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

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.

Figures (5)

• Figure 1: An example of a $5\times 16$ matrix $A$ with two different $5\times 5$ submatrices (blue and red) that share two columns (split blue/red).
• Figure 2: An example of a $9\times 9$ square matrix (black) $A$ with a $4\times 4$ contiguous submatrix $B$ anchored at $(5,7)$ formed by deleting from $A$ all but its red columns and blue rows. The corner decomposition $(\bar{B}, u, v)$ of $B$ is shown to the right in green.
• Figure 3: Square contiguous submatrices anchored at $(i,j)$ shown when $\text{min}(i,j)=6$. The event $\Gamma_{i,j}$ is a function of the matrix elements in the yellow region. Hence these yellow entries are no longer iid when conditioned on $\Gamma_{i,j}$. Each contiguous square submatrix anchored at $(i,j)$ is labeled inside its top-left corner.
• Figure 4: The matrix $M$ shown when $\text{min}(i,j)=6$.
• Figure 5: Plots of the probability of a random $k\times k$ matrix over $\mathbb{F}_{q}$ being super-regular or contiguous super-regular. The red curves are $e^{-\lambda}$. The green curves are counts of $3\times 3$ matrices based on Theorem \ref{['thm:3x3']}(a) for super-regular and (b) for contiguous super-regular. Each blue dot is the fraction of $1000$ randomly selected $k\times k$ matrices over $\mathbb{F}_{q}$ that were super-regular (where $\lambda = \frac{1}{q}\binom{2k}{k}$) or contiguous super-regular (where $\lambda = \frac{1}{3} k^3q^{-1}$), with $k=10$.

Theorems & Definitions (61)

• Lemma 1.1
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof