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arxiv: 2605.17022 · v1 · pith:V4NHCTIZnew · submitted 2026-05-16 · 💻 cs.IT · math.IT

Intermediate Constacyclic Codes and Scalar-Residue Reed--Muller Layers

Yaoran Yang , Yutong Zhang This is my paper

Pith reviewed 2026-05-19 18:33 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords constacyclic codesminimum distanceReed-Muller codesfinite fieldsscalar residue layersgeneralized Reed-MullerBCH bound
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The pith

The minimum distance of intermediate constacyclic codes equals an explicit case formula in the field size q and the parameters a and b of the degree ℓ.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the exact minimum distance for constacyclic codes over finite fields in the intermediate range where the parameter r satisfies 2 less than r less than q minus 1. It proves that this distance equals (q minus 1) over r multiplied by (q minus b plus 1) times q to the power of m minus a minus 2 for a up to m minus 2, and a reduced expression when a equals m minus 1. The work further identifies a dichotomy in the minimum affine supports of scalar-residue layers inside generalized Reed-Muller codes. Readers would care because these distances fix the number of errors the codes can correct.

Core claim

Under the conditions ℓ equals (q minus 1)a plus b less than (q minus 1)m minus 1 with 0 less than or equal to b less than or equal to q minus 2 and b congruent to r minus 1 modulo r, the minimum distance of C(q,m,r,ℓ) equals (q minus 1) over r times (q minus b plus 1) times q to the power m minus a minus 2 when a is at most m minus 2, and equals (q minus b plus r minus 2) over r when a equals m minus 1. The minimum affine support of every non-terminal scalar-residue layer of a generalized Reed-Muller code equals the second Reed-Muller weight except for residue classes 0 and 1.

What carries the argument

The hidden scalar homogeneity of the evaluation model, which enables orbit-counting obstructions on minimum supports and homogeneous pencil constructions that attain the second weight.

Load-bearing premise

The evaluation model for these codes admits a hidden scalar homogeneity that supports orbit counting to obstruct smaller supports and allows homogeneous pencil constructions to reach the second weight.

What would settle it

Direct computation of the minimum Hamming weight in the code C(q,m,r,ℓ) for small concrete values such as q=9, r=4, m=3 and an admissible ℓ, followed by comparison to the predicted distance value.

read the original abstract

A 2024 paper of Sun, Ding and Wang introduced a second class of constacyclic codes over finite fields, denoted $C(q,m,r,\ell)$, with length $(q^m-1)/r$, where $r\mid(q-1)$ and the defining monomials have total $q$-ary degree congruent to $r-1$ modulo $r$. In the non-projective intermediate range $2<r<q-1$ the paper gave a sharp-looking upper bound and a BCH-type lower bound, and left the minimum distance open. We prove that the upper bound is the exact minimum distance for every admissible intermediate parameter. More precisely, if $\ell=(q-1)a+b<(q-1)m-1$, $0\le b\le q-2$, and $b\equiv r-1\pmod r$, then, for every prime power $q$, every divisor $r$ of $q-1$ with $2<r<q-1$, and every $m\ge2$, \[ d(C(q,m,r,\ell))= \begin{cases} \displaystyle \frac{q-1}{r}(q-b+1)q^{m-a-2},&0\le a\le m-2,\\[1mm] \displaystyle \frac{q-b+r-2}{r},&a=m-1. \end{cases} \] The first line settles the open problem of Sun, Ding and Wang; the second line is the terminal case already forced by their BCH bound. We also determine the minimum affine support of every non-terminal scalar-residue layer of a generalized Reed--Muller code. The resulting dichotomy says that the first Reed--Muller weight survives exactly for residue classes $0$ and $1$, while every other residue-matched layer starts at the second Reed--Muller weight. The proof uses the hidden scalar homogeneity of the evaluation model, an orbit-counting obstruction for minimum Reed--Muller supports, and a homogeneous pencil construction that attains the second weight.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves that the minimum distance of the intermediate constacyclic codes C(q,m,r,ℓ) equals the stated piecewise formula: (q-1)/r *(q-b+1)q^{m-a-2} for 0≤a≤m-2 and (q-b+r-2)/r for a=m-1, under the given conditions on ℓ, q, r, m. It additionally determines the minimum affine support of every non-terminal scalar-residue layer of a generalized Reed-Muller code, establishing a dichotomy that the first Reed-Muller weight occurs precisely for residue classes 0 and 1 while all other residue-matched layers begin at the second weight. The proof relies on scalar homogeneity of the evaluation map, an orbit-counting obstruction on supports, and an explicit homogeneous pencil construction attaining the upper bound.

Significance. If the result holds, it closes the open problem left by Sun, Ding and Wang on the exact distance in the non-projective intermediate range 2<r<q-1. The uniform treatment across the non-terminal and terminal cases, together with the new dichotomy for Reed-Muller supports, strengthens the understanding of constacyclic and generalized Reed-Muller codes. The manuscript supplies a complete, self-contained proof with no hidden parameter restrictions, which is a clear strength.

minor comments (2)
  1. [§2.3] §2.3, after Definition 2.4: the notation for the scalar-residue layer could be introduced one sentence earlier to improve flow when the orbit-counting argument begins.
  2. [§4.2] The homogeneous pencil construction in §4.2 is correct but would benefit from an explicit small-field example (e.g., q=7, r=3, m=2) to illustrate how the linear combination of monomials is chosen.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript, for highlighting the resolution of the open problem left by Sun, Ding and Wang, and for recommending acceptance. The positive assessment of the uniform treatment and the new dichotomy on scalar-residue layers is appreciated.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper delivers a direct mathematical proof establishing the exact minimum distance of the intermediate constacyclic codes C(q,m,r,ℓ) for the stated parameter ranges. It invokes the scalar homogeneity of the evaluation map as an intrinsic property of the Reed-Muller evaluation model, then applies an orbit-counting obstruction to obtain the lower bound and constructs an explicit homogeneous pencil of monomials to attain the matching upper bound. Neither direction reduces to a fitted parameter, a self-definitional quantity, or a load-bearing self-citation; the cited prior work of Sun-Ding-Wang supplies only the code definition and the open upper-bound conjecture, while the new arguments are self-contained and uniformly applicable to both the non-terminal and terminal cases. The derivation therefore stands as an independent proof rather than a renaming or reconstruction of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works entirely inside the standard algebraic framework of finite fields, polynomial evaluation, and linear codes. No new free parameters are fitted, no invented entities are postulated, and the axioms invoked are the usual properties of fields and vector spaces.

axioms (1)
  • standard math Standard algebraic properties of finite fields, their multiplicative groups, and polynomial rings over them
    The definitions of constacyclic codes and generalized Reed-Muller codes rest on these background facts about finite fields.

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Reference graph

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