Gröbner basis $G=\{t_{1}^{4}-1,t_{2}^{4}-1\}$. Then, $\mathcal{L}$ is generated by

[TABLE]

and the list of remainder of the elements of $B$ on division by $G$ is

[TABLE]

that is, $\widetilde{\mathcal{L}}$ is generated by $\widetilde{B}$ and $\widetilde{\mathcal{L}}_{T}$ is a standard evaluation code on $T$ relative to $\prec$. Using Theorem 3.4, Proposition 3.5, Theorem 3.9, and Procedure A.2, we obtain that the minimum distance $\delta_{1}(\mathcal{L}_{T})$ of ${\mathcal{L}}_{T}$ is $8$ and the footprint ${\rm fp}_{1}(\mathcal{L}_{X})$ is $4$. The length and the dimension of $\mathcal{L}_{X}$ are $16$ and $6$, respectively.

Example 6.2**.**

Let $K$ be the field $\mathbb{F}_{3}$, let $X=\{(0,0),(1,0),(0,1),(1,1),(0,-1)\}$ be a set of $5$ points on $\mathbb{A}^{2}$, and let $I=I(X)$ be the vanishing ideal of $X$. The ideal $I$ is generated by the binomials $t_{1}^{2}-t_{1},t_{2}^{3}-t_{2},t_{1}t_{2}^{2}-t_{1}t_{2}$. Using Corollary 3.6 and Procedure A.3, we obtain

[TABLE]

$H_{I}^{a}(1)=\dim_{K}(C_{X}(1))=3$, and $H_{I}^{a}(2)=\dim(C_{X}(2)=5$.

Example 6.3**.**

Let $S=\mathbb{Q}[x,y]$ be a polynomial ring and let $\prec$ be the lexicographical order with $x\succ y$. If $I=(g_{1},g_{2})$ is the ideal generated by $g_{1}=y^{40}-y^{2}+1$ and $g_{2}=x-y^{8}$, and $f$ is the polynomial $x^{2}-y^{3}+y$, then $f\in S_{\leq 3}$ but the residue of the division of $f$ by $\{g_{1},g_{2}\}$ is $h=y^{16}-y^{3}+y$ and $h\notin S_{\leq 3}$ (see Procedure A.4). This example shows that the proof of Corollary 3.6 is only valid for graded orders.

Example 6.4**.**

Let $K$ be the field $\mathbb{F}_{3}$ and let $X$ be the following set of $12$ points in $\mathbb{A}^{3}$:

[TABLE]

Using Corollary 3.7 and Procedure A.5, we obtain that ${\rm reg}(H^{a}_{I(X)})=4$ and the minimum distance $\delta(C_{X}(d))$ of $C_{X}(d)$ is given by

[TABLE]

Example 6.5**.**

Let $K$ be the field $\mathbb{F}_{5}$, let $T$ be the affine torus $(\mathbb{F}_{5}^{*})^{2}$, and let $I=I(T)$ be the vanishing ideal of $T$. Using Theorem 3.9 and Procedure A.6, we obtain the inequality ${\rm fp}_{I}(d,r)\leq\delta_{r}(C_{T}(d,r))$ and the following table:

[TABLE]

Example 6.6**.**

Let $S=K[t_{1},t_{2},t_{3}]$ be a polynomial ring over the field $K=\mathbb{F}_{3}$, let $X$ be the following set of $10$ points in $\mathbb{A}^{3}$:

[TABLE]

let $\mathcal{L}$ be the linear space $S_{d}$, and let $\mathcal{L}_{X}$ be the evaluation code on $X$. Using Theorem 3.4, Proposition 3.5, and Procedure A.8, we obtain the following information:

[TABLE]

if $d=2$ the linear space $\widetilde{\mathcal{L}}$ of Proposition 3.5 is generated by $\{t_{2},\,t_{1}t_{2},\,t_{1}t_{3},\,t_{3}^{2},\,t_{2}t_{3},\,t_{1}\}$, the minimum distance $\delta(\mathcal{L}_{X})$ of $\mathcal{L}_{X}$ is given by

[TABLE]

and $\delta_{2}(\mathcal{L}_{X})=9+d-2$ for $d=2,3$.

Example 6.7**.**

Let $S=K[x,y]$ be a polynomial ring over the finite field $K=\mathbb{F}_{25}$ with $25$ elements, let $X=V_{\mathbb{A}^{2}}(g)$ be the Hermitian curve defined by $g=y^{5}+y-x^{6}$ [52, p. 242] and let $I=I(X)$ be its vanishing ideal. If $X\neq\emptyset$, by Proposition 3.12, the vanishing ideal of $X$ can be computed using the equality

[TABLE]

To compute the length of the Reed–Muller-type code $C_{X}(d)$ of degree $d$ over the curve $X$, we use Macaulay$2$ [21] to obtain

[TABLE]

We can compute the minimum distance of $C_{X}(d)$ using Proposition 3.12, Corollary 3.7 and Procedure A.5. The $r$-th generalized Hamming weight $\delta_{r}(C_{X}(d))$ and $r$-th footprint ${\rm fp}_{I}(d,r)$ of $C_{X}(d)$ can be computed using Procedures A.3 and A.6. If $d=1$, we obtain a linear code $C_{X}(1)$ of length $125$, dimension $3$, and minimum distance $119$. If $d=4$, using the footprint lower bound of Corollary 3.10, we obtain a linear code $C_{X}(4)$ of length $125$, dimension $15$, minimum distance at least $40$ because ${\rm fp}_{I}(4,1)=40$, and $\delta_{7}(C_{X}(4)\geq 97$ because ${\rm fp}_{I}(4,7)=97$. We can use this example as a model to estimate the parameters of $C_{X}(d)$ for any affine variety $X=V_{\mathbb{A}^{s}}(G)$ defined by a given finite set $G$ of polynomials of $S$ in $s\geq 2$ variables, by replacing $g$ by $G$.

Example 6.8**.**

Let $C=V_{\mathbb{A}^{2}}(f)\cup\{\mathcal{O}\}$ be the elliptic curve of the polynomial $f=y^{2}-x^{3}+x$ over a finite field $K=\mathbb{F}_{q}$ of ${\rm char}(K)\neq 2$. Elliptic curves have a group structure and they are used in cryptography [29]. If $K=\mathbb{F}_{71}$, to compute the number of zeros of $f$ in $\mathbb{A}^{2}$ notice that $I(\mathbb{A}^{2})=(x^{71}-x,y^{71}-y)$ [27, p. 137]. Then, using Lemma 2.8 and Macaulay$2$ [21], we obtain

[TABLE]

see Procedure A.1. If we apply Theorem 2.12 with the lexicographical order $y\succ x$, we obtain

[TABLE]

For a polynomial $g$ defining an elliptic curve over a finite field $\mathbb{F}_{q}$, the bound of Theorem 2.12 says that $|V_{\mathbb{A}^{2}}(g)|\leq 2q$, which is not a good bound (cf. Hasse’s theorem [29, p. 174]).

Let $X=V_{\mathbb{A}^{2}}(f)$ be the affine variety of $f$ and let $I=I(X)$ be its vanishing ideal. We can compute the minimum distance of the Reed–Muller-type code $C_{X}(d)$, over the elliptic curve $X$, using Proposition 3.12, together with Corollary 3.7 and Procedure A.5. The $r$-th footprint of $C_{X}(d)$ can be computed using Procedure A.6. If $q=5$ and $d=1$, we obtain a linear code $C_{X}(1)$ of length $7$, dimension $3$, minimum distance $4$, and the $1$-st footprint ${\rm fp}_{I}(1,1)$ of $C_{X}(1)$ is $4$. If $q=71$ and $d=1$, we obtain a linear code $C_{X}(1)$ of length $71$, dimension $3$, and minimum distance $68$. If $q=199$ and $d=10$, we obtain a linear code $C_{X}(10)$ of length $199$, dimension $30$, and minimum distance at least $57$ because ${\rm fp}_{I}(10,1)=57$.

Example 6.9**.**

Let $S=K[x,y,z]$ be a polynomial ring over the finite field $K=\mathbb{F}_{4}$, let $\mathbb{P}^{2}$ be the projective space over the field $K$, and let $\mathbb{X}=V_{\mathbb{P}^{2}}(g)$ be the projective variety of the homogeneous polynomial $g=y^{3}+xz^{2}+x^{2}z$. Then, by Theorem 3.13, the homogeneous vanishing ideal $I(\mathbb{X})$ of $\mathbb{X}$ is given by

[TABLE]

Computing the radical on the right with Magma [6] (Procedure A.7) gives

[TABLE]

If $\mathcal{L}=S_{1}$, using Eq. (6.2) and [17, Procedure 7.1, p. 334], we obtain that the standard evaluation code $\mathcal{L}_{\mathbb{X}}$ on $\mathbb{X}$ has length $9$, dimension $3$, and minimum distance $6$. Note that we cannot apply Procedure A.8 since $I(\mathbb{X})$ is not an affine vanishing ideal.

Example 6.10**.**

For $s=4$ and $q=3$, by Theorem 4.5, the list of values of the length, dimension and minimum distance of $\mathcal{C}_{\mathcal{P}}(d)$ are given by the following table.

[TABLE]

Appendix A Procedures for Macaulay$2$

In this section we give procedures for Macaulay$2$ [21] and Magma [6] using finite fields to compute generalized Hamming weights and lower bound footprints of evaluation codes.

Procedure A.1**.**

Computing the number of points of an affine variety over a finite field using Lemma 2.8. This procedure correspond to Example 6.8. To compute other examples just change the finite field and the set of polynomials $F$ that define the affine variety.

q=71 S=ZZ/q[x,y]-- finite field K=ZZ/q I=ideal(x^q-x,y^q-y)--vanishing ideal of K^2 F={y^2-x^3+x} quotient(I,ideal(F))==I--false means F has zeros degree (I+ideal(F))--number of zeros of f

Procedure A.2**.**

Computing the generalized Hamming weights and footprint of an evaluation code $\mathcal{L}_{X}$ using Theorem 3.4, Proposition 3.5, and Theorem 3.9. The input for this procedure is a generating set for $\mathcal{L}$ and the vanishing ideal $I$ of $X$. This procedure corresponds to Example 6.1.

q=5, H=GF(q,Variable=>a), S=H[t1,t2] I=ideal(t1^(q-1)-1,t2^(q-1)-1), M=coker gens gb I r=1--we are computing the r-th generalized Hamming weight G=gb I, div=(x)->x % G --This is the K-basis for L Basis=matrix{{1,t1^3,t1t2^2,t2^3,t1t2,t1^2*t2^4}} --This is the list of remainders on division by G cL=toList set apply(flatten entries Basis,div) --This gives the r-th generalized Hamming weight gmd=degree M-max apply(apply(subsets(toList set apply(toList set(apply(apply(apply(apply(toList ((set(0,a,a^2,a^3,a^4))^(#cL)-(set{0})^(#cL))/deepSplice, toList),x->matrix{cL}*vector x),entries),n->n#0)), m->(leadCoefficient(m))^(-1)*m),r),ideal), x-> if #(set flatten entries leadTerm gens x)==r then degree(I+x) else 0) init=ideal(leadTerm gens gb I) er=(x)-> degree ideal(init,x) --This is the r-th footprint fpr=degree M - max apply(apply(apply(subsets(cL,r), toSequence),ideal),er)

Procedure A.3**.**

Computing the generalized Hamming weights of an affine Reed–Muller-type code using Corollary 3.6. This procedure correspond to Example 6.2. Other examples can be computed changing the finite field, the affine space, and the set of points of $X$.

q=3, G=ZZ/q, S=G[t1,t2]; I1=ideal(t1,t2), I2=ideal(t1-1,t2),I3=ideal(t1,t2-1) I4=ideal(t1-1,t2-1),I5=ideal(t1,t2+1) I=intersect(I1,I2,I3,I4,I5)--this is the vanishing ideal M=coker gens gb I --This is the r-th generalized Hamming weight of C_X(d): genmdaffine=(d,r)->degree M-max apply(apply(subsets(toList set apply(toList set(apply(apply(apply(apply(toList ((set(0..q-1))^(#flatten entries basis(0,d,M))- (set{0})^(#flatten entries basis(0,d,M))) /deepSplice,toList),x->basis(0,d,M)*vector x), entries),n->n#0)), m->(leadCoefficient(m))^(-1)*m),r), ideal), x-> if #(set flatten entries leadTerm gens x)==r then degree(I+x) else 0) --This is the affine Hilbert function of I: #flatten entries basis(0,1,M), #flatten entries basis(0,2,M) genmdaffine(1,1), genmdaffine(1,2),genmdaffine(1,3) genmdaffine(2,1), genmdaffine(2,2)

Procedure A.4**.**

Computing the quotient and the remainder in the multivariate division algorithm [10, Theorem 3, p. 63]. This procedure correspond to Example 6.3

R=QQ[x,y,MonomialOrder=>Lex] I=ideal(y^(40)-y^2+1,x-y^8) f=x^2-y^3+y G=matrix{{y^(40)-y^2+1,x-y^8}} remainder(matrix{{f}},G) quotientRemainder(matrix{{f}},G)

Procedure A.5**.**

Computing the minimum distance of an affine Reed–Muller-type code of degree $d$ using Corollary 3.7. This procedure corresponds to Example 6.4

q=3, S=ZZ/q[t1,t2,t3]; I1=ideal(t1-1,t2,t3), I2=ideal(t1-1,t2,t3-1),I3=ideal(t1-1,t2,t3+1), I4=ideal(t1-1,t2-1,t3),I5=ideal(t1-1,t2-1,t3-1),I6=ideal(t1-1,t2-1,t3+1), I7=ideal(t1,t2,t3), I8=ideal(t1,t2,t3-1),I9=ideal(t1,t2,t3+1), I10=ideal(t1,t2-1,t3),I11=ideal(t1,t2-1,t3-1),I12=ideal(t1,t2-1,t3+1) I=intersect(I1,I2,I3,I4,I5,I6,I7,I8,I9,I10,I11,I12) M=coker gens gb I --This computes the minimum distance of an affine Reed-Muller-type code --of degree d mindisaffine=(d)-> degree M- max apply(apply((toList (set (apply(toList set apply(toList set(apply(apply(apply(apply(toList ((set(0..q-1))^(#flatten entries basis(0,d,M))- (set{0})^(#flatten entries basis(0,d,M)))/deepSplice,toList), x->basis(0,d,M)*vector x),entries),n->n#0)), m->(leadCoefficient(m))^(-1)*m),x-> if degree(x)=={d} then x else t1^0))-set{t1^0})),ideal),x-> degree(I+x)) mindisaffine(1), mindisaffine(2), mindisaffine(3)

Procedure A.6**.**

Computing the footprint of a Reed–Muller-type code $C_{X}(d)$. This procedure corresponds to Example 6.5. To compute other examples just change the finite field and the vanishing ideal $I$ of $X$.

q=5, G=ZZ/q, S=G[t1,t2]; I=ideal(t1^(q-1)-1,t2^(q-1)-1)--vanishing ideal of the affine torus T M=coker gens gb I init=ideal(leadTerm gens gb I) er=(x)-> if not quotient(init,x)==init then degree ideal(init,x) else 0 --This is the r-th footprint: fpraffine=(d,r)->degree M - max apply(apply(apply(subsets (flatten entries basis(0,d,M),r),toSequence),ideal),er) f=(n)->#flatten entries basis(0,n,M), apply(1..6,f) f1=(n)->fpraffine(n,1),apply(1..6,f1) f2=(n)->fpraffine(n,2),apply(1..6,f2) f3=(n)->fpraffine(n,3),apply(1..6,f3)

Procedure A.7**.**

Computing the radical of an ideal over a finite field using Magma [6]. This procedure corresponds to Example 6.9.

P<x,y,z>:=PolynomialRing(FiniteField(2, 2),3); J:= ideal<P| xy^4-x^4y,xz^4-x^4z,yz^4-y^4z,y^3+xz^2+x^2z>; Radical(J);

Procedure A.8**.**

Given an evaluation code $\mathcal{L}_{X}$ on $X$ and a monomial order $\prec$. This procedure computes a linear subspace $\widetilde{\mathcal{L}}$ of $S$ such that $\widetilde{\mathcal{L}}_{X}$ is a standard evaluation code on $X$ and $\widetilde{\mathcal{L}}_{X}=\mathcal{L}_{X}$ (Proposition 3.5). Then it computes the $r$-generalized Hamming weight of a projective Reed–Muller-type code on $X$ of degree $d$ (Theorem 3.4). This procedure corresponds to Example 6.6. To compute $\delta_{r}(\mathcal{L}_{X})$ for an evaluation code $\mathcal{L}_{X}$ replace basis(d,S) by the matrix of a $K$-basis of the linear space $\mathcal{L}$.

q=3, S=ZZ/q[t1,t2,t3]; I1=ideal(t1-1,t2,t3),I2=ideal(t1-1,t2,t3-1),I3=ideal(t1-1,t2,t3+1), I4=ideal(t1-1,t2-1,t3),I5=ideal(t1-1,t2-1,t3-1), I6=ideal(t1-1,t2-1,t3+1),I7=ideal(t1,t2,t3-1),I8=ideal(t1,t2-1,t3), I9=ideal(t1,t2-1,t3-1),I10=ideal(t1,t2-1,t3+1) I=intersect(I1,I2,I3,I4,I5,I6,I7,I8,I9,I10) M=coker gens gb I d=2, r=1, G=gb I div=(x)->x % G --This is the list of residues of S_d after division by G cL=toList set apply(flatten entries basis(d,S),div) --This gives the r-th generalized Hamming weight of --the evaluation code on X defined by S_d gmd=degree M-max apply(apply(subsets(toList set apply(toList set(apply(apply(apply(apply(toList ((set(0..q-1))^(#cL)- (set{0})^(#cL))/deepSplice,toList),x->matrix{cL}*vector x), entries),n->n#0)),m->(leadCoefficient(m))^(-1)*m),r),ideal), x-> if #(set flatten entries leadTerm gens x)==r then degree(I+x) else 0)

Acknowledgments

We thank Nupur Patanker for pointing out an error in the previous statement of Corollary 4.6 and Proposition 5.1. Computations with Magma [6] and Macaulay$2$ [21] were important to give examples and to have a better understanding of evaluation codes.