打破和秩度量中的线性性：来自切换σ-有理正规曲线的MSRD码
Breaking linearity in the sum-rank metric: MSRD codes from switched $σ$-rational normal curves
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- Università degli studi di Perugia（佩鲁贾大学）
- Università di Napoli Federico II（那不勒斯费德里科二世大学）
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中文总结 AI 辅助
该研究通过切换σ-有理正规曲线的范数分量并结合线性化Reed-Solomon伴随映射，构造出首批多块非加性MSRD码，其性能与同参数线性MSRD码相当。
中文摘要 AI 辅助
我们通过切换σ-有理正规曲线的范数分量，并通过线性化Reed-Solomon伴随映射提升所得点集，构造了一类标量闭、非加性的最大和秩距离码（MSRD）。我们通过范数类上的乘性稳定性条件来刻画可允许的切换。特别地，对于1≤n_i≤m、N=n₁+…+n_ℓ且2≤δ≤N-1，每个满足特定条件（称为条件(◇)）的𝔽_q^*划分，都会在𝔽_{q^m}^N中生成一个大小为q^{m(N−δ+1)}、最小和秩距离为δ的码；当保留曲线分量时，该码是非加性的。对于ℓ≥2，这些码似乎是文献中首批非加性MSRD码，此前已知的所有含多个块的族均为𝔽_q线性。此外，该族的每个码都具有与相同参数的𝔽_{q^m}线性MSRD码相同的和秩重量分布，尽管切换集的几何结构使其与线性化Reed-Solomon码相区别。在单块情形下，显式秩等距将该构造与Durante、Grimaldi和Longobardi的锥码等同起来。
英文摘要
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 $1 \leq n_i \leq m$, $N=n_1+\ldots+n_{\ell}$ and $2\leqδ\leq N-1$, 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 $q^{m(N-δ+1)}$ and minimum sum-rank distance $δ$; 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.
