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

Source 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 $1 \leq n_i \leq m$, $N=n_1+\ldots+n_{\ell}$ 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.