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

@misc{neri2026sumrank,
  title = {{Sum-rank analogues of Reed--Muller codes over Abelian Galois extensions}},
  author = {Alessandro Neri and Giuseppe Zampella},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31358},
  eprint = {2610.31358},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31358},
}

TY  - UNPB
TI  - Sum-rank analogues of Reed--Muller codes over Abelian Galois extensions
AU  - Alessandro Neri
AU  - Giuseppe Zampella
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31358
UR  - https://arxiv.org/abs/2610.31358
AB  - Let 𝕃/𝕂 be a finite Abelian Galois extension with a fixed decomposition Gal(𝕃/𝕂)=⟨σ_1⟩×⋯×⟨σ_m⟩. We show that the quotient of the multivariate skew polynomial ring 𝕃[x_1,…,x_m;σ_1,…,σ_m] by suitable central polynomials H_1(x_1),…,H_m(x_m) is isomorphic to a product of t copies of End_𝕂(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 deg H_1,…,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.
ER  -