Breaking linearity in the sum-rank metric: MSRD codes from switched $\sigma$-rational normal curves

Daniele Bartoli; Giovanni Giuseppe Grimaldi; Giovanni Longobardi

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Breaking linearity in the sum-rank metric: MSRD codes from switched $\sigma$-rational normal curves

Daniele Bartoli, Giovanni Giuseppe Grimaldi, Giovanni Longobardi

arXiv:2610.11887v1Oct 8, 2026cs.ITmath.COmath.IT

TL;DR

Known multi-block MSRD constructions were linear, leaving non-additive codes in a genuine sum-rank setting as an open gap. This paper switches norm components of an LRS-associated curve and lifts the switched set through a syndrome map. The resulting codes are scalar-closed and non-additive when a curve component is retained, while matching linear MSRD codes in weight distribution and differing geometrically from LRS codes.

Problem
Known MSRD families with more than one block were linear, and the authors knew of no non-additive MSRD code in that setting.
Source: §1 · p. 2
Method
The construction switches norm components of an LRS-associated σ-rational normal curve and lifts the switched point set through the LRS syndrome map.
Source: §1 · p. 3
Results
Every constructed code has the sum-rank weight distribution of an Fqm-linear MSRD code with the same parameters, while no member with S≠∅ is equivalent to an LRS code.
Source: §1 · p. 4
Takeaways & Limitations
For ℓ≥2, the family provides non-additive MSRD codes in a genuine sum-rank setting, and the single-block construction recovers the cited cone-code family under an explicit rank isometry.
Source: §1 · pp. 2, 4
The stated construction assumes 1≤ni≤m and 2≤δ≤N−1; in the single-block case discussed, Λ={1}, a1=1, and the switching condition requires 1∈T.
Source: §5 · p. 16, §7 · p. 35

Abstract

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We construct a family of scalar-closed, non-additive maximum sum-rank distance codes (MSRD) by switching norm components of a $\sigma$-rational normal curve and lifting the resulting point set through a linearized Reed--Solomon syndrome map. We characterize the admissible switches by a multiplicative stability condition on the norm classes. In particular, for $1 \leq n_i \leq m$, $N=n_1+\ldots+n_{\ell}$ and $2\leq\delta\leq N-1$, every partition of $\mathbb{F}_q^*$ satisfying a specific condition, referred to as Condition $(\diamond)$, yields a code in $\mathbb{F}_{q^m}^{N}$ of size $q^{m(N-\delta+1)}$ and minimum sum-rank distance $\delta$; the code is non-additive whenever a curve component is retained. For $\ell\geq2$ these appear to be the first non-additive MSRD codes in the literature, all previously known families with more than one block being $\mathbb{F}_q$-linear. Moreover, every code of the family has the same sum-rank weight distribution as an $\mathbb{F}_{q^m}$-linear MSRD code with the same parameters, although the geometry of the switched set distinguishes it from linearized Reed--Solomon codes. In the single-block case, an explicit rank isometry identifies the construction with the cone codes of Durante, Grimaldi and Longobardi.

1 Introduction

The paper extends geometric switching constructions from the rank metric to the sum-rank metric, targeting the gap in known multi-block MSRD codes. Switching norm components and lifting the resulting set through an LRS syndrome map yields scalar-closed, non-additive codes whose weight distribution matches that of linear MSRD codes, while their geometry distinguishes them from LRS codes.

1 Introduction

For ℓ≥2, the construction fills a stated gap: the authors knew of no prior non-additive MSRD codes with more than one block.Previously known MSRD families with more than one block were Fq-linear; the paper presents its family as the first in a genuine sum-rank setting.Source: §1 · p. 2
The paper extends earlier geometric constructions of non-linear MRD codes, including cone codes built over maximum exterior sets, from rank metric to sum-rank metric.The earlier rectangular codes are obtained by puncturing the square codes in their matrix representation.Source: §1 · p. 3
Switching selected norm components of an LRS-associated σ-rational normal curve and lifting through its syndrome map produces scalar-closed, non-additive MSRD codes.The switched set and its secants must avoid the low-weight syndrome variety; the syndrome preimage is an affine cone with kernel vertex and a base projectively equivalent to the switched set.Source: §1 · p. 3
Admissible switches require replacing the original norm classes and retaining classes that satisfy a multiplicative stability condition.Theorem 5.8 states that Λ⊆T and SΠ_{δ−1}(Λ)⊆S; equivalently, S is a union of cosets of the subgroup generated by Π_{δ−1}(Λ).Source: §1 · p. 3
The maximal Fqm-linear subcodes are syndrome preimages of switched-set points, contain the syndrome-map kernel, and intersect pairwise exactly in that kernel.The paper also determines which of these maximal linear subcodes are themselves MSRD.Source: §1 · p. 3
Every family member has the sum-rank weight distribution of an Fqm-linear MSRD code with the same parameters, but no member with S≠∅ is equivalent to an LRS code.The switched set's projective equivalence class is an invariant under Fqm-semilinear sum-rank equivalence.Source: §1 · p. 4
For one block with Λ={1} and a1=1, an explicit rank isometry identifies the construction with the square cone codes of Durante, Grimaldi and Longobardi.The switched base is the Donati–Durante exterior set, so the construction recovers this known rank-metric family while extending it to sum-rank.Source: §1 · p. 4

2 Preliminaries and Notation

The preliminaries establish the sum-rank setting and its algebraic representations, then review duality for sum-rank codes. They also state when coefficient duality corresponds to vector duality and how LRS duals relate to LRS codes.

2 Preliminaries and Notation

The sum-rank metric extends both rank and Hamming metrics, reducing to rank for one block and to Hamming when all blocks have length one.A code is an arbitrary subset of the ambient space; it is linear when it is an Fqm-linear subspace.Source: §2 · p. 5
Sum-rank codes admit skew-polynomial, tuples-of-linearized-polynomial, matrix, and block-vector representations; LRS codes are MSRD.In the full-block model, LRS codes recover Reed–Solomon codes when m=1 and generalized Gabidulin codes for a single full block.Source: §2 · p. 8

2.1 The dual of a sum-rank metric code

Under the subgroup hypothesis, coefficient-duality is compatible with vector duality, and the dual of an Fq-linear MSRD code is also Fq-linear MSRD.For LRS codes, coefficient duality yields a code equivalent under a sum-rank isometry to the LRS code of complementary dimension.Source: §2.1 · p. 10
The LRS duality formula applies to any set Λ of pairwise distinct nonzero elements; for one block it specializes to Gabidulin duality, and for m=1 to Reed–Solomon duality.The vector-duality analogy cited in the preliminaries remains under the subgroup hypothesis.Source: §2.1 · p. 10

3 The non-linear MRD codes Cσ,T

This section recalls a square rank-metric construction based on maximum exterior sets and their projective cones. The resulting codes have the stated MRD parameters and are non-linear under the specified conditions, while connecting to earlier geometric constructions.

3 The non-linear MRD codes Cσ,T

The construction takes all nonzero vector representatives of cone points, with each point contributing exactly q^m−1 nonzero linearized polynomials.The cone is formed with vertex Λ⋆ and base ET, and is a maximum exterior set with respect to Ω_{d−2}(Σ_{m,m}).Source: §3 · p. 12
For q>2 and 1∈T⊊Fq*, the constructed code is non-linear, has minimum rank distance d, and has cardinality q^{mk}.The code is an (m,m,q;d)-MRD code, with d=m−k+1.Source: §3 · p. 12
The code contains two distinct subcodes equivalent to the generalized Gabidulin code, distinguishing it from the non-linear Otal–Özbudak family.It is neither equivalent nor adjointly equivalent to that family.Source: §3 · pp. 12–13
Deleting the last k−2 rows gives the rectangular MRD codes of Donati and Durante; special parameter cases recover constructions of Durante–Siciliano and Cossidente–Marino–Pavese.When q=2 or T=Fq*, the construction reduces up to equivalence to the generalized Gabidulin code.Source: §3 · p. 13

4 The syndrome map

The syndrome map connects sum-rank weight bounds to the sets Ω_r(U): vectors of weight at most r map into Ω_{r−1}(U), and every point there is represented by such a vector.

4 The syndrome map

The sets Ω_r(U) are nested and Ω_δ(U)=PG(δ, q^m); when ℓ=1, Ω_r(U) is the r-secant variety of P(U1).Source: §4 · p. 13
wt_SR(z) ≤ r maps its syndrome point into Ω_{r−1}(U), and every point there is represented by a vector outside K of sum-rank weight at most r.Thus the sets Ω_{r−1}(U) describe the projective syndrome images of vectors within the stated weight bound.Source: §4 · p. 14
An exterior set relative to Ω_r(U) has every line joining two distinct points of the set disjoint from Ω_r(U).Source: §4 · p. 16

5 The geometric setting

For 1 ≤ n_i ≤ m and 2 ≤ δ ≤ N−1, the section characterizes when the switched set is exterior to Ω_{δ−2}(U_Λ). Its criterion links admissible switches to multiplicative stability of norm classes.

5 The geometric setting

The row space of H is an MSRD LRS code of dimension δ+1 and length N.Source: §5 · p. 16
Each X_b is a scattered F_q-linear set of rank m, while each J_a is a scattered linear set of rank m of pseudoregulus type.Source: §5 · p. 17
If two distinct points of X_b both lie in H_f, then d_b(f)≥2.Source: §5 · p. 17
Condition (⋄) holds when S is a union of cosets of G contained in F_q^*, with the stated multiplicative stability conditions.For π in Π_{δ−1}(Λ), Sπ⊆S is equivalent to Sπ=S.Source: §5 · p. 19
A partition with nonempty S satisfying Condition (⋄) exists exactly when ΛG≠F_q^*; a nonempty S exists exactly when G is a proper subgroup of F_q^*.Source: §5 · p. 19
The switched set E_{S,T} is exterior to Ω_{δ−2}(U_Λ) if and only if Condition (⋄) holds.Source: §5 · p. 21

6 The family of MSRD codes

This section establishes the MSRD and scalar-closure properties of the syndrome-lifted codes under Condition (⋄), including their non-additivity when S is nonempty. It also examines their structural invariants and shows that their sum-rank weight distribution matches that of a linear MSRD code with the same parameters.

6 The family of MSRD codes

The code C_{S,T} is MSRD and closed under F_{q^m}-multiplication.Its minimum sum-rank distance is δ.Source: §6 · pp. 27–28
If S is nonempty, C_{S,T} is non-additive for every ℓ ≥ 1.Scalar closure would make an additive code an F_{q^m}-subspace, contradicting the switched set’s geometry.Source: §6 · p. 29

6.1 Maximal linear subcodes, a geometric invariant and weight distribution

For S ≠ ∅, C_{S,T} has exactly q^m + 1 maximal F_{q^m}-linear subcodes, namely D_P for P ∈ E_{S,T}.Each such subcode is the preimage of a point under the syndrome map.Source: §6.1 · p. 29
The projective equivalence class of E_{S,T} is invariant under F_{q^m}-semilinear sum-rank equivalence, distinguishing switched codes from the linearized Reed–Solomon case.When S ≠ ∅, E_{S,T} is not a line, whereas E_{∅,F_q^*} is the line AB.Source: §6.1 · p. 33
The sum-rank weight distribution of C_{S,T} depends only on q, m, the block lengths n_1, …, n_ℓ, and δ.It therefore agrees with that of a linear MSRD code with the same parameters.Source: §6.1 · p. 34
The F_q^m-linear MSRD code C∅,F*_q has minimum sum-rank distance δ; its sum-rank weight distribution is determined by its parameters, and (19) gives its minimum-weight codeword count [2, Theorem VI.5 and Remark VI.11].Source: §6.1 · p. 35

7 The case ℓ= 1

In the single-block case, the switching condition reduces to 1 ∈ T, and the switched exterior set matches the Donati–Durante exterior set. The syndrome preimage is identified with the square cone code through an explicit rank-metric isometry.

7 The case ℓ=1

For ℓ = 1, Λ = {1}, and a1 = 1, the switching condition reduces to 1 ∈ T.Here N = n1 = m and 2 ≤ δ ≤ m − 1.Source: §7 · p. 35
The switched base is the Donati–Durante exterior set ET.Under the coordinate embedding that appends m − δ − 1 zeros, ES,T is identified with ET.Source: §7 · pp. 35, 37
The entire syndrome preimage is identified with the square cone code recalled earlier in the paper.The projective cone has vertex Λ⋆ and base ES,T; when δ = m − 1, the vertex is empty and KT = ET.Source: §7 · pp. 36–38
An explicit Fqm-linear rank-metric isometry establishes the equivalence, so the full-block construction recovers the square MRD codes.The single-block sum-rank weight is rank weight, and Ψ preserves rank distances.Source: §7 · pp. 36–37
The trace-based map Ψ is bijective and preserves rank distance in the single-block case.Its image on z equals the Fq-span of z1, …, zm.Source: §7 · p. 37