Generalized Monomial Cartesian Codes

• Generalized Monomial Cartesian Codes are a class of multivariate evaluation codes that assess prescribed monomials on Cartesian products of finite field subsets, unifying classical, quantum, and locally recoverable codes.
• They use twist vectors and Gröbner basis techniques to systematically control key parameters like length, dimension, and minimum distance.
• Their versatile framework enables efficient designs for quantum stabilizer and locally recoverable codes, achieving bounds near optimal Singleton and Gilbert–Varshamov limits.

A Generalized Monomial Cartesian Code (GMCC) is a class of multivariate evaluation codes defined by evaluating a prescribed set of monomials on a Cartesian product of subsets of a finite field, optionally equipped with a weight vector or “twist”. GMCCs are direct generalizations of both Generalized Reed-Solomon (GRS) codes and classical affine Cartesian codes, and serve as a unifying framework for constructing classical, quantum, and locally recoverable codes with rich structural and asymptotic properties (Campion et al., 18 Dec 2025, Barbero-Lucas et al., 2023, López et al., 2019, Galindo et al., 2022).

1. Formal Definition and Construction

Let $\mathbb{F}_{q^2}$ be a finite field. Fix nonempty subsets $$P_X = \{x_1, \ldots, x_{n_X}\} \subset \mathbb{F}_{q^2}$$ and $$P_Y = \{y_1, \ldots, y_{n_Y}\} \subset \mathbb{F}_{q^2}$$; form the Cartesian product $S = P_X \times P_Y$ with $n=|S|=n_X n_Y$. For a finite set $M \subset \mathbb{N}^2$ of exponent pairs, define the degree-filtered polynomial space

$$\mathbb{F}_{q^2}[X,Y]_M = \left\{ f(X,Y) = \sum_{(a,b)\in M} c_{a,b} X^a Y^b \right\}$$

and select a weight array $Q = (Q_{i,j})$ with $Q_{i,j} \in \mathbb{F}_{q^2}^*$. The GMCC is the code

$$C_M(S; Q) = \left\{ (Q_{i,j} f(x_i, y_j))_{1\leq i \leq n_X, 1 \leq j \leq n_Y} : f \in \mathbb{F}_{q^2}[X,Y]_M \right\}.$$

The generator matrix has as its rows the evaluation vectors of the monomials $X^a Y^b$ for $(a,b) \in M$, optionally post-multiplied by the twist $Q$ (Campion et al., 18 Dec 2025, Barbero-Lucas et al., 2023).

This construction admits generalization to $m$ variables, arbitrary coordinate sets, and different choices of vanishing ideals; it encompasses families such as Reed-Muller, affine Cartesian, and toric codes (Barbero-Lucas et al., 2023, López et al., 2019).

2. Fundamental Parameters and Footprint Bounds

For $m=2$, $|S| = n_X n_Y$ and the code dimension is $k = |M|$ under the injectivity of the evaluation map, which is guaranteed for $M \subseteq [0,n_X-1] \times [0,n_Y-1]$ due to the monomials forming a vector space basis over the support.

The minimum Hamming distance $d$ satisfies the “footprint bound”: $$d \geq \delta_{FB}(M; P_X, P_Y) = \min_{(a,b)\in M} (n_X - a)(n_Y - b).$$ If $M$ has hyperbolic shape, i.e., $M = \{(a,b): (a+1)(b+1) < t\}$, then equality holds: $d = \delta_{FB}(M; P_X, P_Y).$ The footprint bound can be established through Gröbner basis techniques on the vanishing ideal of $S$, leveraging standard monomial theory (Campion et al., 18 Dec 2025, Barbero-Lucas et al., 2023, López et al., 2019, Galindo et al., 2022).

For $m \geq 1$ variables, and arbitrary Cartesian sets $$A = A_1 \times \cdots \times A_m \subset \mathbb{F}_q^m$$, similar bounds on length, dimension, and minimum distance can be derived. In particular, for $e = (e_1, \dots, e_m) \in M$,

$d \geq \min_{e\in M} \prod_{j=1}^m (n_j - e_j).$

For code parameters, the following table summarizes the principal invariants:

Parameter	Notation	Value/Expression
Length	$n$	$\prod_{j=1}^m n_j$
Dimension	$k$	$|\Delta|$
Min. dist.	$d$	$\min_{e \in \Delta} \prod_{j=1}^m (n_j - e_j)$

3. Duality, Hermitian Self-Orthogonality, and Quantum Codes

GMCCs provide a systematic platform for constructing self-orthogonal codes under both Euclidean and Hermitian inner products, crucial for quantum stabilizer code constructions. Let $\langle x, y \rangle_H = \sum_k x_k y_k^q$ denote the Hermitian inner product on $\mathbb{F}_{q^2}^n$.

GMCCs constructed via twist vectors or from two GRS codes admit a tensor-factorized Hermitian inner product: $$\langle ev_{Q,S}(X^{e_1}Y^{e_2}), ev_{Q,S}(X^{e'_1}Y^{e'_2}) \rangle_H = \langle ev_X(X^{e_1}), ev_X(X^{e'_1}) \rangle_H \cdot \langle ev_Y(Y^{e_2}), ev_Y(Y^{e'_2}) \rangle_H.$$ Sufficient conditions for Hermitian self-orthogonality can be stated in terms of the exponents’ combinatorial relations (modulo code parameters), as in propositions referencing congruence conditions and dangerous “failure sets” of exponent pairs (Campion et al., 18 Dec 2025, Barbero-Lucas et al., 2023).

From the Hermitian self-orthogonal GMCCs, quantum stabilizer codes can be constructed by the Hermitian construction theorem: if $C \subseteq \mathbb{F}_{q^2}^n$ is $[n,k,d]_{q^2}$ and $C \subseteq C^{\perp_H}$, then there exists a quantum code $[[n, n-2k, \geq d]]_q$ (Campion et al., 18 Dec 2025, Barbero-Lucas et al., 2023).

In favorable parameter regimes, these quantum codes can achieve or exceed known Singleton and Gilbert–Varshamov bounds, and in specific cases yield quantum MDS or Hermitian Almost MDS codes.

4. Broader Framework: Connections to Monomial Cartesian and Related Codes

GMCCs encompass and generalize numerous code classes:

• Affine Cartesian codes: GMCCs with $M$ a box, $A_j = [0,n_j-1]$, yield classical affine Cartesian and Reed-Muller codes (López et al., 2019);
• Toric codes: With $A_j = \mathbb{F}_q^*$, and $M$ corresponding to lattice points in a polytope, these are projective toric codes (López et al., 2019);
• Locally recoverable codes (LRCs): By leveraging the Cartesian structure and code support, GMCCs exhibit natural $(r, \delta)$-locality and are used to construct optimal LRCs and their subfield-subcodes (Galindo et al., 2022);
• Generalized Hamming weights: GMCCs support combinatorial and algebraic techniques for determining higher weights, with specific bounds and explicit formulas available for classes such as affine and nested Cartesian codes (Gonzalez-Sarabia et al., 2017).

5. Applications and Explicit Constructions

GMCCs provide explicit, uniform families of codes with distinguished application areas:

• Quantum error correction: Hermitian self-orthogonal GMCCs, via twist-vector or GRS tensor constructions, generate $[[n, n-2k, d]]_q$ quantum codes—MDS, Almost MDS, or exceeding the Gilbert–Varshamov bound in infinite families (Campion et al., 18 Dec 2025, Barbero-Lucas et al., 2023).
• Locally recoverable codes: Choosing coordinate sets and exponent supports appropriately, GMCCs yield $(r, \delta)$-optimal LRCs, and their subfield-subcodes remain optimal while descending to smaller fields (Galindo et al., 2022).
• Parameter trade-offs: The flexibility in $M$, support sets, and weight/twist design allows optimization of the length, dimension, and minimum distance, drawing upon combinatorial, algebraic, and algorithmic tools (Campion et al., 18 Dec 2025, Barbero-Lucas et al., 2023, López et al., 2019).

Explicit constructions include, for example, quantum MDS codes by taking $m=1$, $a_1 = \lambda(q+1)$, and $\Delta_t = \{0, 1, \ldots, t-2\}$, leading to $[[\lambda(q+1), \lambda(q+1)-2(t-1), \geq t]]_q$ codes (Barbero-Lucas et al., 2023), and bivariate codes with parameters surpassing prior families (Barbero-Lucas et al., 2023, Campion et al., 18 Dec 2025).

6. Algebraic, Combinatorial, and Gröbner Techniques

Computation and structural analysis of GMCCs rely on:

• Vanishing ideals and Gröbner bases: The ideal-theoretic viewpoint is central for describing duals, deriving dimension and minimum distance, and analyzing generalized Hamming weights;
• Footprint methods: The combinatorial footprint function provides tractable lower bounds on distances and higher weights, with algorithms adapted from commutative algebra;
• Integer programming: Lattice-point and integer-linear programming arguments enter into determining the largest sets $M$ supporting self-orthogonality while maintaining other parameters (e.g., $T^*$ bounds for Hermitian codes) (Campion et al., 18 Dec 2025).

7. Generalizations, Open Problems, and Future Directions

Ongoing research directions include:

• Classifications of all exponent sets $M$ supporting Hermitian or Euclidean self-orthogonality beyond hyperbolic shapes;
• Extension of explicit twist/construction methods to higher variable numbers or to other classes of algebraic-geometric or affine-variety codes;
• Optimization of code rate, length, and distance, especially in the quantum regime, including construction of further families violating classical bounds (Campion et al., 18 Dec 2025, Barbero-Lucas et al., 2023);
• Development of efficient algorithms for computing higher generalized Hamming weights and for establishing minimal support properties for LRC or quantum settings;
• Exploration of further connections with locally recoverable codes, LCD codes, and toric codes (López et al., 2019, Galindo et al., 2022).

A plausible implication is that GMCCs serve as a robust and unifying tool for explicit code constructions across classical and quantum error correction, supporting both structural theoretical advances and concrete practical code designs.