--- title: 'Rank-Metric (n,k)-MRD Codes: Theory & Constructions' url: https://www.emergentmind.com/topics/n-k-mrd-codes type: topic --- # Rank-Metric (n,k)-MRD Codes: Theory & Constructions \((n,k)\)-MRD codes are maximum rank-distance codes, the rank-metric analogue of MDS codes. In the matrix model, they are subsets of \(M_{m\times n}(\mathbb{F}_q)\) equipped with the rank distance \(d(A,B)=\operatorname{rank}(A-B)\), and they attain Delsarte’s Singleton-like bound \[ |\mathcal{C}| \leq q^{\max\{m,n\}(\min\{m,n\}-d+1)}. \] In the extension-field model, one studies \([n,k,d]_{q^m/q}\) codes in \(\mathbb{F}_{q^m}^n\); when \(n\le m\), the MRD condition is \(d=n-k+1\). The subject combines linearized polynomials, finite geometry, Moore-type matrices, idealizers, scattered subspaces, and lifted constructions for subspace coding, and the literature surveyed here shows both broad existence results and a rapidly expanding collection of inequivalent families [1904.05813] [2603.27748]. ## 1. Formal framework and parameterizations The basic ambient space for rank-metric coding is \(M_{m\times n}(\mathbb{F}_q)\), with distance measured by the rank of differences. An MRD code is a rank-metric code meeting the Singleton-like upper bound with equality. The classical existence theory is unusually strong: the survey literature states that MRD codes exist for all parameters, in contrast with MDS codes in the Hamming metric, which have parameter restrictions [1904.05813]. A second standard formalism uses \(\mathbb{F}_{q^n}\)-linearized polynomials. Over \(\mathbb{F}_{q^n}\), these are polynomials of the form \[ \sum a_i x^{q^i}, \] and they form a ring under composition. In this language, many linear MRD codes are modeled as spaces of \(q\)-polynomials, and rank is interpreted as the \(\mathbb{F}_q\)-dimension of the image of the associated \(\mathbb{F}_q\)-linear transformation. This representation underlies the constructions of Gabidulin, generalized twisted Gabidulin, and related families [2007.01991]. The literature also uses several parameter notations simultaneously. Matrix-space descriptions emphasize dimensions \(m,n\), code size, and minimum rank distance; extension-field descriptions emphasize \([n,k,d]_{q^m/q}\); and some papers speak of \((n,n,q;d)\)-MRD codes for nonlinear families. This coexistence of notation reflects different but compatible realizations of the same optimality phenomenon [2305.19027]. ## 2. Classical families and the expansion beyond Gabidulin codes The foundational explicit family is the generalized Gabidulin code \[ \mathcal{G}_{k,s}=\left\{a_0X+a_1X^{q^s}+\dots+a_{k-1}X^{q^{s(k-1)}}:a_i\in\mathbb{F}_{q^n}\right\},\qquad \gcd(s,n)=1. \] These are MRD codes for all valid parameters and remain the prototype against which many later constructions are compared [1807.08774]. A broad generalization is the family \[ \mathcal{H}_{k,s}(L_1(x),L_2(x)) = \left\{ L_1(a_0)x+a_1x^{q^s}+\cdots+L_2(a_0)x^{q^{sk}} : a_0,\ldots,a_{k-1}\in\mathbb{F}_{q^n} \right\}. \] If \[ N_{q^n,q}(L_1(a))\neq (-1)^{nk}N_{q^n,q}(L_2(a)) \quad \text{for all } a\in\mathbb{F}_{q^n}^*, \] then \(\mathcal{H}_{k,s}(L_1,L_2)\) is an MRD code. This framework contains Gabidulin codes as the special case \(L_1(x)=x\), \(L_2(x)=0\), \(s=1\), and generalized twisted Gabidulin codes as \(L_1(x)=x\), \(L_2(x)=\eta x^{q^h}\). The same framework also accommodates additive generalized and Trombetti-Zhou codes [2007.01991]. The view that Gabidulin codes exhaust the linear theory is false even in square spaces. The classification of MRD codes in \(\mathbb{F}_q^{n\times n}\) with maximum left and right idealizers shows that for \(n\le 6\) and \(n=9\), such codes are equivalent to generalized Gabidulin codes, but there are additional families for \(n=7\) with \(q\) odd and for \(n=8\) with \(q\equiv 1 \pmod 3\). In particular, the paper exhibits \[ \mathcal{C}_7=\{a_0X+a_1X^{q^2}+a_2X^{q^4}:a_0,a_1,a_2\in\mathbb{F}_{q^7}\} \] and \[ \mathcal{C}_8=\{a_0X+a_1X^{q^2}+a_2X^{q^6}:a_0,a_1,a_2\in\mathbb{F}_{q^8}\}, \] together with their adjoints, as MRD codes not equivalent to Gabidulin codes [1807.08774]. ## 3. Equivalence, duality, nuclei, and code identifiers Equivalence is central because many apparently different polynomial descriptions represent the same rank-metric code. For linearized-polynomial models, two codes \(\mathcal{C},\mathcal{C}'\) are equivalent if there exist bijective \(q\)-polynomials \(\phi_1,\phi_2\) and \(\rho\in\operatorname{Aut}(\mathbb{F}_{q^n})\) such that \[ \mathcal{C}'=\{\phi_1\circ f^\rho\circ \phi_2:f\in\mathcal{C}\}. \] For the families \(\mathcal{H}_{k,s}(L_1,L_2)\), the equivalence problem is characterized explicitly in terms of field automorphisms, shifts, and a bijective \(q\)-polynomial \(T\), thereby extending earlier results on twisted Gabidulin and Trombetti-Zhou codes [2007.01991]. Structural invariants play a major role in distinguishing inequivalent MRD codes. For a code \(\mathcal{C}\subseteq \mathscr{L}_{n,q}[x]\), the right and middle nuclei are \[ \mathcal{N}_r(\mathcal{C})=\{g(x)\in \mathscr{L}_{n,q}[x]: g\circ f\in\mathcal{C}\ \forall f\in\mathcal{C}\}, \] \[ \mathcal{N}_m(\mathcal{C})=\{g(x)\in \mathscr{L}_{n,q}[x]: f\circ g\in\mathcal{C}\ \forall f\in\mathcal{C}\}. \] For \(\mathcal{H}_{k,s}(x,L(x))\), if \(L(x)=\sum_{i=0}^M \eta_i x^{q^{e_i}}\) and \(d=\gcd(e_1,\ldots,e_M,n)\), then \[ \mathcal{N}_r(\mathcal{H})=\mathcal{N}_m(\mathcal{H})=\{ax:a\in\mathbb{F}_{q^d}\}. \] The same paper also computes Delsarte duals, adjoints, and automorphism groups; in particular, if \(\mathcal{C}\) is MRD, then its Delsarte dual \(\mathcal{C}^\perp\) is MRD [2007.01991]. A complementary invariant-based approach uses conjugates of the code under Frobenius. The height \[ h(C)=\max\{\dim(C\cap C^{[j]}): j=1,\ldots,n-1,\ \gcd(j,n)=1\} \] and the Gabidulin index \[ \operatorname{ind}(C)=\max\{\dim(G): G\subseteq C \text{ and } G \sim \text{ a generalized Gabidulin code}\} \] are invariants under equivalence. For generalized Gabidulin codes, \(h(C)=k-1\) and \(\operatorname{ind}(C)=k\); for generalized twisted Gabidulin codes, the characteristic values are \(h(C)=k-2\) and \(\operatorname{ind}(C)=k-1\). The paper also gives a characterization of generalized twisted Gabidulin codes through the dimensions of \(C\cap C^{[s]}\), \(C\cap C^{[s]}\cap C^{[2s]}\), and the existence of a suitable invertible polynomial \(p(x)\) [1807.09476]. ## 4. Geometric and algebraic interpretations One of the deepest structural correspondences links MRD codes to scattered subspaces. For nondegenerate \([n,k]_{q^m/q}\) codes, the cited 2026 work states that \(\mathcal{C}\) is MRD with \(d=n-k+1\) if and only if \(n\le m\) and the associated \(q\)-system \(U\) is \((k-1)\)-scattered. The same source identifies puncturing with inclusion of associated subspaces and concludes that non-extendable MRD codes correspond exactly to maximally \((k-1)\)-scattered subspaces [2603.27748]. This geometric perspective produced new explicit families. A 2017 construction starts from maximum scattered linear sets of pseudoregulus type in \(\mathrm{PG}(3,q^n)\), projects them to \(\mathrm{PG}(1,q^{2n})\), and obtains new MRD codes from the scattered subspaces \[ U_{b,s}=\{(x, b x^{q^s}+x^{q^{n+s}}): x\in\mathbb{F}_{q^{2n}}\}, \] with \(1\le s2\); for \(n=4\), it yields MRD codes with parameters \((8,8,q;7)\) for \(q\) odd. These codes are proved not to be equivalent to generalized Gabidulin or Sheekey’s twisted Gabidulin codes when \(n>2\) [1707.08487]. The algebraic geometry of minimum-rank codewords can also be made explicit. A 2021 note describes the elements of minimum rank in generalized Gabidulin codes via Grassmann coordinates, characterizes linearized polynomials of rank at most \(n-k\), and gives parametric equations for MRD codes of distance \(d=n-k+1\). In particular, if \(L(x)\) is a linearized polynomial, then \(L(x)\) has rank at most \(n-k\) if and only if the appropriate minors of its Dickson matrix vanish. This supplies an intrinsic criterion for rank defect and places MRD code classification in a Grassmannian framework [2106.15708]. ## 5. Nonlinear, switched, and base-field constructions Although much of the classical theory is additive or \(\mathbb{F}_{q^n}\)-linear, the current landscape includes substantial nonlinear theory. A 2023 construction uses cones over maximum exterior sets with respect to secant varieties in \(\mathrm{PG}(n-1,q^n)\) to build nonlinear \((n,n,q;d)\)-MRD codes \(\mathcal{C}_{\sigma,T}\) for every \(2\le d\le n-1\). The construction is geometric: points of a cone \(K(A^*,\mathcal{E})\) correspond to \(\sigma\)-linearized polynomials or, equivalently, rank-\(n\times n\) matrices over \(\mathbb{F}_q\). The resulting codes are closed under scalar multiplication, are generally non-additive, and are shown to be neither equivalent nor adjointly equivalent to the non-linear Otal–Özbudak codes \(\mathcal{C}_{n,k,\sigma,I}\), except in the cases \(q=2\) or \(T=\mathbb{F}_q^*\), where they coalesce to standard Gabidulin codes [2305.19027]. Switching gives a very different source of abundance. In this method, one replaces special MRD subcodes by other subcodes with the same parameters while preserving the global MRD property. The construction applies to punctured twisted Gabidulin codes and direct-product codes, and it yields a huge class of MRD codes whose cardinality grows doubly exponentially in \(m\) when \(n\), \(q\), and the code distance are fixed. The same work constructs MRD codes with different affine ranks and aperiodic MRD codes, showing that the combinatorial diversity of MRD codes is far greater than a family-by-family list would suggest [2211.00298]. A further departure from the classical extension-field paradigm is the 2026 circular-shift construction. It produces \((J\times n, q^{Jk}, d)\) MRD codes entirely over \(\mathbb{F}_q\), with \(J=\varphi(L)\) and \(\gcd(q,L)=1\), and avoids arithmetic over \(\mathbb{F}_{q^J}\). The paper proves that when \(J=m_L\), with \(m_L\) the multiplicative order of \(q\) modulo \(L\), the constructed codes coincide with Gabidulin codes; when \(J\ne m_L\), a family of parameter settings yields codes different from any Gabidulin code and any twisted Gabidulin code. For \(q=2\), \(L\) prime, and \(n\le m_L\), encoding a codeword requires \(O(nkL)\) XOR operations for the proposed circular-shift codes, compared with \(O(nkL^2)\) XOR operations for customary Gabidulin encoding [2602.12766]. ## 6. Liftings, subspace codes, and current structural directions MRD codes are a principal engine for constructing constant-dimension subspace codes. If \(Q_q(n,k,d)\) is an MRD code of \(k\times n\) matrices, the lifted construction \[ \operatorname{lift}(A)=\operatorname{rowspace}\begin{bmatrix}I_k & A\end{bmatrix} \] produces \(k\)-dimensional subspaces of \(F_q^{n+k}\), and the minimum subspace distance becomes \(2d\). This construction is a standard bridge from rank-metric coding to random network coding [1911.00154]. Recent work has pushed this bridge well beyond single-block lifting. One 2019 paper generalizes from two parallel lifted MRD codes to \(s+1\) parallel blocks in ambient dimension \((s+1)k+n\). The new bounds for \(A_q(n+k,k,d)\) and \(A_q((s+1)k+n,d,k)\) depend on Delsarte’s theorem for the rank distribution of MRD codes and on subsets of MRD codes with bounded rank. The counting argument uses codewords of restricted rank to preserve minimum subspace distance across different block positions [1911.00154]. A closely related construction, also from 2019, develops two several-block methods: a refined linkage construction and a multi-block lifting construction. In both, subsets of MRD codes with bounded ranks are essential, Delsarte’s theorem is used to count admissible matrices, and the resulting codes improve more than 110 previously best known lower bounds for constant-dimension subspace codes [1908.03804]. A major current structural direction concerns extendability. The 2026 paper on non-extendable MRD codes introduces the first infinite family of non-extendable \([4,2,3]_{q^5/q}\) MRD codes, for \(q=3^{2h+1}\), built from the subspaces \[ U_\delta=\{(x, x^q+\delta x^{q^4}) : x\in H_1\}, \qquad H_1=\ker(\operatorname{Tr}_{q^5/q}), \] with \(\operatorname{N}_{q^5/q}(\delta)=1\). These codes are self-dual up to equivalence and are not obtainable by puncturing longer MRD codes [2603.27748]. The survey literature leaves several questions open: full classification up to equivalence, the construction of non-Gabidulin MRD codes with efficient decoding algorithms, asymptotic enumeration, and the classification of symmetric and alternating MRD codes. Taken together with the newer work on switching, nonlinear geometric constructions, and non-extendability, these open problems indicate that \((n,k)\)-MRD codes are no longer a theory dominated by a single paradigm, but a large and structurally varied class of optimal objects in rank-metric geometry [1904.05813]. Source: https://www.emergentmind.com/topics/n-k-mrd-codes