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arxiv: 2605.17371 · v2 · pith:YOOXTG47new · submitted 2026-05-17 · 💻 cs.IT · math.IT

Algebraic Resolutions of Seven Open Problems on Cyclic and Negacyclic Codes Supporting Designs

Yutong Zhang , Yaoran Yang This is my paper

Pith reviewed 2026-05-20 13:24 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords cyclic codesnegacyclic codesconstacyclic codesovoid codesMDS codescombinatorial designsprojective geometryBaer sublines
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The pith

Cayley parametrization reduces trace-zero conditions in cyclic codes to semilinear equations whose solutions are exactly the sublines of the projective line over a subfield.

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

This paper supplies algebraic resolutions for seven open problems on cyclic, negacyclic, and constacyclic codes that support combinatorial designs. A Cayley parametrization of the unit circle converts the trace-zero condition for the code with parameters ((p^s-1)/2, (p^s+1)/2) into a semilinear equation on PG(1,q), proving that the large root sets coincide with the F_{p^{gcd(m,s)}}-sublines and produce the complementary design overline{S(3,q_0+1,q+1)}. For negacyclic codes of length q^2+1, a quotient transport between unit groups together with a unit-circle parametrization shows that the minimum zero sets are precisely the Baer sublines of PG(1,q^2), which correspond to the complements of non-tangent plane sections of the elliptic quadric Q^-(3,q). The work also supplies an exact existence criterion for constacyclic ovoid codes and explicit constructions of consecutive-root negacyclic MDS codes that yield complete simple 5-designs.

Core claim

For the cyclic code C((p^s-1)/2,(p^s+1)/2), a Cayley parametrization of the unit circle reduces the trace-zero condition to a semilinear equation on PG(1,q) whose large root sets are exactly the F_{p^{gcd(m,s)}}-sublines, yielding the complementary design overline{S(3,q_0+1,q+1)}. For the length q^2+1 negacyclic code, a quotient transport from U_{2(q^2+1)} to U_{q^2+1} and unit-circle parametrization prove that the minimum zero sets are the Baer sublines of PG(1,q^2), so the support design is the complement of the non-tangent plane sections of the elliptic quadric Q^-(3,q). The exact existence criterion for lambda-constacyclic ovoid codes of length q^2+1 over F_q is that lambda belongs to F_

What carries the argument

Cayley parametrization of the unit circle that converts the trace-zero condition into a semilinear equation on PG(1,q), together with quotient transport between unit groups and a corrected projective-order congruence that determines the order of theta F_q^*.

Load-bearing premise

The large root sets of the cyclic code C((p^s-1)/2,(p^s+1)/2) after Cayley parametrization are exactly the F_{p^{gcd(m,s)}}-sublines of PG(1,q), and the analogous geometric identifications hold for Baer sublines and non-tangent sections in the negacyclic and ovoid cases.

What would settle it

Finding a square lambda in F_q^* for which a lambda-constacyclic ovoid code of length q^2+1 still exists, or verifying that the constructed negacyclic [11,5,7] code over F_23 has minimum distance exactly 7 and that its minimum supports form the complete 5-(11,7,15) design.

read the original abstract

This paper gives a unified algebraic solution to seven open problems of Wang, Tang and Ding on cyclic, negacyclic and constacyclic codes supporting designs. For the cyclic code \[ C\left(\frac{p^s-1}{2},\frac{p^s+1}{2}\right), \] a Cayley parametrization of the unit circle reduces the trace-zero condition to a semilinear equation on \(\PG(1,q)\). Its large root sets are exactly the \(\F_{p^{\gcd(m,s)}}\)-sublines, yielding the complementary design \[ \overline{S(3,q_0+1,q+1)}. \] For the length \(q^2+1\) negacyclic code, a quotient transport from \(\U_{2(q^2+1)}\) to \(\U_{q^2+1}\) and a unit-circle parametrization show that the minimum zero sets are precisely the Baer sublines of \(\PG(1,q^2)\). Equivalently, the corresponding support design is the complement of the non-tangent plane sections of an elliptic quadric \(\Q^-(3,q)\). For constacyclic ovoid codes of length \(q^2+1\) over \(\F_q\), the exact existence criterion is \[ \lambda\in\F_q^*,\qquad \exists\ \lambda\text{-constacyclic ovoid code} \Longleftrightarrow \lambda\notin(\F_q^*)^2. \] In particular, negacyclic ovoid codes exist exactly when \(q\equiv3\pmod4\). The proof uses the corrected projective-order congruence \[ a=(q+1)c,\qquad c\equiv b\pmod{q-1},\qquad \operatorname{ord}(\theta\F_q^*)=\frac{q^2+1}{\gcd(q^2+1,c)}. \] The paper also derives a universal weight enumerator for lifted ovoid codes over extension fields, independent of the chosen ovoid. Finally, consecutive-root negacyclic MDS codes are constructed to give complete simple \(5\)-designs, including a proper negacyclic \([11,5,7]_{23}\) code whose minimum supports form the complete \(5-(11,7,15)\) design.

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

2 major / 2 minor

Summary. This paper claims to provide a unified algebraic solution to seven open problems of Wang, Tang and Ding on cyclic, negacyclic and constacyclic codes supporting designs. For the cyclic code C((p^s-1)/2, (p^s+1)/2), a Cayley parametrization reduces the trace-zero condition to a semilinear equation on PG(1,q) whose large root sets are exactly the F_{p^{gcd(m,s)}}-sublines, yielding the complementary design S(3,q_0+1,q+1) bar. For the length q^2+1 negacyclic code, quotient transport from U_{2(q^2+1)} to U_{q^2+1} and unit-circle parametrization identify minimum zero sets with Baer sublines of PG(1,q^2), equivalently the complement of non-tangent plane sections of the elliptic quadric Q^-(3,q). For constacyclic ovoid codes of length q^2+1, the exact existence criterion is given as lambda in F_q^* with lambda not a square (hence negacyclic ovoid codes exist precisely when q ≡ 3 mod 4), using a corrected projective-order congruence a=(q+1)c, c≡b mod (q-1) for the order formula. The paper also derives a universal weight enumerator for lifted ovoid codes independent of the ovoid and constructs consecutive-root negacyclic MDS codes yielding complete simple 5-designs, including the explicit [11,5,7]_{23} code whose minimum supports form the complete 5-(11,7,15) design.

Significance. If the central root-set identifications hold, the manuscript resolves multiple longstanding open problems at the interface of algebraic coding theory and combinatorial design theory by supplying explicit geometric characterizations, corrected congruences, and parameter-free constructions. The universal weight enumerator and the explicit MDS-code example for a 5-design are concrete strengths that could facilitate further work on optimal codes and designs over finite fields.

major comments (2)
  1. [§3] §3 (cyclic-code section), after the Cayley parametrization: the reduction of the trace-zero condition to a semilinear equation on PG(1,q) is load-bearing for the claim that large root sets are exactly the F_{p^{gcd(m,s)}}-sublines; the manuscript must supply an explicit argument (e.g., counting or field-trace analysis) ruling out extraneous solutions to the semilinear equation, as any mismatch would invalidate the supported complementary S(3,q_0+1,q+1) design.
  2. [§5] §5 (constacyclic-ovoid section), the corrected projective-order congruence a=(q+1)c with c≡b mod (q-1): the derivation of the existence criterion λ∉(F_q^*)^2 from the order formula ord(θ F_q^*)=(q^2+1)/gcd(q^2+1,c) requires a self-contained verification that the congruence is applied without hidden assumptions on the generator θ, since this criterion is central to the resolution of the ovoid-code problem.
minor comments (2)
  1. [Preliminaries] The notation for the cyclic code C((p^s-1)/2,(p^s+1)/2) would benefit from an explicit reminder of the defining zeros or the generator polynomial in the preliminaries section to aid readers unfamiliar with the Wang-Tang-Ding problems.
  2. [Figures] Figure captions for the geometric illustrations of sublines and Baer sublines could include coordinate labels or explicit point counts to improve clarity when comparing algebraic root sets to geometric objects.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive report. The comments identify key points where additional explicit verification will strengthen the algebraic arguments. We address each major comment below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (cyclic-code section), after the Cayley parametrization: the reduction of the trace-zero condition to a semilinear equation on PG(1,q) is load-bearing for the claim that large root sets are exactly the F_{p^{gcd(m,s)}}-sublines; the manuscript must supply an explicit argument (e.g., counting or field-trace analysis) ruling out extraneous solutions to the semilinear equation, as any mismatch would invalidate the supported complementary S(3,q_0+1,q+1) design.

    Authors: We agree that an explicit argument is required to confirm that the semilinear equation admits no extraneous solutions of the requisite cardinality. In the revised version we will insert a self-contained counting lemma immediately after the Cayley parametrization. The lemma will use the fact that the equation is a linear fractional transformation over the subfield F_{p^{gcd(m,s)}} together with a direct enumeration of fixed points and orbits under the Frobenius action, showing that any solution set of size q_0+1 must coincide with an F_{p^{gcd(m,s)}}-subline. This establishes the identification rigorously and preserves the complementary design claim. revision: yes

  2. Referee: [§5] §5 (constacyclic-ovoid section), the corrected projective-order congruence a=(q+1)c with c≡b mod (q-1): the derivation of the existence criterion λ∉(F_q^*)^2 from the order formula ord(θ F_q^*)=(q^2+1)/gcd(q^2+1,c) requires a self-contained verification that the congruence is applied without hidden assumptions on the generator θ, since this criterion is central to the resolution of the ovoid-code problem.

    Authors: We appreciate the referee’s request for a fully self-contained verification. The congruence a=(q+1)c, c≡b mod (q-1) arises from equating the projective order in the quotient group F_{q^2+1}^*/F_q^* with the order of the image of a primitive element θ. In the revision we will add a short proposition that derives the order formula directly from the structure of the cyclic group of order q^2+1 and the kernel of the norm map to F_q^*, without assuming any special property of θ beyond it being a generator of the multiplicative group. The argument uses only the standard formula ord(gH)=ord(g)/gcd(ord(g),|H|) for a cyclic group G with subgroup H, thereby eliminating any hidden assumptions and confirming the criterion λ∉(F_q^*)^2. revision: yes

Circularity Check

0 steps flagged

No circularity; algebraic derivations are self-contained using standard finite-field and projective-geometry facts

full rationale

The paper reduces trace-zero and minimum-zero conditions via explicit Cayley and unit-circle parametrizations to semilinear equations over PG(1,q) or PG(1,q²), then proves that the solution sets coincide exactly with the indicated sublines or Baer sublines by direct verification of the resulting equations. These steps invoke only the field trace, norm, and order formulas together with the geometry of quadrics; no parameter is fitted to data and then relabeled a prediction, no result is defined in terms of itself, and no load-bearing claim rests on a self-citation chain. The constacyclic existence criterion follows from the corrected projective-order congruence, which is derived internally rather than imported. The constructions of consecutive-root MDS codes and the universal weight enumerator are likewise obtained by direct algebraic manipulation. Consequently the central claims do not collapse to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard background from finite field theory and projective geometry with no apparent free parameters fitted to data or new invented entities; the corrected congruence is presented as a fix rather than an ad-hoc addition.

axioms (1)
  • standard math Standard properties of finite fields, projective lines PG(1,q), and unitary groups U_n(q) hold as background.
    Invoked for parametrizations, root sets, and order calculations throughout the described solutions.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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