Random Linear Rank-Metric Codes

• Random linear rank-metric codes are F₍q₎-linear subspaces with the rank metric, offering strong error correction that closely meets the Gilbert–Varshamov bound.
• They exhibit sharply concentrated minimum rank distance and optimal list-decodability, making them robust for network coding, cryptography, and distributed storage.
• Their performance analysis leverages Gaussian coefficients, MacWilliams identities, and duality theory to predict weight distribution and design thresholds.

Random linear rank-metric codes are linear subspaces of matrices (or equivalently, vectors over extension fields) equipped with the rank metric, where the code is selected uniformly at random within the ensemble of all such linear spaces. These codes are vital in network coding, cryptography, and distributed storage due to their strong error-correcting capabilities, their combinatorial structure, and their connections to optimal objects such as maximum rank distance (MRD) codes. The paper of random linear rank-metric codes centers on parameters such as minimum rank distance, list-decodability, weight distribution, and their asymptotic behavior, especially in relation to the Gilbert–Varshamov bound and the sphere-packing bound.

1. Definition and Structural Properties

A random linear rank-metric code is defined as an F₍q₎-linear subspace 𝒞 ⊆ F₍q₎^{m×n}, chosen uniformly at random among all such subspaces of a fixed dimension. The rank metric on F₍q₎^{m×n} is given by

$d_R(X, Y) = \operatorname{rk}(X-Y)$

where X, Y ∈ F₍q₎^{m×n}.

Key parameters of such codes include:

• Minimum Rank Distance:

$$d_{min}(𝒞) = \min\{\operatorname{rk}(X) : X \in 𝒞,\ X \ne 0\}$$

• Code Rate:

$R = \frac{\log_q |𝒞|}{mn}$

The distribution of minimum rank distance, as well as other rank-based invariants such as the rank distribution, are sharply concentrated for random codes due to high symmetry of the ensemble [0610057]. Most random linear rank-metric codes are not structured MRD codes; in fact, MRD codes are sparse among all linear codes of the same parameters as field size grows (Gruica et al., 2023).

2. Minimum Rank Distance and Existence Bounds

The main combinatorial benchmark for the parameters of random rank-metric codes is the Gilbert–Varshamov (GV) bound. In the rank-metric setting, this asserts that there exist codes of rate R and minimum rank distance δn satisfying:

$R \leq (1-\delta)(1-b\delta)$

where $b = \lim_{n \to \infty} n/m$ (Ding, 2014). Explicitly, for an F₍q₎-linear code in F₍q₎^{m \times n} of rate R and normalized minimum rank distance δ, nearly all random codes satisfy minimum distance at least δ_GV(R), i.e., attain the GV bound with high probability. The asymptotic behavior of the minimum distance and its probability distribution concentrate sharply around this value [0610057]. No perfect codes exist in the rank metric, but for characteristic 2, there exist quasi-perfect codes close to the sphere-packing bound [0610057].

3. List Decodability and Capacity

Random linear rank-metric codes are now known to achieve essentially optimal list-decodability. If rate $R < (1-\rho)(1-b\rho)$, then with high probability a random F₍q₎-linear code is $(\rho, O(1/\epsilon))$-list-decodable for any arbitrarily small $\epsilon > 0$ (Guruswami et al., 2017, Ding, 2014, Liu et al., 13 Mar 2025).

Key facts:

• For non-linear codes, the list size obeys $O(1/\epsilon)$; for F₍q₎-linear codes initially analysis gives $O(\exp(1/\epsilon))$, but careful correlation bounds reduce this to $O(1/\epsilon)$ for most rates (Liu et al., 13 Mar 2025).
• The list-decoding capacity is thus $1 - ( \rho + b\rho - b\rho^2 )$ for a radius $\rho$ and normalized parameter $b = n / m$ (Liu et al., 13 Mar 2025).
• Approaching this bound ($\epsilon \to 0$) implies that, for rates exceeding the bound, exponentially large list sizes are unavoidable (Ding, 2014).
• The existence of explicit codes meeting these combinatorial guarantees for efficient list decoding beyond half the minimum distance is an active area; random constructions provably achieve the information-theoretic limits (Guruswami et al., 2017).

4. Weight Distributions, MacWilliams Identities, and Duality

For random F₍q₎-linear rank-metric codes, the weight (rank) distribution is governed by Gaussian coefficients and relates strictly to the code parameters:

• For a linear code $C$, the rank distribution $W_i(C)$ (number of codewords of rank $i$) is sharply determined by the minimum distance and the code dimension (Gorla et al., 2017).
• Duals of random codes are also random, and the MacWilliams identities relate the rank distribution of a code and its dual via $q$-binomial combinatorics. For $C \leq F_q^{k \times m}$:

$$W_i(C^{\perp}) = \frac{1}{|C|} \sum_{u=0}^k (-1)^{i-u} q^{\binom{i-u}{2}} {k-u \brack k-i}_q W_u(C)$$

(cf. MacWilliams identity (Gorla et al., 2017)).

The combinatorial structure and duality theory remain robust for random linear codes; weight distribution is critical for performance predictions and bounds on covering radius (Gruica et al., 2023).

5. Covering Radius and Density of MRD Codes

The covering radius $\rho(C)$ of a random linear rank-metric code (the maximal distance from any matrix to the code) is strongly influenced by the dual minimum distance, and external distance (number of nonzero ranks in $C^\perp$) (Gruica et al., 2023).

MRD codes, which are extremal (i.e., meet the Singleton-type bound with equality), exist for all parameters but are extremely sparse among all codes of fixed dimension when $q$ or $m$ are large (Gruica et al., 2023). Specifically, given

$$\delta_q(n \times m, d) = \frac{|\{ C \leq F_q^{n \times m} : \dim C = m(n-d+1),\ d(C) = d \}|}{\binom{mn}{m(n-d+1)}_q}$$

one has $\delta_q(n \times m, d) \in O(q^{-(d-1)(n-d+1)+1})$ as $q \to \infty$ for $d \geq 2$, $n \geq 3$. Thus, most random linear codes are not MRD for large parameters.

6. Quasi-perfect Codes and Special Properties in Characteristic 2

Although perfect codes do not exist in the rank metric [0610057], the existence of quasi-perfect codes in characteristic 2 is established. Such codes nearly saturate the sphere-packing (volume) bound—every element of the ambient space is within distance $d$ or $d+1$ of some codeword. For certain parameters, random codes over characteristic 2 fields attain quasi-perfect properties and thus provide near-optimal error correction.

7. Applications and Significance

The theoretical properties of random linear rank-metric codes translate to high-performing and robust codes in practice. Notably:

• In random network coding, large random subspace codes built from rank-metric codes enable correction of packet errors and erasures with high probability (0807.2440).
• In code-based cryptography, the intractability of syndrome decoding for random linear rank-metric codes forms a foundation for designing post-quantum secure primitives (Gaborit et al., 2016, Burle et al., 2023).
• The existence of combinatorially good codes via randomness (e.g., achieving list decoding capacity) guides explicit code constructions and algorithm development across related fields (Guo et al., 20 Apr 2024).

Table: Critical List-Decoding Tradeoffs for Random Linear Rank-Metric Codes

Parameterization	Rate $R$	List Size $L$
Non-Linear Codes	$R < 1 - \rho - \epsilon$, $n/m \leq \epsilon$	$O(1/\epsilon)$
F₍q₎-Linear Codes	$R < (1-\rho)(1-b\rho) - \epsilon$, $b=n/m$	$O(1/\epsilon)$
Above list-decoding capacity	$R > (1-\rho)(1-b\rho) + \epsilon$	Exponential in $mn$

All results above are “with high probability” and $\epsilon > 0$ is arbitrarily small.

The probabilistic behavior of minimum rank distance, list size, and weight structure places random linear rank-metric codes as benchmarks for both theoretical limits and practical design, especially in settings where explicit optimal codes are either unknown or structurally infeasible.