Counting contiguous superregular $4 \times 4$ matrices

Abstract: This short paper has two goals. First, explaining a simple procedure (which is essentially folklore) that, sometimes, makes it possible to obtain a formula for the number of solutions to a system of multivariate polynomial inequalities over a finite field. Second, applying that procedure to prove a formula for the number of contiguous superregular $4 \times 4$ matrices over a finite field. The formula was previously conjectured by Appuswamy, Bazzani, Connelly, Ekaireb, Congero, and Zeger [Probability of super-regular matrices and MDS codes over finite fields, arXiv:2603.20983]. In addition, the same procedure is used to provide formulas for the number of contiguous superregular $3 \times 4$, $3 \times 5$, and $3 \times 6$ matrices over a finite field.

• The paper presents a rigorous derivation of an explicit counting formula for contiguous superregular 4×4 matrices over finite fields.
• It employs a recursive symbolic computation method with linear substitutions and inclusion-exclusion to solve multivariate polynomial inequalities.
• The results strengthen the connection between algebraic combinatorics and coding theory by confirming conjectural enumerations and extending to rectangular cases.

Counting Contiguous Superregular $4 \times 4$ Matrices: Formal Enumeration Over Finite Fields

Introduction and Context

This essay addresses the enumeration of contiguous superregular $4 \times 4$ matrices over finite fields, a problem closely intertwined with the structure of MDS (Maximum Distance Separable) codes and the algebraic theory of polynomial systems over $\mathbb{F}_q$. The focus is on explicit counting formulas for such matrices, leveraging recursive symbolic computation and polynomial inequalities, and providing rigorous confirmation of a previously conjectured formula for $C_4^{\text{CSR}}(q)$—the number of contiguous superregular $4 \times 4$ matrices over $\mathbb{F}_q$—as conjectured in "Probability of super-regular matrices and MDS codes over finite fields" (Appuswamy et al., 22 Mar 2026).

Superregular and Contiguous Superregular Matrices

Let a contiguous superregular $k \times k$ matrix over $\mathbb{F}_q$ be defined by invertibility of all square contiguous submatrices. Such matrices are foundational in coding theory, having direct relevance to generator matrices of MDS codes. The enumeration problem reduces to counting the solutions to certain systems of multivariate polynomial inequalities over finite fields, an approach that, while conceptually simple, is nontrivial in execution due to the combinatorial complexity and algebraic structure imposed.

Enumeration via Polynomial Inequalities and Recursive Substitution

The core computational approach is a recursive procedure, employing linear substitutions and inclusion-exclusion to reduce the dimension and degree of systems of polynomial inequalities over $\mathbb{F}_q$. Fundamental steps include:

• Recursive Linear Substitution: Identification of variables for which a polynomial is linear, enabling reduction of the system via substitution and recursive application of inclusion-exclusion.
• Splitting Into Irreducible Factors: Factoring polynomials to handle characteristics of $\mathbb{F}_q$ and further simplify the inequalities.
• Base Cases: Handling constant and univariate polynomial factors directly, providing termination for the recursion in tractable cases.

The output is characterized by dependence on $4 \times 4$0, certain prime characteristics of the field, and counts of roots of auxiliary polynomials, encapsulated in a polynomial $4 \times 4$1 that computes the solution count for any prime power $4 \times 4$2.

Main Result: Explicit Formula for $4 \times 4$3

The main achievement is the rigorous derivation of the formula

$4 \times 4$4

which matches the conjecture in (Appuswamy et al., 22 Mar 2026). The method involves normalizing the matrix to reduce degrees and symmetries, rewriting the conditions into explicit polynomial inequalities, and employing automated symbolic computation to confirm the formula. Notably, even after field-specific adjustments (such as row and column scaling to fix entries), the complexity of the inequalities persists until the variable transformation, which facilitates tractable enumeration.

Extension: Contiguous Superregular $4 \times 4$5 Matrices

By applying the same methodology, explicit formulas are provided for contiguous superregular $4 \times 4$6, $4 \times 4$7, and $4 \times 4$8 matrices:

• $4 \times 4$9
• $\mathbb{F}_q$0
• $\mathbb{F}_q$1

Each formula results from systematizing the polynomial inequalities arising from contiguous invertibility constraints and normalizing the matrix entries for computational tractability.

Numerical Strength and Theoretical Implications

The formula for $\mathbb{F}_q$2 exhibits polynomial growth but also substantial combinatorial suppression for small $\mathbb{F}_q$3, consistent with the scarcity of superregular structures over small fields. The results solidify the link between algebraic combinatorics and coding theory, confirming the conjectural enumeration and enabling precise analysis of the probability of random matrices being superregular as a function of $\mathbb{F}_q$4.

The methodology provides a template for rigorously enumerating structures governed by high-degree polynomial constraints over finite fields, potentially extending to code theory, cryptography, and combinatorial designs. The explicit formulas enable probabilistic analysis and asymptotic estimation of MDS code prevalence, sharpening theoretical bounds on their existence as $\mathbb{F}_q$5 increases.

Practical and Theoretical Implications, Future Directions

Practically, these results streamline the computation for code designers evaluating the likelihood of certain matrix configurations in field-based constructions. Theoretically, the recursive symbolic procedure exposes the limitations of general solution methods for higher-dimensional or more complex polynomial systems, indicating boundary cases where such explicit enumeration is tractable.

Future research may extend the strategy to non-contiguous superregularity, larger matrices, or incorporate additional algebraic or combinatorial symmetries to further optimize enumeration. Additionally, improving or generalizing the recursive procedure via advanced symbolic computation and memoization may enable deeper exploration of related matrix classes.

Conclusion

The paper delivers a rigorous, explicit enumeration for contiguous superregular $\mathbb{F}_q$6 matrices over finite fields, confirming prior conjectures for $\mathbb{F}_q$7 and extending the results to rectangular cases. The algebraic techniques underscore the tractability of matrix enumeration under polynomial constraints and illuminate foundational combinatorial properties underpinning MDS codes and related coding theoretic constructs (2606.14296).