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arxiv: 2606.13307 · v1 · pith:4VRGGDJSnew · submitted 2026-06-11 · 🧮 math.NT

Discrete Fourier Transform Approach to Cyclically Covering Subspaces of mathbb{F}^n_q

Yangcheng Li , Pingzhi Yuan This is my paper

Pith reviewed 2026-06-27 05:35 UTC · model grok-4.3

classification 🧮 math.NT
keywords cyclically covering subspacesdiscrete Fourier transformcyclic codesfinite fieldscodimensionfull-weight codewordsprimitive roots
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The pith

When gcd(q,n)=1, h_q(n)=0 exactly when cyclic codes over F_q^n contain full-weight codewords

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

The paper establishes necessary and sufficient conditions for the maximum codimension h_q(n) of a cyclically covering subspace in F_q^n to equal zero. Using discrete Fourier transforms on the cyclic shifts, it proves this vanishing occurs precisely when full-weight codewords exist in the cyclic codes, but only when q and n share no common factors. The same approach yields a complete picture of h_q(n) for the case of prime q and n where q is a primitive root modulo n. These links connect the geometry of covering subspaces directly to the weight distribution of cyclic codes.

Core claim

When gcd(q,n)=1, h_q(n)=0 if and only if there exist full-weight codewords in cyclic codes of F_q^n. This equivalence follows from applying the discrete Fourier transform to the indicator functions of the subspace and its shifts. For primes q and n with n>q and q a primitive root modulo n, one obtains h_2(n) at least 2 together with h_q(n)=0 whenever q is an odd prime.

What carries the argument

Discrete Fourier transform converting the cyclic covering condition on subspaces into a weight condition on codewords of the associated cyclic codes

If this is right

  • The codimension satisfies the inequalities h_q(mn) at least the maximum of h_q(m) and h_q(n), and at least their sum.
  • Galois descent yields the inequality h_{q^m}(n) at most h_q(n).
  • Explicit constructions reach codimension floor of log_q of n.
  • Under the generalized Riemann hypothesis there exist infinitely many primes n where h_q(n)=0 for fixed prime q greater than 3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The DFT equivalence may let bounds from coding theory transfer to estimates of covering codimensions in other finite-field vector spaces.
  • The same transform technique could be tested on covering problems for non-cyclic group actions.
  • The average lower bounds obtained under GRH indicate that h_q(n) typically grows at least logarithmically with n.

Load-bearing premise

The coprimeness of q and n is required for the discrete Fourier transform to produce the stated equivalence.

What would settle it

For a small coprime pair such as q=2 and n=5, compute whether h_2(5) is zero and check whether the cyclic codes over F_2^5 contain any full-weight codeword; mismatch disproves the claimed equivalence.

read the original abstract

Let $q$ be a prime power and $n$ a positive integer. A subspace \( U \subseteq \mathbb{F}_q^n \) is called cyclically covering if the union of all its cyclic shifts covers the whole space \( \mathbb{F}_q^n \). Let \( h_q(n) \) denote the maximum possible codimension of such a subspace. When \(\gcd(q,n)=1\), we derive necessary and sufficient conditions for \(h_q(n)=0\) via Discrete Fourier Transforms, and prove this equality is equivalent to the existence of full-weight codewords in cyclic codes of \(\mathbb{F}_q^n\). We also characterize codimension-$k$ cyclically covering subspaces. Based on these results, we give a unified characterization of \(h_q(n)\) in the case where $q$ and $n$ are primes with \(n>q\) and $q$ being a primitive root modulo $n$. Specifically, \(h_2(n) \geq 2\) and \(h_q(n) = 0\) for \(q \neq 2\). We prove that \(h_3(n) \ge 1\) for every prime \(n > 3\) with odd \(\operatorname{ord}_n(3)\). Moreover, for any prime \(q > 3\), the Generalized Riemann Hypothesis implies the existence of infinitely many primes \(n > q\) such that $q$ is not a primitive root modulo $n$ and \(h_q(n) = 0\). We provide algebraic interpretations for the inequalities \(h_q(mn)\ge\max\{h_q(m),h_q(n)\}\) and \(h_q(mn)\ge h_q(m)+h_q(n)\). Using Galois descent, we prove \(h_{q^m}(n)\le h_q(n)\). Furthermore, we generalize a class of constructions that achieve the upper bound \(\lfloor\log_q(n)\rfloor\). Finally, under the Generalized Riemann Hypothesis, we obtain average lower bounds of \(h_q(n)\) for $q=2,3$.

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 / 3 minor

Summary. The manuscript defines h_q(n) as the maximum codimension of a cyclically covering subspace U of F_q^n (i.e., the cyclic shifts of U cover the whole space). When gcd(q,n)=1 it derives necessary and sufficient conditions for h_q(n)=0 via Discrete Fourier Transforms over extension fields and proves equivalence to the existence of full-weight codewords in cyclic codes of F_q^n; it also characterizes all codimension-k cyclically covering subspaces. For prime q,n with n>q and q a primitive root mod n it obtains the explicit values h_2(n)≥2 and h_q(n)=0 (q≠2). Further results include h_3(n)≥1 for primes n>3 with odd ord_n(3), a GRH-conditional infinitude statement for primes n where q is not primitive mod n and h_q(n)=0, algebraic interpretations of the inequalities h_q(mn)≥max{h_q(m),h_q(n)} and h_q(mn)≥h_q(m)+h_q(n), the Galois-descent inequality h_{q^m}(n)≤h_q(n), a generalization of constructions attaining ⌊log_q n⌋, and GRH-based average lower bounds on h_q(n) for q=2,3.

Significance. If the derivations hold, the DFT approach supplies a clean algebraic criterion linking a geometric covering property directly to the weight distribution of cyclic codes, which is a substantive contribution at the interface of finite geometry and coding theory. The explicit prime-case characterizations, the Galois-descent result, and the generalized constructions that meet the known upper bound are concrete strengths. The GRH-conditional infinitude and average-bound statements are noteworthy for exhibiting how arithmetic hypotheses control the function h_q(n).

minor comments (3)
  1. §2 (or the section introducing the DFT setup): the precise extension field containing the primitive n-th root of unity should be named explicitly when the basis change is performed, to make the separability argument fully self-contained.
  2. The statement of the main equivalence (h_q(n)=0 iff full-weight codewords exist) is correctly conditioned on gcd(q,n)=1, but a short remark reminding the reader why the DFT is undefined or non-bijective when the gcd>1 would prevent any misreading.
  3. In the GRH-conditional infinitude theorem, the precise form of the Dirichlet density or the effective version of the hypothesis being invoked should be stated once, even if only by reference, so that the logical dependence is transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed and positive summary of our work, the assessment of its significance at the interface of finite geometry and coding theory, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives necessary and sufficient conditions for h_q(n)=0 via Discrete Fourier Transforms (conditioned explicitly on gcd(q,n)=1) and proves equivalence to existence of full-weight codewords in cyclic codes of F_q^n as a derived algebraic property, not a definitional restatement. Prime-case characterizations (h_2(n)>=2, h_q(n)=0 for q!=2 when n>q and q primitive root mod n), the h_3(n)>=1 result, GRH-conditional infinitude statements, inequalities like h_q(mn)>=max{h_q(m),h_q(n)}, Galois descent h_{q^m}(n)<=h_q(n), and constructions achieving floor(log_q(n)) are all presented as consequences of the DFT framework or external tools/hypotheses, with no self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work by the same authors. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard axioms of finite field theory, discrete Fourier analysis over finite fields, and Galois theory for the descent result, plus the Generalized Riemann Hypothesis for the infinitude statements. No free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard properties of the discrete Fourier transform over finite fields when gcd(q,n)=1
    Invoked to derive the necessary and sufficient conditions for h_q(n)=0
  • domain assumption Generalized Riemann Hypothesis
    Used to obtain the existence of infinitely many primes n with h_q(n)=0 for prime q>3

pith-pipeline@v0.9.1-grok · 5922 in / 1550 out tokens · 32456 ms · 2026-06-27T05:35:06.677468+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

19 extracted references

  1. [1]

    Aaronson, C

    J. Aaronson, C. Groenland and T. Johnston,Cyclically covering subspaces inF n 2 , J. Comb. Theory, Ser. A 181 (2021) 105436

    2021

  2. [2]

    T. M. Apostol,Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York

    1976

  3. [3]

    Bourbaki,Algebra II: Chapters 4–7, Elements of Mathematics, (1990) Springer

    N. Bourbaki,Algebra II: Chapters 4–7, Elements of Mathematics, (1990) Springer. 38

    1990

  4. [4]

    Ballot,Density of prime divisors of linear recurrences, Memoirs of the A.M.S., vol

    C. Ballot,Density of prime divisors of linear recurrences, Memoirs of the A.M.S., vol. 115, Nu. 551 (1995)

    1995

  5. [5]

    Ballot,Counting monic irreducible polynomialsPinF q[X]for which order ofX(modP)is odd, J

    C. Ballot,Counting monic irreducible polynomialsPinF q[X]for which order ofX(modP)is odd, J. Th´ eor. Nombres Bordeaux 19 (2007), 41–58

    2007

  6. [6]

    P. J. Cameron,Problems from the Thirteenth British Combinatorial Conference, Discrete Math. 125, (1994) 407–417

    1994

  7. [7]

    P. J. Cameron, P. Frankl and W. M. Kantor,Intersecting families of finite sets and fixed-point-free 2-elements, Eur. J. Comb. 10 (1989) 149–160

    1989

  8. [8]

    P. J. Cameron, D. Ellis and W. Raynaud,Smallest cyclically covering subspaces ofF q n, and lower bounds in Isbell’s conjecture, Eur. J. Comb. 81 (2019) 242–255

    2019

  9. [9]

    D. R. Heath-Brown,Artin’s conjecture for primitive roots, Q. J. Math. 37 (1) (1986) 27–38

    1986

  10. [10]

    Hooley,On Artin’s conjecture, J

    C. Hooley,On Artin’s conjecture, J. Reine Angew. Math. 225 (1967) 209–220

    1967

  11. [11]

    Huang,On trivial cyclically covering subspaces ofF n q , Finite Fields Appl

    J. Huang,On trivial cyclically covering subspaces ofF n q , Finite Fields Appl. 96 (2024) 102423

    2024

  12. [12]

    J. R. Isbell,Homogeneous games, Math. Stud. 25 (1957) 123–128

    1957

  13. [13]

    J. R. Isbell,Homogeneous games, II, Proc. Amer. Math. Soc. 11 (1960) 159–161

    1960

  14. [14]

    H. W. Lenstra Jr., P. Moree and P. Stevenhagen,Character sums for primitive root densities, Math. Proc. Cambridge Philos. Soc. 157 (2014) 489–511

    2014

  15. [15]

    Li and P

    S. Li and P. Yuan,On trivial cyclically covering subspaces ofF n q in non-coprime characteristic, 2025, arXiv:2512.24301

    arXiv 2025

  16. [16]

    Y. C. Li, P. Z. Yuan, S. Li and Y. P. Zeng,On cyclically covering subspaces ofF n q , 2026, arXiv:2602.04558v1

    arXiv 2026

  17. [17]

    Luh,On the representation of vector spaces as a finite union of subspaces, Acta Math

    J. Luh,On the representation of vector spaces as a finite union of subspaces, Acta Math. Acad. Sci. Hungar. 23 (1972), 341–342

    1972

  18. [18]

    Moree,Near-primitive roots, Funct

    P. Moree,Near-primitive roots, Funct. Approx. Comment. Math. 48 (2013) 133–145

    2013

  19. [19]

    M. Sun, C. Ma and L. Zeng,Cyclically covering subspaces ofF qn, Finite Fields Appl. 106 (2025) 102625

    2025