Three-Weight Rank-Metric Codes

• Three-weight rank-metric codes are F₍qᵐ₎/F₍q₎-linear codes characterized by exactly three nonzero rank weights, linking algebraic coding theory with finite geometry.
• Explicit constructions employ geometric structures such as scattered subspaces, linear sets, and i-clubs to achieve precise weight distributions and code parameters.
• MacWilliams identities and combinatorial bounds are used to analyze and classify these codes, providing insights for applications in network coding and algebraic combinatorics.

Three-weight rank-metric codes are a class of $\mathbb{F}_{q^m}/\mathbb{F}_q$-linear codes characterized by the property that the rank-weight distribution has exactly three nonzero values. These codes, constructed through deep connections between finite geometry, specifically linear sets such as scattered subspaces and clubs, and algebraic coding theory, have explicit, tractable combinatorial and geometric structures. Their paper intersects several active areas, including projective geometry, MacWilliams identities for the rank metric, and classification problems for linear sets.

1. Fundamental Definitions and Parameters

A rank-metric code $\mathcal{C} \subseteq \mathbb{F}_{q^m}^n$ is an $\mathbb{F}_{q^m}$-subspace of dimension $k$. The ambient distance is

$$d(x, y) = \operatorname{rk}(x - y) = \dim_{\mathbb{F}_q} \langle x - y\rangle_{\mathbb{F}_q},$$

and the minimum rank distance is the smallest nonzero rank among codewords: $$d = \min_{0 \neq v \in \mathcal{C}} \operatorname{rk}(v).$$ An $[n, k, d]_{q^m/q}$ code denotes such a code over $\mathbb{F}_{q^m}$ of length $n$, dimension $k$, and minimum distance $d$.

The weight distribution is given by $$A_i = |\{v \in \mathcal{C}: \operatorname{rk}(v) = i\}|$$ for $0 \leq i \leq n$. A three-weight rank-metric code is one whose nonzero weights occur exactly in three values, i.e., the set $\{i: A_i > 0, i \neq 0\}$ has size three (Mannaert et al., 1 Aug 2025).

2. Geometric Constructions: Linear Sets, Clubs, and Scattered Subspaces

The geometric underpinning of three-weight rank-metric codes is the theory of $\mathbb{F}_q$-linear sets in projective spaces. For a $k$-dimensional ambient space over $\mathbb{F}_{q^m}$, any $\mathbb{F}_q$-subspace $U \subseteq \mathbb{F}_{q^m}^k$ of dimension $n$ defines

$$L_U = \left\{ \langle u\rangle_{\mathbb{F}_{q^m}} : u \in U\setminus \{0\}\right\} \subseteq \mathrm{PG}(k-1, q^m),$$

an $\mathbb{F}_q$-linear set of rank $n$. Points' weights are $$\dim_{\mathbb{F}_q}(U \cap \langle v\rangle_{\mathbb{F}_{q^m}})$$.

A key structure is the $i$-club: an $\mathbb{F}_q$-linear set of rank $n$ in which exactly one point has weight $i>1$, with all others of weight $1$ (Mannaert et al., 1 Aug 2025). Scattered subspaces are those such that every $\mathbb{F}_{q^m}$-line in $\mathbb{F}_{q^m}^k$ meets $U$ in at most a $1$-dimensional $\mathbb{F}_q$-space.

A crucial geometric-algebraic correspondence is that to every nondegenerate $[n, k, d]_{q^m/q}$ code, one can associate an $\mathbb{F}_q$-system (i.e., a specific $U$) and vice versa [(Lia et al., 2023), Theorem 2.1]. If $L_U$ is an $i$-club, then the associated code has exactly three rank weights.

3. Explicit Constructions and Weight Distributions

An explicit family of such codes arises from scattered $\mathbb{F}_q$-subspaces in $\mathbb{F}_{q^m}^3$. Fix an $\mathbb{F}_q$-basis $a_1, \dots, a_m$ of $\mathbb{F}_{q^m}$ and set

$$U_\sigma = \left\{ \left(x,\, x^{q^s} + a,\, x^{q^{2s}} + b\right) : x \in \mathbb{F}_{q^m},\, a, b \in \mathbb{F}_q\right\},$$

with $\gcd(s, m) = 1$ and $m \geq 5$ odd. Under mild arithmetic hypotheses on $(q, m)$, $U_\sigma$ has dimension $m+2$ and is 1-scattered (Lia et al., 2023).

The corresponding generator matrix is

$$G = \begin{pmatrix} a_1 & \cdots & a_m & 0 & 0 \ a_1^{q^s} & \cdots & a_m^{q^s} & 1 & 0 \ a_1^{q^{2s}} & \cdots & a_m^{q^{2s}} & 0 & 1 \end{pmatrix},$$

and the generated rank-metric code has parameters $[m+2, 3, m-2]_{q^m/q}$.

The rank weight distribution for this code is explicitly

$w_1 = m-2, \quad w_2 = m-1, \quad w_3 = m,$

with multiplicities

$$\begin{aligned} A_{w_3} &= \#\{v\in C\,:\,\operatorname{rk}(v)=m \} \ &= \frac{1}{q^4(q+1)(q-1)^2}\left[q^7 + q^{m+1}(q^6-q^5-q^4-1) + q^{2m}(q^7-q^6-q^5+q^2+1)\right], \ A_{w_2} &= \frac{(q^{m-1}-1)\left(q^5 + q^m(q^4 - q^3 - 1)\right)}{q^4(q-1)^2}, \ A_{w_1} &= \frac{(q^{m-1}-1)(q^{m-4}-1)}{(q+1)(q-1)^2}. \end{aligned}$$

The sum $A_{w_1} + A_{w_2} + A_{w_3} = q^{2m} + q^m + 1$ is the total count of nonzero codewords.

4. Extensions via Clubs, Parameter Regimes, and Further Constructions

For code parameters arising from $i$-clubs, if $L_U \subseteq \mathrm{PG}(k-1, q^m)$ is an $i$-club of rank $n$ ($i<m$), then the associated code $\mathcal{C}$ has

$[n = km-n,\, k,\, d = m-i]_{q^m/q}$

with nonzero weights $w_1 = m-i$, $w_2 = m-1$, $w_3 = m$. The respective multiplicities are

$$A_{w_1}=q^m-1, \quad A_{w_2}=(q^m-1) \sum_{j=i}^n q^j, \quad A_{w_3}=q^{mk}-1-A_{w_1}-A_{w_2}.$$

General construction techniques are as follows (Mannaert et al., 1 Aug 2025):

• Cone construction: Adjoin an $i$-dimensional space $U''$ to a scattered subspace in a hyperplane, yielding an $i$-club of prescribed rank.
• Lifting: For odd $k=2s+1$, lift scattered sets via explicit coordinate maps, forming clubs for $i \geq m/2$.
• Specific $k=4$ construction: For $i=m/2$, use $$U=\{(x, \operatorname{Tr}_{q^m/q^{m/2}}(x), y, y^q) \mid x, y \in \mathbb{F}_{q^m}\}$$, yielding a non-cone club.
• Classification for $i=m-1$: Every maximal $(m-1)$-club of specific rank is equivalent to a cone over a maximum scattered set [(Mannaert et al., 1 Aug 2025), Thm 6.3].

Table: Key Construction Families for Three-Weight Codes

Method	Parameter Range	Structural Features
Cone	$i \geq m/2$, all $k, m$ even	Adjoins $i$-dim. space to hyperplane
Lifting	$k$ odd, $i \geq m/2$	Uses explicit field-lifting mappings
Direct sum	Even $k\ge 6$, $i \leq m-3$	Non-equivalent to cone, more varied
Trace-based	$k=4$, $i=m/2$	Involves field trace functions

5. Theoretical Analysis and Constraints

MacWilliams identities in the rank metric establish relationships between the weight enumerators of a code and its dual [(Mannaert et al., 1 Aug 2025), §5.1]. For $\mathcal{C}$ and $\mathcal{C}^\perp$,

$$W_{\mathcal{C}}(X,Y) = \frac{1}{|\mathcal{C}|} W_{\mathcal{C}^\perp}(X + (q^m-1)Y, X-Y).$$

This relationship imposes bounds on the possible parameters of $i$-clubs and thus on the families of three-weight codes. Specifically, if $n$ is the rank of the $i$-club, then

$$n \leq \begin{cases} \frac{mk}{2}, & i \leq \frac{m}{2} \text{ or } k=2, \ \frac{m(k-1)}{2} + i, & i \geq \frac{m}{2},\, k>2. \end{cases}$$

These bounds are tight for $i \geq m/2$ due to the existence of the corresponding club constructions.

6. Examples and Explicit Weight Enumerators

Consider $q=2$, $m=4$, $k=3$, $i=2$. Via the cone or lifting construction, the $i$-club has rank

$n = \frac{(3-1)\cdot 4}{2} + 2 = 6,$

so the code parameters are $[6, 3, 2]_{2^4/2}$. Its three nonzero weights are $2, 3, 4$. The weight multiplicities are

$$A_2 = 15, \quad A_3 = 15 \cdot (2^6 + 2^5 + 2^4 + 2^3 + 2^2) = 1860, \quad A_4 = 4095 - 15 - 1860 = 2220.$$

The weight enumerator is

$$W_C(X, Y) = X^6 + 15 X^4 Y^2 + 1860 X^3 Y^3 + 2220 X^2 Y^4.$$

A general pattern emerges: the nonzero weights are $\{m-i, m-1, m\}$, with $A_{m-i} = q^m - 1$ and $A_{m-1}$ scaling with sums over $q^j$ for $j$ from $i$ to $n$.

7. Classification, Existence, and Open Questions

The existence of three-weight rank-metric codes associated to $i$-clubs is completely determined for $i \geq m/2$, with explicit constructions achieving the bounds from MacWilliams identities [(Mannaert et al., 1 Aug 2025), §6.1]. For $i \leq m-2$, multiple non-equivalent $i$-clubs (arising via cone and lifting constructions) exist, yielding inequivalent codes. When $i = m-1$, all maximal $(m-1)$-clubs are equivalent to the cone form under $\Gamma L$.

However, for $i < m/2$, the classification is incomplete. The precise geometric and algebraic structure of non-cone clubs, as well as the parameters of associated codes, remain partly open. This suggests that further advances in the theory of linear sets and the projective classification of $i$-clubs may yield new families of three-weight rank-metric codes, with potential impact on finite geometry and network coding theory.

8. Relationship to Broader Research Directions

Three-weight rank-metric codes form a nexus between finite geometry (through scattered spaces and linear sets), algebraic combinatorics (via MacWilliams-type dualities), and the construction of codes with prescribed combinatorial properties. Their explicit enumerators permit fine-grained analysis, and the interrelations with geometrical invariants (such as intersection numbers of clubs with lines and hyperplanes) further enrich their theoretical significance (Lia et al., 2023, Mannaert et al., 1 Aug 2025). The paper of two-weight and three-weight rank-metric codes also interfaces with multiplicity enumerators, minimality, and classification problems central to the theory of rank-metric and classical codes.