Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes

TL;DR

This paper introduces monomial-Cartesian codes, generalizes existing code families, describes their duals, and explores applications in quantum error correction and locally recoverable codes with multiple recovery options.

Contribution

It provides a dual code description for monomial-Cartesian codes and demonstrates their applications in constructing quantum and locally recoverable codes.

Findings

Duals of monomial-Cartesian codes are characterized using vanishing ideals.

Existence of quantum error-correcting codes derived from these codes.

Product codes exhibit local recoverability with multiple recovery options.

Abstract

A monomial-Cartesian code is an evaluation code defined by evaluating a set of monomials over a Cartesian product. It is a generalization of some families of codes in the literature, for instance toric codes, affine Cartesian codes and $J$-affine variety codes. In this work we use the vanishing ideal of the Cartesian product to give a description of the dual of a monomial-Cartesian code. Then we use such description of the dual to prove the existence of quantum error correcting codes and MDS quantum error correcting codes. Finally we show that the direct product of monomial-Cartesian codes is a locally recoverable code with $t$-availability if at least $t$ of the components are locally recoverable codes.

Equations

$$\mathcal{S}:={\mathcal{S}}_1\times \cdots \times {\mathcal{S}}_m\subseteq {\mathcal{K}}^m.$$

$$\mathcal{L}(\mathcal{A})={\mathrm{Span}}_{\mathcal{K}}\{{\mathbf{x}}^{\mathbf{a}}:\mathbf{a}\in \mathcal{A}\}\subset \mathcal{R}.$$

$$\begin{array}[]{lclll}{\operatorname{ev}_{\cal S}}\colon&\cal L(A)&\to&K^{|\mathcal{\cal S}|}\\ &f&\mapsto&\left(f(\bm{s}_{1}),\ldots,f(\bm{s}_{n})\right).\end{array}$$

$$\delta (C)=\min \{|\mathrm{supp}(c)|:0\ne c\in C\},$$

$$\delta \left(C({\mathcal{S}}_1\times {\mathcal{S}}_2,\mathcal{A}\times \mathcal{B})\right)=\delta \left(C({\mathcal{S}}_1,\mathcal{A})\right)\delta \left(C({\mathcal{S}}_2,\mathcal{B})\right).$$

$$C^{\perp }=\{\mathbf{w}\in K^n:\mathbf{w}\cdot \mathbf{c}=0\text{ for all }\mathbf{c}\in C\},$$

$$k\le n-2d+2.$$

$$L_i(x_i):=\prod _{\begin{array}{c}{s}_j\in S_i\end{array}}\left(x_i-s_j\right).$$

$${\mathrm{Res}}_{\mathbf{s}}f=f(\mathbf{s}){\left(\prod _{i=1}^m\prod _{\begin{array}{c}{s}_i^{\prime }\in S_i\setminus \{s_i\}\end{array}}\left(s_i-s_i^{\prime }\right)\right)}^{-1}.$$

$${\mathrm{Res}}_{\mathcal{S}}f=\left({\mathrm{Res}}_{{\mathbf{s}}_1}f,\dots ,{\mathrm{Res}}_{{\mathbf{s}}_n}f\right).$$

$$C(\mathcal{S},\bm{0})^{\perp}=\operatorname{Span}_{K}\Big{\{}\operatorname{Res}_{\cal S}f:\deg(f)<\sum_{i=1}^{m}(n_{i}-1),\deg_{x_{i}}(f)<n_{i}\Big{\}}.$$

$$\sum _{i=1}^n{\mathrm{Res}}_{{\mathbf{s}}_i}f=0,\text{ for }f\in R\text{ with }\mathrm{deg}(f)<\sum _{i=1}^m(n_i-1)\text{ and }{\mathrm{deg}}_{x_i}(f)<n_i.$$

$$L_i=x_i^j{q}_{i,j-1}+r_{i,j-1},$$

$$Q_{\mathbf{b}}(\mathbf{x}):=\prod _{i=1}^m{q}_{i,b_i}(x_i).$$

$$\left({\mathbf{s}}_1^{\mathbf{a}},\dots ,{\mathbf{s}}_n^{\mathbf{a}}\right)\cdot {\mathrm{Res}}_{\mathcal{S}}{Q}_{\mathbf{b}}=\sum _{i=1}^n{\mathbf{s}}_i^{\mathbf{a}}{\mathrm{Res}}_{{\mathbf{s}}_i}{Q}_{\mathbf{b}}=\sum _{i=1}^n{\mathrm{Res}}_{{\mathbf{s}}_i}{\mathbf{x}}^{\mathbf{a}}{Q}_{\mathbf{b}}=\sum _{i=1}^n{\mathrm{Res}}_{{\mathbf{s}}_i}\overline{{\mathbf{x}}^{\mathbf{a}}{Q}_{\mathbf{b}}}.$$

$$\deg_{x_{i}}\big{(}\overline{\bm{x}^{\bm{a}}Q_{\bm{b}}(\bm{x})}\big{)}\leq\left\{\begin{array}[]{cll}a_{i}+n_{i}-(b_{i}+1)&\mbox{ if }a_{i}\leq b_{i},\\ a_{i}-1&\mbox{ if }a_{i}>b_{i}.\end{array}\right.$$

$$\deg_{x_{i}}\big{(}\overline{\bm{x}^{\bm{a}}Q_{\bm{b}}(\bm{x})}\big{)}=\deg_{x_{i}}\big{(}\overline{x_{i}^{a_{i}}q_{i,b_{i}}(x_{i})}\big{)}=\deg_{x_{i}}\big{(}x_{i}^{a_{i}-(b_{i}+1)}r_{i,b_{i}}(x_{i})\big{)}<a_{i}.$$

$$\deg_{x_{j}}\big{(}\overline{\bm{x}^{\bm{a}}Q_{\bm{b}}(\bm{x})}\big{)}<n_{j}-1.$$

$$C(\mathcal{S},(2,3){)}^{\perp }={\text{Span}}_K\left\{{\mathrm{Res}}_{\mathcal{S}}{Q}_{\mathbf{b}}:b\in \{0,1,2\}\times \{0,1,2,3,4\},b\ne (2,3)\right\}.$$

$$C(\mathcal{S},\{(0,1),(3,5)\}{)}^{\perp }={\text{Span}}_K\left\{{\mathrm{Res}}_{\mathcal{S}}{Q}_{\mathbf{b}}:b\in \{0,1,2\}\times \{0,1,2,3,4\},b\notin \{(0,1),(3,5)\}\right\}.$$

$$F(\mathbf{x})=\sum _{\left(s_1,\dots ,s_m\right)\in \mathcal{S}}\frac{{\prod }_{i=1}^m{\prod }_{\begin{array}{c}{s}_i^{\prime }\in S_i\setminus \{s_i\}\end{array}}\left(x_i-s_i^{\prime }\right)}{{\left({\prod }_{i=1}^m{\prod }_{\begin{array}{c}{s}_i^{\prime }\in S_i\setminus \{s_i\}\end{array}}\left(s_i-s_i^{\prime }\right)\right)}^2}.$$

$$F^{\prime }(\mathbf{x}):=\frac{{\prod }_{i=1}^m{\prod }_{\begin{array}{c}{s}_i^{\prime }\in K\setminus S_i\end{array}}\left(x_i-s_i^{\prime }\right)}{{\left(-1\right)}^m}.$$

$$F^{\prime }(\mathbf{s})=\frac{{\prod }_{\begin{array}{c}{s}_i^{\prime }\in K\setminus S_i\end{array}}\left(s_1-s_i^{\prime }\right)}{-1}\cdots \frac{{\prod }_{\begin{array}{c}{s}_i^{\prime }\in K\setminus S_i\end{array}}\left(s_m-s_i^{\prime }\right)}{-1}={\left(\prod _{i=1}^m\prod _{\begin{array}{c}{s}_i^{\prime }\in S_i\setminus \{s_i\}\end{array}}\left(s_i-s_i^{\prime }\right)\right)}^{-1}.$$

Full text

The second author was supported by NSF DMS-1855136.

1. Introduction

Let $K=\mathbb{F}_{q}$ be a finite field with $q$ elements and $R=K[x_{1},\ldots,x_{m}]$ the polynomial ring over $K$ in $m$ variables. We write $K^{*}=K\setminus\{0\}$ for the multiplicative group of $K$. Given a lattice point $\bm{a}\in\mathbb{Z}_{\geq 0}^{m}$ we use $\bm{x}^{\bm{a}}$ to denote the corresponding monomial in $R$, i.e. $\bm{x}^{\bm{a}}=x_{1}^{a_{1}}\cdots x_{m}^{a_{m}}$ for $\bm{a}=(a_{1},\dots,a_{m})$. Given a positive integer $\ell,$ we define $\left[\ell\right]:=\left\{1,\ldots,\ell\right\}.$

A monomial-Cartesian code is defined as follows. Fix non-empty subsets $S_{1},\ldots,S_{m}$ of $K$. Define their Cartesian product as

[TABLE]

Furthermore, let $A\subset\mathbb{Z}_{\geq 0}^{m}$ be a finite lattice set and $\cal L(A)$ the subspace of polynomials of $R$ that are $K$-linear combinations of monomials with exponents in $A$:

[TABLE]

Fix a linear order of the points in $\cal S=\{\bm{s}_{1},\ldots,\bm{s}_{n}\},$ $\bm{s}_{1}\prec\cdots\prec\bm{s}_{n}$. This defines the evaluation map

[TABLE]

In what follows, $n_{i}:=|S_{i}|$, the cardinality of $S_{i}$ for $i\in[m].$ From now on, we assume that $A\subset\{0,\ldots,n_{1}-1\}\times\cdots\times\{0,\ldots,n_{m}-1\},$ that is the degree of each $f\in\cal L(A)$ in $x_{i}$ is less than $|S_{i}|$. In this case the evaluation map $\operatorname{ev}_{\cal S}$ is injective (see the proof of Proposition 2.1).

Definition 1.1**.**

Let $\cal S\subset K^{m}$ and $A\subset\mathbb{Z}_{\geq 0}^{m}$ be as above. The image $\operatorname{ev}_{\cal S}(\cal L(A))\subset K^{|\cal S|}$ is called the monomial-Cartesian code associated with $\cal S$ and $A$. We denote it by $C(\cal S,A)$. By an abuse of notation, if $\bm{a}\in A$ then $C(\cal S,\bm{a})$ denotes the code $C(\cal S,\{\bm{a}\}).$

The monomial-Cartesian code has the following parameters (Proposition 2.1). Its length and dimension are given by $n=|\cal S|$ and $k=\dim_{K}C(\cal S,A)=|A|$, respectively. Recall that the minimum weight of a code $C$ is given by

[TABLE]

where $\operatorname{supp}(c)$ denotes the support of $c$, that is the set of all non-zero entries of $c$. Unlike the case of the length and the dimension, in general, there is no explicit formula for $\delta C(\cal S,A)$ in terms of $\cal S$ and $A$. For toric codes, some explicit formulas appear in [35] and non-trivial bounds appear in [34] when $m=2$. In the following proposition we mention a simple formula for the direct product of monomial-Cartesian codes. We consider them in Section 4.

Proposition 1.2**.**

Let $C(\cal S_{1},A)$ and $C(\cal S_{2},B)$ be monomial-Cartesian codes and consider their direct product $C(\cal S_{1}\times\cal S_{2},A\times B)$. Then

[TABLE]

Previous result can be proven doing a slight modification to the proof of [35, Theorem 2.1] or because [42, Theorem 3 c)]. There is also an inductive lower bound for $\delta\left(C(\cal S,A)\right)$ in terms minimum weights of monomial-Cartesian codes corresponding to projections and fibers of $A$ along coordinate subspaces. It is stated in [36, Theorem 4.1] in the case of generalized toric codes, but the statement and the proof can be easily adapted to arbitrary monomial-Cartesian codes.

The dual of the code $C$ is defined by

[TABLE]

where $\bm{w}\cdot\bm{c}$ represents the Euclidean inner product. The code $C$ is called a linear complementary dual (LCD) [30] if $C\cap C^{\perp}=\{\bm{0}\},$ and is called a self-orthogonal code if $C^{\perp}\subset C.$ In [10], Carlet, Mesnager, Tang, Qi, and Pellikaan show that any linear code over $\mathbb{F}_{q}$ with $q>3$ is equivalent to an LCD code; even so, explicit constructions can be elusive. In this paper, we provide a characterization for monomial-Cartesian codes which are LCD, thus providing explicit constructions of LCD codes.

Instances of monomial-Cartesian codes for particular families of lattice sets $A$ and Cartesian products $\cal S$ have been extensively studied in the literature. For example, a Reed-Muller code of order $r$ in the sense of [39, p. 37] is the monomial-Cartesian code $C(K^{m},A_{r}),$ where $A_{r}=\{(a_{1},\ldots,a_{m})\in\mathbb{Z}^{m}_{\geq 0}:a_{1}+\dots+a_{m}\leq r\}$. Note that in this case $\cal L(A_{r})=R_{\leq r}$, the set of all polynomials of degree at most $r$.

Another example of a monomial-Cartesian code is a toric code $C(\left(K^{*}\right)^{m},A_{P}),$ where $A_{P}=P\cap\mathbb{Z}^{m}$ is the set of lattice points of a convex lattice polytope $P\subset\mathbb{R}^{m}$ and $\left(K^{*}\right)^{m}$ is the Cartesian product with $S_{1}=\cdots=S_{m}=K^{*}.$ Good references for toric codes are [23, 25, 35].

An affine Cartesian code of order $r$ is a monomial-Cartesian code $C(\cal S,A_{r}),$ where $A_{r}$ is as above and $\cal S$ is an arbitrary Cartesian set. This family of codes appeared first time in [20] and then independently in [28]. In [20], the authors study the basic parameters of Cartesian codes, they determine optimal weights for the case when $A_{r}$ is the Cartesian product of two sets, and then present two list decoding algorithms. In [28] the authors study the vanishing ideal $I(\mathcal{S})$. Using commutative algebra tools such as regularity, degree, and Hilbert function, the authors determine the basic parameters of Cartesian codes in terms of the size of the components of the Cartesian product. In [11], the author shows some results on higher Hamming weights of Cartesian codes and gives a different proof for the minimum distance using the concepts of Gröbner basis and footprint of an ideal. In [12] the authors find several values for the second least weight of codewords, also known as the next-to-minimal Hamming weight. In [2] the authors find the generalized Hamming weights and the dual of Cartesian codes. In [27] the authors study the dual of a generalized Cartesian product and the property of being LCD, i.e., when the code and the dual have zero intersection.

Let $\cal S\subset K^{m}$ and $A\subset\mathbb{Z}_{\geq 0}^{m}$ be as above. In this work we are interested in the properties and applications of the monomial-Cartesian code $C(\cal S,A)$. In Section 2 we give a nice description of the dual of the code $C(\cal S,A)$ in terms of the complement of the set $A$ and the vanishing ideal of the set of points $\cal S.$ Our main theorem generalizes some results of [3, 19, 18] and [33], where the dual of toric codes, $J$-affine variety codes and generalized toric codes are studied. The representation for the dual gives rise to a Goppa representation for $C(\cal S,A)$, which may open the path for an efficiently decoding algorithm, because such representation is the key to decode the well-known Reed-Solomon codes. It is important to remark that there are decoding algorithms in the literature that can be used to decode particular cases of monomial-Cartesian codes, but the complexity is not as good as the one for the Reed-Solomon codes. For instance, the decoding algorithm developed by [17] depends of finding a Gröbner basis for each received codeword, and it would decode monomial-Cartesian codes on the case when $\cal S$ is arbitrary and $A\subset\mathbb{Z}_{\geq 0}^{m}$ are the smallest elements for a fixed monomial order in $Z_{\geq 0}^{m}.$ Excellent references about how to decode linear codes using Gröbner basis are [4, 5, 6] and [7].

The monomial-Cartesian code construction provides the flexibility needed for some applications, such as that of quantum error-correcting codes and locally recoverable codes. Quantum codes support resilience of quantum information by correcting bit and phase flip errors in qudits, quantum digits, which is fundamental to fault-tolerant quantum computation. While the goal of quantum codes is similar to that of linear codes, new techniques are needed for their construction due to the inability to duplicate quantum information. Even so, there is a link between quantum codes and classical linear codes, due to independent work of Calderbank and Shor [8] and Steane [37]. Indeed, the CSS construction uses linear codes which contain their duals to construct quantum codes. A family of codes called $J$-affine variety codes were introduced and studied in [19] and [18], respectively. This family of codes can be seen as monomial-Cartesian codes $C(\cal S,A)$ with the condition that $n_{i}-1$ divides $q-1.$ Inspired by those works, where the authors use $J$-affine variety codes to prove the existence of quantum error correcting codes we use monomial-Cartesian codes in Section 3 to prove the existence of quantum error correcting codes with certain parameters. An $[[n,k,d]]_{q}$ quantum code satisfies the quantum Singleton bound [26]

[TABLE]

If $k=n-2d+2$ then the quantum code is called quantum maximum-distance-separable (MDS) code. We obtain quantum MDS codes from monomial-Cartesian codes, making use of knowledge of the dual.

The idea of a locally recoverable code is that every coordinate depends of a few other coordinates. By “depends” we mean that if one of the coordinates is erased, then that coordinate can be recovered using some coordinates. Of course, it is desirable that “some” is small. The concept of $t$-availability means that for any coordinate there are $t$ pair disjoint subsets of a few coordinates each in such a way that the each subset can be used to recover such coordinate. Traditionally, for locality and availability it is assumed that the received coordinates are correct, but it may happens in practice that the received coordinates that are not erased contain also errors. Previous situation with errors gives rise to the codes known as locally recoverable codes with local error detection, which was introduced recently in [32]. Section 4 we study local properties for direct product of monomial-Cartesian codes.

More information about basic theory for coding theory can be found in [24, 29, 40]. More constructions of evaluation codes can be seen in [13, 14, 21, 31]. Excellent references for theory of vanishing ideals and its properties are [15, 16, 22, 41].

2. Dual of Monomial-Cartesian codes

Denote the variables $x_{1},\ldots,x_{m}$ by $\bm{x}.$ An important characteristic for monomial-Cartesian codes and evaluation codes in general is the fact that we can use commutative algebra methods to study them. The kernel of the evaluation map ${\rm ev}{S}$ is precisely $\cal{L}\left(A\right)\cap I(\mathcal{S})$, where $I(\mathcal{S})$ is the vanishing ideal of $\mathcal{S}$ consisting of all polynomials of $R$ that vanish on ${\mathcal{S}}$. Thus, algebraic properties of $R/\left(\cal{L}\left(A\right)\cap I(\mathcal{S})\right)$ are related to the basic parameters of $C(\mathcal{S},A)$. For each $i\in[m],$ define the polynomial

[TABLE]

The vanishing ideal of the Cartesian product $\mathcal{S}$ is given by $I(\mathcal{S})=\left(L_{1}(x_{1}),\ldots,L_{m}(x_{m})\right),$ [28, Lemma 2.3]. Moreover, let $\prec$ be the graded-lexicographic order on the set of monomials of $R.$ This order is defined in the following way: $x_{1}^{a_{1}}\cdots x_{m}^{a_{m}}\prec x_{1}^{b_{1}}\cdots x_{m}^{b_{m}}$ if and only if $\sum_{i=1}^{m}a_{i}<\sum_{i=1}^{m}b_{i}$ or $\sum_{i=1}^{m}a_{i}=\sum_{i=1}^{m}b_{i}$ and the leftmost nonzero entry in $(b_{1}-a_{1},\ldots,b_{m}-a_{m})$ is positive. From now on we fix the order $\prec$. Then, according to [15, Proposition 4], $\left\{L_{1}(x_{1}),\ldots,L_{m}(x_{m})\right\}$ is a Gröbner basis of $I(\mathcal{S})$, relative to the order $\prec$.

Proposition 2.1**.**

The dimension and the length of the monomial-Cartesian code $C(\mathcal{S},A)$ are given by $|A|$ and $|\cal{S}|,$ respectively.

Proof.

It is enough to show that the evaluation map ${\rm ev}_{\mathcal{S}}\colon\cal{L}\left(A\right)\ \to K^{|\mathcal{S}|}$ is injective. By above $\operatorname{Ker}({\rm ev}_{\mathcal{S}})=\cal{L}\left(A\right)\cap I(\mathcal{S})$. On one hand, by assumption $\deg_{x_{i}}(f)<n_{i}$ for every $f\in\cal L(A)$ and $i\in[m]$. On the other hand, $I(\mathcal{S})$ has a Gröbner basis $\left\{L_{1}(x_{1}),\ldots,L_{m}(x_{m})\right\}$ with $\deg_{x_{i}}(L_{i})=n_{i}$ for each $i\in[m]$. Therefore, $\cal{L}\left(A\right)\cap I(\mathcal{S})$ is trivial. ∎

Definition 2.2**.**

For $\bm{s}=\left(s_{1},\ldots,s_{m}\right)\in\mathcal{S}$ and $f\in R,$ define the residue of $f$ at $\bm{s}$ as

[TABLE]

For simplicity, we introduce the following notation for the residues vector

[TABLE]

Remark 2.3**.**

Note that $\operatorname{Res}_{\cal S}:R\to K^{|\cal S|}$ is a linear map which is injective on the subspace of polynomials $f$ satisfying $\deg_{x_{i}}(f)<n_{i}$. This follows from the definition of the residue and the proof of Proposition 2.1.

By [2, Theorem 5.7] or [27, Theorem 2.3], the dual of the monomial-Cartesian code $C(\mathcal{S},\bm{0})\subset K^{|\mathcal{S}|},$ where $\bm{0}$ is the zero vector in $\mathbb{Z}^{m}_{\geq 0},$ is given by

[TABLE]

This implies

[TABLE]

By the division algorithm there are polynomials $q_{i,j}$ and $r_{i,j}$ in $K[x_{i}]$ for $i\in[m]$, such that

[TABLE]

and $\deg(r_{i,j-1})<j$. For every $\bm{b}=\left(b_{1},\ldots,b_{m}\right)$ in $\mathbb{Z}^{m}_{\geq 0}$ define the polynomial

[TABLE]

These polynomials $Q_{\bm{b}}\in R$ help to describe the dual of a monomial-Cartesian code.

Lemma 2.4**.**

Let $B=\{0,\ldots,n_{1}-1\}\times\cdots\times\{0,\ldots,n_{m}-1\}.$ For any $\bm{a}\in B$, the set $\left\{\operatorname{Res}_{\cal S}Q_{\bm{b}}:\bm{b}\in B,\bm{b}\neq\bm{a}\right\}$ forms a basis for the dual $C(\mathcal{S},\bm{a})^{\perp}$ of the monomial-Cartesian code $C(\mathcal{S},\bm{a})$.

Proof.

By definition, $\deg_{x_{i}}(Q_{\bm{b}})=n_{i}-(b_{i}+1).$ This implies that the $Q_{\bm{b}}$ for $\bm{b}\in B$ have pairwise distinct multidegrees (with respect to the graded-lexicographic order). Thus the set $\{Q_{\bm{b}}:\bm{b}\in B,\bm{b}\neq\bm{a}\}$ is linearly independent. Furthermore, by Remark 2.3, its image under the residue map $\left\{\operatorname{Res}_{\cal S}Q_{\bm{b}}:\bm{b}\in B,\bm{b}\neq\bm{a}\right\}$ spans a subspace of dimension $\sum_{i=1}^{m}(n_{i}-1)-1=\dim C(\bm{a},\mathcal{S})^{\perp}$.

Now we check the inner product. Let $\overline{f}$ denote the normal form of $f$ with respect to the Gröbner basis $\left\{L_{1}(x_{1}),\ldots,L_{m}(x_{m})\right\}$. Note that $\operatorname{Res}_{\bm{s}_{i}}f=\operatorname{Res}_{\bm{s}_{i}}\overline{f}$ for any $i\in[n]$ and $f\in R$. Therefore,

[TABLE]

It remained to show that $\sum_{i=1}^{n}\operatorname{Res}_{\bm{s}_{i}}\overline{\bm{x}^{\bm{a}}Q_{\bm{b}}}=0$. For this we check the conditions in Equation (2.3). We have

[TABLE]

Indeed, the first inequality is clear. For the second one, when $a_{i}>b_{i}$ the devision algorithm and Equation 2.4 provide

[TABLE]

Now, since $\bm{a}\neq\bm{b}$, there is $j\in[m]$ such that $a_{j}\neq b_{j}.$ Then (2.6) implies

[TABLE]

Also, (2.6) provides \deg_{x_{i}}\big{(}\overline{\bm{x}^{\bm{a}}Q_{\bm{b}}(\bm{x})}\big{)}<n_{i} for all $i\in[m]\setminus\{j\}$. Therefore, both conditions of (2.3) are satisfied which shows that $\sum_{i=1}^{n}\operatorname{Res}_{\bm{s}_{i}}\overline{\bm{x}^{\bm{a}}Q_{\bm{b}}}=0$. ∎

Example 2.5**.**

Let $K=\mathbb{F}_{7}$ and assume $\cal S=\{1,3,4,5\}\subset K.$ On this case $L_{1}(x_{1})=(x_{1}-1)(x_{1}-3)(x_{1}-4)(x_{1}-5)$ and

[TABLE]

Then we have the following duals of $C(\mathcal{S},a)$ for $a\in A$.

[TABLE]

Example 2.6**.**

Let $K=\mathbb{F}_{7}$. Consider the following Cartesian set: $\cal S=\{0,2,3\}\times\{0,1,3,5,6\}\subset K^{2}.$ On this case $L_{1}(x_{1})=x_{1}(x_{1}-2)(x_{1}-3)$ and $L_{2}(x_{2})=x_{2}(x_{2}-1)(x_{2}-3)(x_{2}-5)(x_{2}-6)$. Then we have

[TABLE]

Then, the dual of $C(\mathcal{S},\bm{a})$ for $\bm{a}=(2,3)$ is given by

[TABLE]

In other words, we take the residue of all the products $q_{1,i}q_{2,j}$ except when $(i,j)$ is the given point $(2,3).$

Theorem 2.7**.**

Let $\cal S=S_{1}\times\cdots\times S_{m}\subseteq K^{m}$ and $B=\{0,\ldots,n_{1}-1\}\times\cdots\times\{0,\ldots,n_{m}-1\}\subset\mathbb{Z}^{m}$. For any $A\subseteq B$, the set $\left\{\operatorname{Res}_{\cal S}Q_{\bm{b}}:\bm{b}\in B\setminus A\right\}$ forms a basis for the dual $C(\mathcal{S},A)^{\perp}$ of the monomial-Cartesian code $C(\mathcal{S},A).$

Proof.

As for any two points $\bm{a}_{1},\bm{a}_{2}\in A$ we have that $C(\mathcal{S},\{\bm{a}_{1},\bm{a}_{2}\})^{\perp}=C(\mathcal{S},\bm{a}_{1})^{\perp}\cap C(\mathcal{S},\bm{a}_{2})^{\perp},$ the result is a consequence of Lemma 2.4. ∎

Example 2.8**.**

Let $K=\mathbb{F}_{7}$ and assume $\cal S=\{1,3,4,5\}\subset K$ as in Example 2.5. As before we have $L_{1}(x_{1})=(x_{1}-1)(x_{1}-3)(x_{1}-4)(x_{1}-5)$ and

[TABLE]

Then we obtain the following dual codes. $C(\mathcal{S},\{2,3\})^{\perp}=\text{Span}_{K}\left\{\operatorname{Res}_{\cal S}q_{0},\operatorname{Res}_{\cal S}q_{1}\right\},$

[TABLE]

Example 2.9**.**

Let $K=\mathbb{F}_{7}$. Consider the following Cartesian set: $\cal S=\{0,2,3\}\times\{0,1,3,5,6\}\subset K^{2}.$ On this case $L_{1}(x_{1})=x_{1}(x_{1}-2)(x_{1}-3)$ and $L_{2}(x_{2})=x_{2}(x_{2}-1)(x_{2}-3)(x_{2}-5)(x_{2}-6)$. We have

[TABLE]

Then, the dual of the code $C(\mathcal{S},\{(0,1),(3,5)\})$ is given by

[TABLE]

In other words, we take the residue of all the products $q_{1,i}q_{2,j}$ except when $(i,j)$ is either $(0,1)$ or $(3,5).$

3. Quantum error correcting codes

In this section, we give some applications of Monomial-Cartesian codes to quantum error correcting codes. Our main result shows how to use monomial-Cartesian codes to find quantum error correction codes and MDS quantum error correction codes. We continue using same notation than previous sections, in particular $\cal S=S_{1}\times\cdots\times S_{m}\subseteq K^{m},$ $n_{i}:=|S_{i}|$, $A\subset\{0,\ldots,n_{1}-1\}\times\cdots\times\{0,\ldots,n_{m}-1\},$ and $\cal L(A)=\operatorname{Span}_{K}\{\bm{x}^{\bm{a}}:\bm{a}\in A\}\subset R.$

First, we provide a slightly different representation for the dual of a monomial-Cartesian code.

Definition 3.1**.**

Let $F(\bm{x})$ be the unique element in $R$ such that $\deg_{x_{i}}F(\bm{x})<n_{i}$ and $F(\bm{s})=\left(\prod_{i=1}^{m}\prod_{\begin{subarray}{c}s_{i}^{\prime}\in S_{i}\setminus\{s_{i}\}\end{subarray}}\left(s_{i}-s_{i}^{\prime}\right)\right)^{-1}$ for every $\bm{s}=\left(s_{1},\ldots,s_{m}\right)\in\mathcal{S}.$

Observe that the polynomial $F(\bm{x})$ can be found using interpolation:

[TABLE]

Theorem 3.2**.**

Let $\cal S=S_{1}\times\cdots\times S_{m}\subseteq K^{m}$ and $B=\{0,\ldots,n_{1}-1\}\times\cdots\times\{0,\ldots,n_{m}-1\}\subset\mathbb{Z}^{m}$. Let $F(\bm{x})$ be as defined in Definition 3.1. For any $A\subseteq B$, the set $\left\{\operatorname{ev}_{\cal S}(FQ_{\bm{b}}):\bm{b}\in B\setminus A\right\}$ forms a basis for the dual $C(\mathcal{S},A)^{\perp}$ of the monomial-Cartesian code $C(\mathcal{S},A).$

Proof.

Because the definition of $F(\bm{x})$ and $\operatorname{Res}_{\cal S}Q_{\bm{b}},$ it is clear that $\operatorname{Res}_{\cal S}Q_{\bm{b}}=\operatorname{ev}_{\cal S}(FQ_{\bm{b}}).$ ∎

Lemma 3.3**.**

Let $f_{1},\ldots,f_{k},g_{1},\ldots,g_{\ell}\in\cal L(A).$ Then $\operatorname{Span}_{K}\{\operatorname{ev}_{\cal S}(f_{1}),\ldots,\operatorname{ev}_{\cal S}(f_{k})\}\subset\operatorname{Span}_{K}\{\operatorname{ev}_{\cal S}(g_{1}),\ldots,\operatorname{ev}_{\cal S}(g_{\ell})\}$ if and only if $\operatorname{Span}_{K}\{f_{1},\ldots,f_{k}\}\subset\operatorname{Span}_{K}\{g_{1},\ldots,g_{\ell}\}.$

Proof.

This is a consequence to the fact that the evaluation function $\operatorname{ev}_{\cal S}$ is injective. ∎

Using previous result we can give conditions to determine if a monomial-Cartesian code is self-orthogonal or LCD. An important application for LCD codes can be found in [9].

Theorem 3.4**.**

*Let $\cal S=S_{1}\times\cdots\times S_{m}\subseteq K^{m}$ and $A\subseteq B=\{0,\ldots,n_{1}-1\}\times\cdots\times\{0,\ldots,n_{m}-1\}\subset\mathbb{Z}^{m}$. Let $F(\bm{x})$ be as defined in Definition 3.1. Then

(a) $C(\mathcal{S},A)^{\perp}\subset C(\mathcal{S},A)$ if and only if $\operatorname{Span}_{K}\left\{\overline{FQ_{\bm{b}}}:\bm{b}\in B\setminus A\right\}\subset\operatorname{Span}_{K}\left\{\bm{x}^{\bm{a}}:\bm{a}\in A\right\}.$ (b) $C(\mathcal{S},A)$ is LCD if and only if $\operatorname{Span}_{K}\left\{\overline{FQ_{\bm{b}}}:\bm{b}\in B\setminus A\right\}\bigcap\operatorname{Span}_{K}\left\{\bm{x}^{\bm{a}}:\bm{a}\in A\right\}=0.$ Where $\overline{FQ_{\bm{b}}}$ denotes the normal form of the polynomial $FQ_{\bm{b}}$ with respect to the Gröbner basis $\left\{L_{1}(x_{1}),\ldots,L_{m}(x_{m})\right\}.$*

Proof.

The result is a consequence of Lemma 3.3 and Theorem 3.2. ∎

Next, we describe some properties for the polynomial $F(\bm{x})$ in order to find conditions that satisfy part (a) from Theorem 3.4.

Proposition 3.5**.**

If $q>n_{i}\geq q/2$ for all $i\in[m],$ then $\deg_{x_{i}}(F(\bm{x}))\leq q-n_{i}.$

Proof.

Define

[TABLE]

Observe that if $\bm{s}=\left(s_{1},\ldots,s_{m}\right)\in\mathcal{S},$ then

[TABLE]

Last equality is true because for every $i\in[m]$ we have $-1=\prod_{\begin{subarray}{c}s_{i}^{\prime}\in K\setminus\{s_{i}\}\end{subarray}}\left(s_{i}-s_{i}^{\prime}\right).$ If $n_{i}>q/2,$ then $\deg_{x_{i}}F^{\prime}(\bm{x})=q-n_{i}<n_{i}.$ Thus $F(\bm{x})=F^{\prime}(\bm{x}),$ because $F(\bm{x})-F^{\prime}(\bm{x})\in I(\mathcal{S}).$ If $n_{i}=q/2,$ then defining $F$ by interpolation we get $\deg_{x_{i}}F<n_{i}=q-n_{i}.$ ∎

The following theorems gives a path to construct quantum and MDS quantum codes.

Theorem 3.6**.**

Let $\cal S=S_{1}\times\cdots\times S_{m}\subseteq K^{m}$ such that $q>n_{i}=|S_{i}|\geq q/2$ for all $i\in[m].$ For every $\bm{t}=(t_{1},\ldots,t_{m})\in\{0,\ldots,n_{1}-\lceil{\frac{q}{2}}\rceil\}\times\cdots\times\{0,\ldots,n_{m}-\lceil{\frac{q}{2}}\rceil\}\subset\mathbb{Z}^{m}$ define the set $A_{\bm{t}}=\{0,\ldots,n_{1}-1-t_{1}\}\times\cdots\times\{0,\ldots,n_{m}-1-t_{m}\},$ then $C(\mathcal{S},A_{\bm{t}})^{\perp}\subset C(\mathcal{S},A_{\bm{t}}).$

Proof.

Define $B=\{0,\ldots,n_{1}-1\}\times\cdots\times\{0,\ldots,n_{m}-1\}\subset\mathbb{Z}^{m}$ and take $\bm{b}\in B\setminus A_{\bm{t}}.$ By Theorem 3.4 (a) we just need to check that $\overline{FQ_{\bm{b}}}\in\operatorname{Span}_{K}\left\{\bm{x}^{\bm{a}}:\bm{a}\in A_{\bm{t}}\right\}.$ By definition $Q_{\bm{b}}(\bm{x})=\prod_{i=1}^{m}q_{i,b_{i}}(x_{i}),$ where $L_{i}(x_{i})=x_{i}^{b_{i}+1}q_{i,b_{i}}(x_{i})+r_{i,b_{i}}(x_{i}).$ It means $q_{i,b_{i}}(x_{i})\in\operatorname{Span}_{K}\left\{1,\ldots,x_{i}^{n_{i}-b_{i}-2}\right\}.$ As $\bm{b}\in B\setminus A_{\bm{t}},$ then $n_{i}-1-t_{i}<b_{i},$ thus $n_{i}-b_{i}-2<t_{i}-1.$ We obtain $\deg q_{i,b_{i}}(x_{i})<t_{i}-1.$ By Proposition 3.5 $\deg_{x_{i}}(F(\bm{x}))\leq q-n_{i}.$ Thus $\deg_{x_{i}}FQ_{\bm{b}}<q-n_{i}+t_{i}-1\leq n_{i}-1-t_{i}.$ Last inequality holds because $t_{i}\leq n_{i}-\lceil{\frac{q}{2}}\rceil$ and implies that $\overline{FQ_{\bm{b}}}=FQ_{\bm{b}}\in\operatorname{Span}_{K}\left\{\bm{x}^{\bm{a}}:\bm{a}\in A_{\bm{t}}\right\}.$ ∎

Now we state an important result to construct stabilizer codes.

Lemma 3.7**.**

[1, Lemma 17]** If there exists a classical linear $[n,k,d]_{q}$ code $C$ such that $C^{\perp}\subset C$, then there exists an $[[n,2k-n,d]]_{q}$ stabilizer code that is pure to $d.$ If the minimum distance of $C^{\perp}$ exceeds $d,$ then the stabilizer code is pure and has minimum distance $d.$

Theorem 3.8**.**

Let $\cal S=S_{1}\times\cdots\times S_{m}\subseteq K^{m}$ such that $q>n_{i}=|S_{i}|\geq q/2$ for all $i\in[m].$ For every $\bm{t}=(t_{1},\ldots,t_{m})\in\{0,\ldots,n_{1}-\lceil{\frac{q}{2}}\rceil\}\times\cdots\times\{0,\ldots,n_{m}-\lceil{\frac{q}{2}}\rceil\}\subset\mathbb{Z}^{m}$ there exists an $[[\prod_{i=1}^{m}n_{i},2\prod_{i=1}^{m}(n_{i}-t_{i})-n,\prod_{i=1}^{m}(t_{i}+1)]]_{q}$ stabilizer code that is pure to $t_{1}\cdots t_{m}.$

Proof.

The idea is to apply Lemma 3.7 to Theorem 3.6. By Theorem 3.6 we have that for $A_{\bm{t}}=\{0,\ldots,n_{1}-1-t_{1}\}\times\cdots\times\{0,\ldots,n_{m}-1-t_{m}\},$ $C(\mathcal{S},A_{\bm{t}})^{\perp}\subset C(\mathcal{S},A_{\bm{t}}).$ It is clear that the length and dimension of $C(\mathcal{S},A_{\bm{t}})$ are given by $n$ and $\prod_{i=1}^{m}(n_{i}-t_{i}),$ respectively. Finally, the minimum distance comes from Proposition 1.2. ∎

Previous result gives a very simple path to prove the existence of quantum error correcting codes with certain parameters.

Example 3.9**.**

Let $K=\mathbb{F}_{49}$ and take $n_{1}=35,n_{2}=40,t_{1}=5$ and $t_{2}=8.$ By Theorem 3.8 we have that there exist the following quantum error correcting codes: $\displaystyle[[35,25,6]]_{49}$, $\displaystyle[[40,24,9]]_{49}$ and $\displaystyle[[1400,320,54]]_{49}.$

Observe that the first two of the previous examples are quantum MDS codes. Actually it is possible to prove the existence of more of them.

Corollary 3.10**.**

For every $q>n\geq q/2$ and every $0\leq t\leq n-\lceil{\frac{q}{2}}\rceil$ there exists an MDS quantum code $[[n,n-2t,t+1]]_{q}.$

Proof.

This is the particular case of Theorem 3.8 when $m=1.$ ∎

4. Local properties of direct products

Local properties for linear codes have been studied extensively in the context of distributed storage. The idea is that every coordinate of a linear code can be used to save the information of a server, so $n$ servers store a linear code of length $n.$ Informally speaking, a linear code is said to have locality $r$ if for all elements of the code, every coordinate $i$ is a function of other $r$ coordinates. It is important to remark that the set of these $r$ coordinates depend of $i,$ but not of the codeword. In terms of distributed storage, locality $r$ means that if one of the $n$ servers fails, then the information of the failed server can be recovered by accessing $r$ other servers (rather than $n-1$). If one of these $r$ servers also fails, local recovery might not be possible. For that reason it is useful to have availability. A linear code with availability $t$ means that every coordinate can be recovered from $t$ pairwise disjoint sets. Formal definitions follow.

Definition 4.1**.**

A linear code $C$ of length $n$ over $K$ is a locally recoverable code with locality $r$ if for every position $i\in[n]$ there exist a set $\mathcal{R}_{i}\subseteq[n]\setminus\{i\}$ and a function $\phi_{i}:K^{r}\to K$ such that $|\mathcal{R}_{i}|=r$ and for all $c=(c_{1},\ldots,c_{n})$ in $C,$ $c_{i}=\phi_{i}(c\mid_{\mathcal{R}_{i}}).$ This definition represents that every coordinate $c_{i}$ for any codeword $c$ can be recovered by the coordinates $c_{j},$ where $j\in\mathcal{R}_{i}.$ The set $\mathcal{R}_{i}$ is called a recovery set for the $i$-th position.

Definition 4.2**.**

A linear code $C$ is said to have $t$-availability with locality $\left(r_{1},\ldots,r_{t}\right)$ if every position $i\in[n]$ has $t$ pairwise disjoint recovery sets $\mathcal{R}_{i1},\ldots,\mathcal{R}_{it}$ with $|\mathcal{R}_{ij}|=r_{j},$ for $j\in[t].$

Lemma 4.3**.**

Let $C(\cal S_{1},A)$ and $C(\cal S_{2},B)$ be locally recoverable monomial-Cartesian codes with localities $r_{1}$ and $r_{2},$ respectively. The direct product $C(\cal S_{1}\times\cal S_{2},A\times B)$ has $2$-availability with locality $(r_{1},r_{2}).$

Proof.

Observe that the coordinates of a monomial-Cartesian code are indexed by the elements of the Cartesian product, for this reason every position will be given in terms of the elements of the Cartesian product. Let $\bm{s}_{1}$ and $\bm{s}_{2}$ be elements of $\cal S_{1}$ and $\cal S_{2},$ respectively. Let $\mathcal{R}_{\bm{s}_{1}}$ be a recovery set for $\bm{s}_{1}$ of cardinality $r_{1}$ and $\mathcal{R}_{\bm{s}_{2}}$ a recovery set for $\bm{s}_{2}$ of cardinality $r_{2},$ which exist because $C(\cal S_{1},A)$ and $C(\cal S_{2},B)$ are locally recoverable monomial-Cartesian codes with locality $r_{1}$ and $r_{2},$ respectively. In the code $C(\cal S_{1}\times\cal S_{2},A\times B),$ we claim the position $(\bm{s}_{1},\bm{s}_{2})$ has recovery sets $\mathcal{R}_{\bm{s}_{1}}\times\{\bm{s}_{2}\}$ and $\{\bm{s}_{1}\}\times\mathcal{R}_{\bm{s}_{2}}.$

Let $c$ be an element of $C(\cal S_{1}\times\cal S_{2},A\times B).$ By definition of the direct product, there is a polynomial $f(\bm{x},\bm{y})\in\cal L(A\times B)\subset K[x_{1},\dots,x_{m_{1}},y_{1},\dots,y_{m_{2}}]$ such that $c=\left(f\left(\bm{s},\bm{s}^{\prime}\right)\right)|_{\left(\bm{s},\bm{s}^{\prime}\right)\in\cal S_{1}\times\cal S_{2}}.$ As $f(\bm{x},\bm{s}_{2})\in\cal L(A)\subset K[x_{1},\dots,x_{m_{1}}],$ we can use the set $\{f(\bm{s},\bm{s}_{2})\mid\bm{s}\in\mathcal{R}_{\bm{s}_{1}}\}$ to recover the value $f(\bm{s}_{1},\bm{s}_{2}).$ Thus $\mathcal{R}_{\bm{s}_{1}}\times\{\bm{s}_{2}\}$ is a recovery set for $(\bm{s}_{1},\bm{s}_{2}).$ In analogous way, $\{\bm{s}_{1}\}\times\mathcal{R}_{\bm{s}_{2}}$ is a second recovery set for the same position $(\bm{s}_{1},\bm{s}_{2}).$ ∎

We come to the main result of this section, which shows how locally recoverable monomial-Cartesian codes give rise to codes with availability.

Theorem 4.4**.**

Let $C(\cal S_{1},A_{1}),\ldots,C(\cal S_{t},A_{t})$ be locally recoverable monomial-Cartesian codes with localities $r_{1},\ldots,r_{t},$ respectively. The direct product $C(\cal S_{1}\times\cdots\times\cal S_{t},A_{1}\times\cdots\times A_{t})$ has $t$-availability with locality $(r_{1},\ldots,r_{t}).$

Proof.

This is a consequence of Lemma 4.3 because the product of two monomial-Cartesian codes is again a monomial-Cartesian code. ∎

Remark 4.5**.**

As a corollary of Theorem 4.4 we obtain the family of codes obtained on [38, Construction 4], which are direct products of sub-codes of Reed-Solomon codes.