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

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 $1\le n_i\le m$, $N=n_1+\dots +n_{\mathrm{\ell }}$ and $2\le \delta \le N-1$, every partition of ${\mathbb{F}}_q^{\ast }$ 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 $\mathrm{\ell }\ge 2$ 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.