Breaking linearity in the sum-rank metric: MSRD codes from switched σ-rational normal curves
Abstract
We construct a family of scalar-closed, non-additive maximum sum-rank distance codes (MSRD) by switching norm components of a σ-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 ≤ ni ≤ m, N=n1+…+nℓ and 2≤δ≤ N−1, every partition of 𝔽q* satisfying a specific condition, referred to as Condition (diamond), yields a code in 𝔽qmN of size qm(N−δ+1) and minimum sum-rank distance δ; the code is non-additive whenever a curve component is retained. For ℓ≥2 these appear to be the first non-additive MSRD codes in the literature, all previously known families with more than one block being 𝔽q-linear. Moreover, every code of the family has the same sum-rank weight distribution as an 𝔽qm-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.
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BibTeX
@misc{bartoli2026breaking,
title = {{Breaking linearity in the sum-rank metric: MSRD codes from switched $σ$-rational normal curves}},
author = {Daniele Bartoli and Giovanni Giuseppe Grimaldi and Giovanni Longobardi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11887},
eprint = {2610.11887},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11887},
}
RIS
TY - UNPB
TI - Breaking linearity in the sum-rank metric: MSRD codes from switched σ-rational normal curves
AU - Daniele Bartoli
AU - Giovanni Giuseppe Grimaldi
AU - Giovanni Longobardi
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11887
UR - https://arxiv.org/abs/2610.11887
AB - We construct a family of scalar-closed, non-additive maximum sum-rank distance codes (MSRD) by switching norm components of a σ-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 ≤ n_i ≤ m, N=n_1+…+n_ℓ and 2≤δ≤ N−1, every partition of 𝔽_q^* satisfying a specific condition, referred to as Condition (diamond), yields a code in 𝔽_q^m^N of size q^m(N−δ+1) and minimum sum-rank distance δ; the code is non-additive whenever a curve component is retained. For ℓ≥2 these appear to be the first non-additive MSRD codes in the literature, all previously known families with more than one block being 𝔽_q-linear. Moreover, every code of the family has the same sum-rank weight distribution as an 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.
ER -