Sum-rank analogues of Reed--Muller codes over Abelian Galois extensions
Let $\mathbb L/\mathbb K$ be a finite Abelian Galois extension with a fixed decomposition $\mathrm{Gal}(\mathbb L/\mathbb K)=\langleσ_1\rangle\times\cdots\times\langleσ_m\rangle$. We show that the quotient of the multivariate skew polynomial ring $\mathbb L[x_1,\ldots,x_m;σ_1,\ldots,σ_m]$ by suitable central polynomials $H_1(x_1),\ldots,H_m(x_m)$ is isomorphic to a product of $t$ copies of $\mathrm{End}_{\mathbb K}(\mathbb L)$, 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 $\mathrm{deg}\,H_1,\ldots,\mathrm{deg}\,H_m$; 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.