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arxiv: 2607.02239 · v1 · pith:Q6Q2A32Onew · submitted 2026-07-02 · 🧮 math.NT

Cyclic Codes and Cyclically Covering Subspaces over Finite Fields

Yangcheng Li , Pingzhi Yuan This is my paper

Pith reviewed 2026-07-03 06:39 UTC · model grok-4.3

classification 🧮 math.NT
keywords cyclic codescyclically covering subspacesfinite fieldsfull-weight codewordscodimensioncyclic shiftsweight bounds
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The pith

h_q(n) is zero exactly when every nonzero cyclic code over F_q of length n has a full-weight codeword.

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

The paper connects the covering behavior of subspaces under cyclic shifts to the weight distribution of cyclic codes. It proves that the largest codimension h_q(n) of a cyclically covering subspace equals zero if and only if every nonzero cyclic code in F_q^n contains a codeword of weight n. When h_q(n) is positive the paper supplies sharp upper bounds on the largest weight of any cyclic code that avoids weight n, together with explicit examples attaining the bounds. It also determines the exact number of cyclic codes without full-weight codewords over F_2, gives lower bounds over F_3, and uses the latter to show that h_q((q^m + 1)/2) is positive for every odd prime q at least 3 and every integer m at least 4.

Core claim

A subspace U of F_q^n is cyclically covering when the union of its cyclic shifts equals the whole space. The quantity h_q(n) is the maximum codimension of any such subspace. The paper shows h_q(n) equals zero if and only if every nonzero cyclic code in F_q^n contains a codeword of weight n. When h_q(n) exceeds zero it derives sharp bounds on the maximum weight attained by cyclic codes lacking any weight-n word and supplies explicit attaining examples; it further counts the cyclic codes without full-weight words completely over F_2 and gives lower bounds over F_3, from which it deduces that h_q((q^m + 1)/2) > 0 whenever q is an odd prime at least 3 and m is an integer at least 4.

What carries the argument

The equivalence between the codimension of cyclically covering subspaces and the presence of full-weight codewords in cyclic codes, realized through the linear cyclic shift operator on F_q^n.

If this is right

  • Sharp bounds hold for the maximum weight of cyclic codes without full-weight codewords precisely when h_q(n) is positive.
  • The number of cyclic codes without full-weight codewords is completely determined over F_2.
  • Lower bounds on the number of cyclic codes without full-weight codewords hold over F_3.
  • h_q((q^m + 1)/2) is positive for every odd prime q at least 3 and every integer m at least 4.

Where Pith is reading between the lines

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

  • The same correspondence may allow weight-distribution techniques from coding theory to decide covering questions for other linear group actions on vector spaces.
  • The explicit existence result for lengths of the form (q^m + 1)/2 indicates that positive h_q(n) occurs for many n that are neither prime nor a power of the field characteristic.
  • The counting formulas over small fields supply a way to estimate the density of cyclic codes that avoid full weight, which could be compared against random-code heuristics.
  • Direct computation of h_q(n) and the cyclic-code weight sets for small q and n would give an immediate check of the claimed equivalence.

Load-bearing premise

The cyclically covering property of subspaces is completely characterized by the weight properties of the associated cyclic codes under the standard cyclic shift operator.

What would settle it

An explicit nonzero cyclic code of length n over F_q that contains no weight-n codeword in a case where h_q(n) is asserted to be zero, or a cyclically covering subspace of positive codimension in a case where every nonzero cyclic code is asserted to have a full-weight codeword.

read the original abstract

Let \(q\) be a power of a prime \(p\), and let \(n\) be a positive integer. A subspace \(U\subseteq \mathbb F_q^n\) is called cyclically covering if the union of all its cyclic shifts covers \(\mathbb F_q^n\), and \(h_q(n)\) denotes the maximum possible codimension of such a subspace. This paper studies cyclically covering subspaces via cyclic codes. We first prove that \(h_q(n)=0\) if and only if every nonzero cyclic code in \(\mathbb F_q^n\) contains a full-weight codeword. We also relate \(h_q(n)\) to the maximum weights of cyclic codes. In particular, when \(h_q(n)>0\), we obtain sharp bounds for the maximum weight of cyclic codes without full-weight codewords and provide explicit examples attaining these bounds. Moreover, we study the number of cyclic codes containing no full-weight codeword. We determine this number completely over \(\mathbb F_2\), and give lower bounds over \(\mathbb F_3\). From this, we prove that if \(q\ge 3\) is an odd prime and \(m\ge 4\) is an integer, then \(h_q\left(\frac{q^m+1}{2}\right)>0\).

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 defines cyclically covering subspaces U of F_q^n (union of cyclic shifts covers the whole space) and studies h_q(n), the maximum codimension of such a U. It proves that h_q(n)=0 if and only if every nonzero cyclic code in F_q^n has a full-weight codeword, relates h_q(n) to the maximum weights of cyclic codes without full-weight words, derives sharp bounds and explicit examples when h_q(n)>0, determines the number of cyclic codes without full-weight words completely over F_2 and gives lower bounds over F_3, and proves that h_q((q^m+1)/2)>0 whenever q>=3 is an odd prime and m>=4.

Significance. If the stated equivalence and bounds hold, the work supplies a clean dictionary between cyclically covering subspaces and weight properties of cyclic codes, together with explicit constructions, complete enumeration over F_2, and a concrete existence result for odd-prime fields. These are concrete, falsifiable statements in finite-field coding theory that could be used to test covering-radius questions or to construct constant-weight codes.

minor comments (2)
  1. The abstract states that the number of cyclic codes without full-weight codewords is determined completely over F_2; the corresponding theorem statement and proof should be cross-referenced to the section that contains the enumeration formula.
  2. Notation for the cyclic shift operator and the associated code is introduced early; a single displayed definition (perhaps as Eq. (1) or (2)) would help readers who skip the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for recommending acceptance. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

Derivation is self-contained with no circular reductions

full rationale

The paper defines cyclically covering subspaces directly via unions of cyclic shifts and proves the equivalence h_q(n)=0 iff every nonzero cyclic code has a full-weight codeword as a theorem using standard properties of the cyclic shift operator on F_q^n. Bounds on maximum weights, explicit examples, and the count of codes without full-weight codewords (including the lower bound for h_q((q^m+1)/2)>0) are all derived from this relation and field arithmetic without any fitted parameters, self-referential constructions, or load-bearing self-citations. The central claims rest on explicit constructions and linear algebra over finite fields rather than reducing to inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper rests entirely on standard axioms from linear algebra and finite-field theory; no free parameters or new entities are introduced.

axioms (3)
  • standard math F_q^n is a vector space over the finite field F_q with the standard cyclic shift operator that rotates coordinates.
    Defines the cyclic shifts and covering union central to h_q(n).
  • standard math Cyclic codes are linear subspaces of F_q^n that are invariant under the cyclic shift, equivalently ideals in the quotient ring F_q[x]/(x^n-1).
    Enables the translation between subspaces and codes used throughout.
  • domain assumption A full-weight codeword is a nonzero vector with every coordinate nonzero.
    Definition required for the equivalence and weight bounds.

pith-pipeline@v0.9.1-grok · 5750 in / 1556 out tokens · 52301 ms · 2026-07-03T06:39:10.945681+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

14 extracted references · 3 canonical work pages · 1 internal anchor

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