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

1 Introduction

• A quotient of the multivariate skew-polynomial ring represents all blocks of the sum-rank ambient space with one polynomial class.Evaluating the class at the commuting operators αi,jiσi yields the blocks, and its sum-rank weight is the sum of their ranks.Source: §1 · p. 2
• The sum-rank Reed–Muller-type code of order r spans quotient monomials indexed by u in ∆(ν) with |u| ≤ r.The quotient has a standard monomial basis indexed by ∆(ν), with |u| = u1 + ··· + um.Source: §1 · p. 2
• The code dimensions and minimum distances have the same combinatorial formulas as affine Cartesian codes with side lengths νi = niti.The distance bounds use Dickson and generalized Moore matrices whose ranks equal sum-rank weight; leading-monomial arguments give lower bounds attained by products of univariate right divisors.Source: §1 · p. 3
• For every nonempty lower-set support D, the associated monomial code has dimension #D and an exact minimum distance.Weighted-degree constructions are included, and the lower-set distance theorem applies to standard linearized Reed–Muller and adapted lattice-simplex codes.Source: §1 · p. 3
• The construction specializes to linearized Reed–Solomon, Gabidulin, rank-metric Reed–Muller, affine Cartesian, and Reed–Solomon codes.These cases follow from choices including m = 1, t = 1, or N = 1.Source: §1 · p. 3
• After a monomial change of variables, every Berardini–Caruso linearized Reed–Muller code is represented as a monomial code in this framework.This determines their dimensions; exact minimum distances follow when the support is a lower set, while the two families overlap but neither contains the other in general.Source: §1 · p. 3

2.1 Notation and sum-rank metric codes

• The sum-rank metric adds the ranks of matrix blocks, specializing to rank metric for one block and Hamming weight for single-column blocks.The section also recalls the Singleton-type bound and the MSRD name for codes attaining it.Source: §2.1 · p. 4

2.2 Classical polynomial evaluation codes

• Affine Cartesian codes evaluate reduced monomials of bounded total degree on a finite grid, with dimension from monomial counting and distance tied to polynomial zero bounds.They specialize to generalized Reed–Muller codes on full finite-field grids and Reed–Solomon codes in one variable.Source: §2.2 · p. 5

2.3 Abelian Galois extensions and skew group algebras

• A finite Abelian Galois extension decomposes into commuting cyclic factors, and its skew group algebra acts on the extension as a K-algebra.Choosing a K-basis realizes the action by matrices and equips tuples of such actions with a sum-rank weight.Source: §2.3 · p. 5

2.4 Multivariate skew polynomials and truncated norms

• The multivariate skew polynomial variables commute with one another while twisting coefficients by the corresponding Galois automorphisms; suitable central relations yield a quotient isomorphic to the skew group algebra.The cited operator-evaluation result identifies the quotient by x_i^n_i − 1 with L[G] [4].Source: §2.4 · p. 6

2.5 Reference families in rank and sum-rank metrics

• The recalled families include Gabidulin codes, linearized Reed–Solomon codes, and rank-metric Reed–Muller codes; the paper’s construction represents all sum-rank blocks with one polynomial class.The cited Gabidulin and linearized Reed–Solomon constructions attain their stated MRD and MSRD parameters [4] [20].Source: §2.5 · p. 7

3 A skew-polynomial representation of the sum-rank ambient space

• The central polynomials H_i(x_i) are built from distinct norm values, and their generated ideals are pairwise comaximal two-sided ideals.The choices require enough distinct nonzero norms in each intermediate extension; over K = F_q, the text gives t_i ≤ q − 1 as an example bound.Source: §3 · p. 8
• The quotient has L-dimension Nt, with standard monomials indexed by exponents in Δ(ν) forming an L-basis.The relations have leading monomials x_i^ν_i and form a reduced Gröbner basis for every monomial order.Source: §3 · p. 8
• The Representation Theorem identifies the quotient by HΛ with (L[G])^t through a surjective homomorphism whose kernel is HΛ.After choosing a K-basis of L, this gives t copies of End_K(L).Source: §3 · p. 10
• The worked example yields a sum-rank ambient space consisting of four blocks, with each block corresponding to a copy of the skew group algebra L[G].Source: §3 · p. 11

4.3 The Dickson–Moore rank correspondence

• The sum-rank weight of Φ(F) equals the L-rank of its Dickson matrix and of the generalized Moore matrix formed from Φ(F) evaluated on a K-basis of L.This rank correspondence is the tool used for the distance bounds in Section 5.Source: §4 · p. 11, §4.3 · pp. 16–17

4.1 Dickson matrices associated with right multiplication

• The Dickson matrix represents right multiplication by F, and its L-rank is the dimension of the image of that multiplication map.Its rank does not depend on the chosen basis.Source: §4.1 · p. 11

4.2 Generalized Moore matrices and the sum-rank weight

• For a block vector v, the generalized Moore matrix has L-rank equal to the sum-rank weight of v.The proof reduces the matrix to the nonzero columns from each block and establishes their linear independence.Source: §4.2 · pp. 14, 16

4.3 The Dickson–Moore rank correspondence

• The Moore matrix’s first row is v, and its four blocks have K-ranks 1, 2, 2, and 4, respectively.The block ranks are rk_K(v(1,1)) = 1, rk_K(v(1,2)) = 2, rk_K(v(2,1)) = 2, and rk_K(v(2,2)) = 4.Source: §4.3 · p. 19

5 Sum-rank Reed–Muller-type codes

• Order-r codes consist of residue classes of skew polynomials of total degree at most r; their L-dimension counts exponent vectors i with 0 ≤ i_h < ν_h and |i| ≤ r.These are L-linear codes in the sum-rank ambient space (L[G])^t.Source: §5 · p. 20

5.1 Degree bounds for the sum-rank weight

• For reduced representatives of degree r, the Alon–Füredi-type bound gives sum-rank weight at least d(r, ν), and the minimum distance of the order-r code attains this bound.The exact minimum distance is d(r, ν), defined by minimizing the product of ν_i − u_i over admissible exponent vectors u with |u| ≤ r.Source: §5.1 · pp. 21, 23, 24

5.2 Lower-set sum-rank Reed–Muller-type codes

• For any nonempty lower-set support D, the corresponding code is spanned by its monomial classes and has L-dimension #D.The degree-bound and sharp-distance arguments extend because they require only that the support be closed under componentwise decrease.Source: §5.2 · p. 25
• The minimum sum-rank distance of a lower-set code is exactly d(D, ν), with the lower bound attained.The proof combines a lower bound for every nonzero codeword with a construction attaining it.Source: §5.2 · p. 25
• Positive weighted-degree supports D_ω(c) are lower sets, so weighted sum-rank Reed–Muller-type codes inherit the lower-set minimum-distance result.The weighted code is defined using these supports, and the corollary applies the lower-set theorem to them.Source: §5.2 · p. 26
• With unit weights and c = r, D_ω(r) = D_r, so the weighted construction includes the codes of Definition 5.2 as a special case.The corresponding codes satisfy L/K(r, m) = SRRM_L/K(r, m).Source: §5.2 · p. 26

• The section compares the proposed code family with classical constructions in the Hamming, rank, and sum-rank metrics.Source: §6 · p. 26

• For m = 1, the construction gives linearized Reed–Solomon codes with t blocks, each of length N.Source: §6.1 · p. 27
• For t = 1, it coincides with the σ-Reed–Muller code of order r, also identified as a rank-metric Reed–Muller code.Source: §6.1 · p. 27
• For N = 1, the skew polynomial ring becomes commutative and the code is an affine Cartesian evaluation code with the Hamming metric.Source: §6.1 · pp. 27–28
• For m = 1 and 0 ≤ r < Nt − 1, the resulting linearized Reed–Solomon code attains the sum-rank Singleton bound.Source: §6.1 · p. 27
• When m = t = 1, the construction gives a Gabidulin code of dimension r + 1 over the cyclic Galois extension L/K.Source: §6.1 · p. 28
• When N = m = 1, it gives a Reed–Solomon code with parameters [t, r + 1, t − r]_K.Source: §6.1 · p. 28

6.2 Comparison with linearized Reed–Muller codes

• After a monomial change of variables, every Berardini–Caruso code [8] is the image of a monomial code, determining its dimension and giving exact minimum distance when its support is a lower set.This reduces comparison of the two families to comparison of their monomial supports.Source: §6.2 · p. 28

6.2.2 Adapted bases and the identification

• If N is a prime power, every adapted basis has orders (1, . . . , 1, N), up to reordering the variables.For every extension, an adapted basis with these orders exists.Source: §6.2.2 · p. 29
• For any finite support S, an adapted basis gives a sum-rank equivalence between LRM(e; S) and a quotient-algebra monomial code; its dimension counts classes of S modulo (q − 1)Le.The equivalence acts blockwise by conjugation, which preserves rank.Source: §6.2.2 · pp. 29–30

6.2.3 Standard codes for rectangular central lattices

• For rectangular central lattices, standard codes have exact dimension and minimum-distance formulas, and the Berardini–Caruso bound [8] is exact when N is a prime power.For m = 2, the improved bound in [8, Remark 3.2] is sharp in the stated range c ≤ q − 2.Source: §6.2.3 · pp. 30–31

6.2.4 Lattice-simplex codes

• Lattice-simplex codes have exact minimum distance at every degree, while their dimension counts reduced support points rather than the full Ehrhart count.The parameters are determined for every degree over a fixed Fq; [8, Theorem 3.3] requires q ≥ c + 2.Source: §6.2.4 · p. 32

6.2.5 Hyperbolic supports

• For almost-commutative lattice-simplex codes, a hyperbolic-support code can have the same length and minimum distance but dimension at least as large.The hyperbolic supports are lower sets, and their codes correspond to compact convex supports that are generally not simplices.Source: §6.2.5 · p. 32

6.2.6 Differences between the two families

• Every LRM(e; S) is equivalent to a monomial code in the quotient-algebra framework, providing the basis for comparing the two families.The passage introduces differences between the families but does not specify them here.Source: §6.2.6 · p. 33
• Unlike [8], the construction applies over infinite fields and to finite Abelian Galois extensions; [8] uses all characters, while norm grids allow products of subsets of blocks.The extended almost-commutative codes of [8, Section 3.4.1], which evaluate commutative variables at 0, are not included because the norm grids lie in K∗.Source: §6.2.6 · p. 33
• Exact minimum distance remains open when reduced supports are not lower sets; for N = 2 and e = (1, 1), some standard-code supports fail this condition.For 1 ≤ c ≤ q − 2, the adapted basis b = ((1, −1), (0, 1)) gives S_b = {(a, s) | 0 ≤ a ≤ s ≤ c}, which is not a lower set.Source: §6.2.6 · p. 33
• Every linearized Reed–Muller code of [8] is a monomial code here, with dimension given in every degree and exact minimum sum-rank distance for adapted-coordinate lower-set supports.This extends [8, Theorems 3.1 and 3.3] beyond c ≤ q − 2.Source: §6.2.6 · p. 33
• Exact minimum distance is established for standard codes on rectangular central lattices and lattice-simplex codes associated with adapted bases, including the almost-commutative family.The results extend [8, Theorems 3.1 and 3.3] beyond the range c ≤ q − 2; lower-set codes with hyperbolic supports have larger dimension than almost-commutative lattice-simplex codes of the same length and minimum distance.Source: §6.2.6 · p. 33