First-Order Generalized Reed–Muller Codes

• First-Order Generalized Reed–Muller Codes are q-ary linear codes defined by evaluating affine functions on F_q^m, with parameters including length q^m, dimension m+1, and minimum distance q^(m-1)(q-1).
• Their algebraic structure supports multiple formulations—affine-invariant, cyclic, and modular-algebra—which enable explicit parameter determination and tailored decoding strategies.
• These codes exhibit distinctive design and permutation decoding properties, serving as a benchmark for duality, covering radius analysis, and information-set constructions in coding theory.

First-order generalized Reed–Muller codes are the $q$-ary linear codes obtained by evaluating all affine functions on $\mathbb{F}_q^m$. In one standard notation,

$\mathrm{RM}_q(1,m)=\{\Eval(f): f\in \mathbb{F}_q[x_1,\dots,x_m],\ \deg(f)\le 1\},$

while equivalent formulations write the same family as $R_q(1,m)$ or $C_1(m,q)$, depending on whether the emphasis is on evaluation, affine-invariant, cyclic, or modular-algebra realizations. In the full-length form the code has length $q^m$, dimension $m+1$, and minimum distance $q^{m-1}(q-1)$; in the binary specialization $\mathrm{RM}(1,m)$ these become $[2^m,1+m,2^{m-1}]$, and the punctured or primitive version of length $\mathbb{F}_q^m$0 is frequently used in cyclic and affine-invariant treatments (Abdon et al., 2015, Bernal, 2024, Andriatahiny, 2016).

1. Definition, parameters, and notation

Let $\mathbb{F}_q^m$1. A first-order generalized Reed–Muller codeword is the evaluation vector of an affine function

$\mathbb{F}_q^m$2

over all $\mathbb{F}_q^m$3. The resulting code is the set of all affine maps on $\mathbb{F}_q^m$4, and its most basic parameters are therefore determined by the $\mathbb{F}_q^m$5 linear coefficients together with one constant coefficient: the dimension is $\mathbb{F}_q^m$6, the length is $\mathbb{F}_q^m$7, and the minimum distance is $\mathbb{F}_q^m$8 (Abdon et al., 2015, Andriatahiny, 2016).

Several notational conventions coexist. The full evaluation code is commonly written $\mathbb{F}_q^m$9. In affine-invariant and cyclic formulations, the primitive version is written $\mathrm{RM}_q(1,m)=\{\Eval(f): f\in \mathbb{F}_q[x_1,\dots,x_m],\ \deg(f)\le 1\},$0 and has length $\mathrm{RM}_q(1,m)=\{\Eval(f): f\in \mathbb{F}_q[x_1,\dots,x_m],\ \deg(f)\le 1\},$1. In that setting the code is described as an affine-invariant code in $\mathrm{RM}_q(1,m)=\{\Eval(f): f\in \mathbb{F}_q[x_1,\dots,x_m],\ \deg(f)\le 1\},$2, where $\mathrm{RM}_q(1,m)=\{\Eval(f): f\in \mathbb{F}_q[x_1,\dots,x_m],\ \deg(f)\le 1\},$3 is the additive group of $\mathrm{RM}_q(1,m)=\{\Eval(f): f\in \mathbb{F}_q[x_1,\dots,x_m],\ \deg(f)\le 1\},$4, with defining set

$\mathrm{RM}_q(1,m)=\{\Eval(f): f\in \mathbb{F}_q[x_1,\dots,x_m],\ \deg(f)\le 1\},$5

dimension $\mathrm{RM}_q(1,m)=\{\Eval(f): f\in \mathbb{F}_q[x_1,\dots,x_m],\ \deg(f)\le 1\},$6, minimum distance $\mathrm{RM}_q(1,m)=\{\Eval(f): f\in \mathbb{F}_q[x_1,\dots,x_m],\ \deg(f)\le 1\},$7, and dual

$\mathrm{RM}_q(1,m)=\{\Eval(f): f\in \mathbb{F}_q[x_1,\dots,x_m],\ \deg(f)\le 1\},$8

This punctured/primitive description is the one used in recent work on information sets and permutation decoding (Bernal et al., 15 Sep 2025).

The binary specialization is exceptionally rigid. The first-order Reed–Muller code $\mathrm{RM}_q(1,m)=\{\Eval(f): f\in \mathbb{F}_q[x_1,\dots,x_m],\ \deg(f)\le 1\},$9 is a binary linear $R_q(1,m)$0 code, and any binary linear code with these parameters is equivalent to the first-order Reed–Muller code. The punctured code is the binary Simplex code, its dual is the Hamming code, and the full code has weight distribution consisting of one all-zero codeword, one all-one codeword, and $R_q(1,m)$1 codewords of weight $R_q(1,m)$2 (0901.2062).

2. Algebraic realizations and structural descriptions

A central feature of first-order generalized Reed–Muller codes is that they admit several mutually compatible algebraic descriptions. In the affine-invariant viewpoint, the code is determined by its defining set and is naturally related to punctured cyclic codes and multidimensional abelian codes. For first-order codes the defining set structure simplifies substantially: in recent constructions of information sets, the first-order case is singled out precisely because the defining set is governed by indices of $R_q(1,m)$3-weight less than $R_q(1,m)$4, and this permits direct constructions in terms of the basic parameters $R_q(1,m)$5, $R_q(1,m)$6, and factorizations of $R_q(1,m)$7 (Bernal, 2024, Bernal et al., 2024).

A second formulation places generalized Reed–Muller codes inside a modular algebra

$R_q(1,m)$8

whose radical $R_q(1,m)$9 is generated by the classes $C_1(m,q)$0. Over a prime field $C_1(m,q)$1, the classical Berman–Charpin identification holds: $C_1(m,q)$2 Hence, in the prime-field case the first-order code satisfies

$C_1(m,q)$3

Over non-prime fields $C_1(m,q)$4 with $C_1(m,q)$5, however, the identification with radical powers fails in general: only

$C_1(m,q)$6

remain valid, and $C_1(m,q)$7 in general. This distinction is one of the principal algebraic differences between prime-field and non-prime-field first-order generalized Reed–Muller codes (Andriatahiny, 2016).

A closely related group-algebra formulation uses

$C_1(m,q)$8

where $C_1(m,q)$9 is the additive group of a Galois ring. In this setting the first-order code $q^m$0 is the image, under a linear map $q^m$1, of the ideal generated by

$q^m$2

and a basis is given by

$q^m$3

The same framework also provides the Jennings monomials

$q^m$4

which organize the radical powers and clarify why the first-order code generally fails to coincide with the full radical outside the prime-field case (Andriatahiny et al., 2016).

3. Enumerators, Jacobi polynomials, and design-theoretic properties

The shells

$q^m$5

of first-order generalized Reed–Muller codes have been studied through refined enumerator polynomials. For a code $q^m$6 of length $q^m$7 and a subset $q^m$8, the Jacobi polynomial is

$q^m$9

where $m+1$0 count zero and nonzero entries on $m+1$1, and $m+1$2 count zero and nonzero entries on the complement of $m+1$3. For linear codes there is a MacWilliams-type relation

$m+1$4

and a shell $m+1$5 is a combinatorial $m+1$6-design if and only if the coefficient of $m+1$7 in $m+1$8 is independent of $m+1$9 with $q^{m-1}(q-1)$0 (Yamaguchi, 2023).

For first-order $q^{m-1}(q-1)$1-ary generalized Reed–Muller codes, explicit Jacobi polynomials have been obtained for $q^{m-1}(q-1)$2. The principal consequence is a sharp design-theoretic boundary. The shells are combinatorial $q^{m-1}(q-1)$3-designs for any weight, but if $q^{m-1}(q-1)$4 and $q^{m-1}(q-1)$5, then for every $q^{m-1}(q-1)$6 the shell $q^{m-1}(q-1)$7 is not a combinatorial $q^{m-1}(q-1)$8-design. The same work also identifies refined nonuniform design behavior: for the shell of weight $q^{m-1}(q-1)$9, the shell is a combinatorial $\mathrm{RM}(1,m)$0-design and similarly a combinatorial $\mathrm{RM}(1,m)$1-design, but not a uniform $\mathrm{RM}(1,m)$2- or $\mathrm{RM}(1,m)$3-design (Yamaguchi, 2023).

In the binary case, Jacobi polynomials and harmonic weight enumerators have been computed explicitly for $\mathrm{RM}(1,m)$4 and for its dual, the extended Hamming code $\mathrm{RM}(1,m)$5. The harmonic weight enumerator associated with a harmonic function $\mathrm{RM}(1,m)$6 of degree $\mathrm{RM}(1,m)$7 is

$\mathrm{RM}(1,m)$8

and satisfies a MacWilliams relation of the form

$\mathrm{RM}(1,m)$9

These tools recover the previously known fact that the shells of $[2^m,1+m,2^{m-1}]$0 support $[2^m,1+m,2^{m-1}]$1-designs via the Assmus–Mattson theorem, and they prove that no shell of $[2^m,1+m,2^{m-1}]$2 or of $[2^m,1+m,2^{m-1}]$3 supports a combinatorial $[2^m,1+m,2^{m-1}]$4-design (Miezaki et al., 2023).

4. Distance to affine codewords and covering radius

Because first-order generalized Reed–Muller codes are precisely the affine functions, distance computation to the code can be organized as a simultaneous distance-to-affinity problem. For a function $[2^m,1+m,2^{m-1}]$5, define

$[2^m,1+m,2^{m-1}]$6

so that

$[2^m,1+m,2^{m-1}]$7

The transform-based method developed for this problem works in the group algebra $[2^m,1+m,2^{m-1}]$8: with $[2^m,1+m,2^{m-1}]$9, the transform

$\mathbb{F}_q^m$00

has coefficients exactly equal to the numbers $\mathbb{F}_q^m$01. The double transform yields a linear system with $\mathbb{F}_q^m$02 equations in the $\mathbb{F}_q^m$03 unknowns $\mathbb{F}_q^m$04, augmented by the normalization equations

$\mathbb{F}_q^m$05

The resulting Cramer system has a unique solution, and therefore all Hamming distances from $\mathbb{F}_q^m$06 to all codewords of $\mathbb{F}_q^m$07 can be computed simultaneously (Abdon et al., 2015).

The covering radius problem has also been analyzed directly. Writing $\mathbb{F}_q^m$08 for the covering radius of the first-order generalized Reed–Muller code and

$\mathbb{F}_q^m$09

one has the explicit formula

$\mathbb{F}_q^m$10

The stated general bounds are

$\mathbb{F}_q^m$11

and when $\mathbb{F}_q^m$12 is even,

$\mathbb{F}_q^m$13

For $\mathbb{F}_q^m$14 the paper further gives $\mathbb{F}_q^m$15 and $\mathbb{F}_q^m$16, obtained through explicit computation and recursive arguments (Leducq, 2011).

5. Information sets and permutation decoding

First-order generalized Reed–Muller codes occupy a particularly tractable position among affine-invariant codes because recent work constructs information sets directly from the defining set. A typical setup factors

$\mathbb{F}_q^m$17

and fixes an isomorphism $\mathbb{F}_q^m$18. One construction defines

$\mathbb{F}_q^m$19

and then

$\mathbb{F}_q^m$20

which is an information set for $\mathbb{F}_q^m$21. A related construction starts from a set of check positions for the dual code and obtains the information set as $\mathbb{F}_q^m$22. For first-order codes these constructions are described as valid for any $\mathbb{F}_q^m$23 (Bernal et al., 15 Sep 2025, Bernal, 2024).

These information sets support strengthened permutation-decoding algorithms. In the binary case, a variation of classical permutation decoding is formulated for affine-invariant codes and applied to first-order Reed–Muller codes through $\mathbb{F}_q^m$24-PD-like sets. If $\mathbb{F}_q^m$25 is a suitable information set and $\mathbb{F}_q^m$26 an $\mathbb{F}_q^m$27-PD-like set, Algorithm II corrects up to $\mathbb{F}_q^m$28 errors. For first-order binary Reed–Muller codes, the group generated by the automorphism $\mathbb{F}_q^m$29 forms the relevant PD-like set, and the number of correctable errors is

$\mathbb{F}_q^m$30

The paper reports that this improves considerably the number of errors that can be corrected in comparison with known results (Bernal et al., 2023).

The q-ary extension studies the same idea for first-order generalized Reed–Muller codes. With the information set above and the translation group generated by $\mathbb{F}_q^m$31, the punctured code $\mathbb{F}_q^m$32 admits an $\mathbb{F}_q^m$33-PD-like set with

$\mathbb{F}_q^m$34

The stated comparison bound for previous translation-based permutation decoding is

$\mathbb{F}_q^m$35

and the newer construction achieves $\mathbb{F}_q^m$36 for many parameter choices. The same paper also analyzes when a smaller subgroup generated by $\mathbb{F}_q^m$37 suffices probabilistically, with success probability

$\mathbb{F}_q^m$38

where $\mathbb{F}_q^m$39 is the set of $\mathbb{F}_q^m$40-subsets whose second-coordinate projection does not cover all of $\mathbb{F}_q^m$41 (Bernal et al., 15 Sep 2025).

6. Associated algebraic invariants and exceptional features

First-order generalized Reed–Muller codes also appear as a distinguished case in the commutative-algebraic study of code invariants. To a linear code $\mathbb{F}_q^m$42 one associates a simplicial complex and its Stanley–Reisner ring

$\mathbb{F}_q^m$43

The minimal graded free resolution of $\mathbb{F}_q^m$44 is called pure when, in each homological degree, all nonzero graded Betti numbers occur in a single shift. For generalized Reed–Muller codes $\mathbb{F}_q^m$45, the complete characterization is that the resolution is pure if and only if $\mathbb{F}_q^m$46, $\mathbb{F}_q^m$47, $\mathbb{F}_q^m$48, $\mathbb{F}_q^m$49, or $\mathbb{F}_q^m$50. Thus the first-order case $\mathbb{F}_q^m$51 is uniformly pure for all $\mathbb{F}_q^m$52 and all $\mathbb{F}_q^m$53, and its shifts are the generalized Hamming weights of the code (Ghorpade et al., 2021).

The first-order level is therefore exceptional in several independent senses. It is the point at which the code remains describable by affine functions, retains dimension $\mathbb{F}_q^m$54, supports direct information-set constructions from the defining set, admits explicit simultaneous distance computation to all codewords, and yields strong negative design-theoretic results that are sharp at low $\mathbb{F}_q^m$55: $\mathbb{F}_q^m$56-designs do occur in the $\mathbb{F}_q^m$57-ary setting, $\mathbb{F}_q^m$58-designs do not for $\mathbb{F}_q^m$59, and in the binary setting $\mathbb{F}_q^m$60-designs occur while $\mathbb{F}_q^m$61-designs do not (Yamaguchi, 2023, Miezaki et al., 2023). Over prime fields it also coincides with a radical power of the modular algebra, whereas over non-prime fields that identification fails in general, which marks a fundamental algebraic boundary inside the generalized Reed–Muller hierarchy (Andriatahiny, 2016).

Within the broader Reed–Muller family, the first-order generalized code is thus both the simplest nontrivial member and a structurally singular one: its geometry is controlled by affine hyperplanes, its algebra is unusually explicit, and a large part of the modern literature uses it as the baseline case for duality, design theory, information-set constructions, covering-radius analysis, and permutation decoding.