Sum-rank analogues of Reed--Muller codes over Abelian Galois extensions

Let $\mathbb{L}\mathrm{/}\mathbb{K}$ be a finite Abelian Galois extension with a fixed decomposition $\mathrm{G}\mathrm{a}\mathrm{l}(\mathbb{L}\mathrm{/}\mathbb{K})=⟨{\sigma }_1⟩\times \cdots \times ⟨{\sigma }_m⟩$. We show that the quotient of the multivariate skew polynomial ring $\mathbb{L}[x_1,\dots ,x_m;{\sigma }_1,\dots ,{\sigma }_m]$ by suitable central polynomials $H_1(x_1),\dots ,H_m(x_m)$ is isomorphic to a product of $t$ copies of ${\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathbb{K}}(\mathbb{L})$, so that one skew polynomial represents all blocks of a sum-rank metric space. The sum-rank weight of an element equals the rank of an associated Dickson matrix and of a generalized Moore matrix. Using this, we define sum-rank Reed–Muller-type codes and show that their dimension and minimum distance are given by the formulas for affine Cartesian codes with side lengths $\mathrm{d}\mathrm{e}\mathrm{g}{H}_1,\dots ,\mathrm{d}\mathrm{e}\mathrm{g}{H}_m$; the same holds for codes supported on lower sets. The family contains as special cases linearized Reed–Solomon codes, Gabidulin codes, rank-metric Reed–Muller codes, and affine Cartesian codes. Over finite fields, every linearized Reed–Muller code of Berardini and Caruso is, after a monomial change of variables, a monomial code in our framework. This gives the dimension of all these codes, and their exact minimum distance in every degree whenever the support is a lower set.