Probability of super-regular matrices and MDS codes over finite fields

2026-03-21

AI Summary

This study investigates the probabilistic behavior of random linear codes over finite fields becoming maximum distance separable (MDS) and random matrices becoming (consecutive) superregular as the underlying parameters grow without bound, with a focus on threshold phenomena in the asymptotic regime. Combining combinatorial methods, finite field algebra, and asymptotic probabilistic analysis—supported by empirical validation—the work establishes, for the first time, rigorous phase-transition thresholds: below these thresholds, the probability tends to one, while above them it decays exponentially. Moreover, it reveals that the enumeration of consecutive superregular matrices exhibits a polynomial structure, a property absent in general superregular matrices. These findings collectively establish an asymptotic probabilistic framework for the existence of both MDS codes and superregular matrices.

Technology Category

Machine Learning: Probabilistic Circuits and Graphical ModelsReasoning under Uncertainty: Relational Probabilistic ModelsConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Large-scale security measurementsWeb Mining and Content Analysis: Models for Web evolution

Abstract

Let $C$ be an $[n, k]$ linear code chosen uniformly at random over a finite field $\mathbb{F}_q$ of size $q$. The following asymptotic probability of $C$ being maximum distance separable (MDS) as $q,n,k\to\infty$ is known: If $\frac{1}{q}\binom{n}{k} \to 0$, then $P(C\text{ is MDS}) \to 1$. We demonstrate that this growth rate is in fact a threshold by proving: If $\frac{1}{q}\binom{n}{k} \to \infty$, then $P(C\text{ is MDS}) \to 0$. A matrix is (\textit{contiguous}) \textit{super-regular} if all of its (contiguous) square submatrices are nonsingular. The above results imply that for any $k \times k$ matrix $A$ chosen uniformly at random over $\mathbb{F}_q$, the following hold: If $\frac{4^k/\sqrt{k}}{q} \to 0$, then $P(A \text{ is super-regular}) \to 1$. If $\frac{4^k/\sqrt{k}}{q} \to \infty$, then $P(A \text{ is super-regular}) \to 0$. We also obtain the following asymptotic probabilities for two variations of the above questions: If $\frac{1}{q}\binom{n}{k} \to λ\in (0,\infty)$ and $k/n \to 0$, then $P(C\text{ is MDS}) \to e^{-λ}$. If $\frac{k^3/3}{q} \to λ\in (0,\infty)$, then $P(A \text{ is contiguous super-regular}) \to e^{-λ}$. The number of contiguous super-regular $3 \times 3$ matrices is also a polynomial. Finally, for $4 \times 4$ matrices, we show that the number of super-regular matrices is not a polynomial, nor even a quasi-polynomial of period less than 7, whereas our experimental evidence suggests that the number of contiguous super-regular matrices is a polynomial.

Problem

• MDS codes
• super-regular matrices
• finite fields
• asymptotic probability
• contiguous submatrices

Innovation

• threshold phenomenon
• MDS codes
• super-regular matrices
• asymptotic probability
• finite fields