Pith Number

pith:YOOXTG47

pith:2026:YOOXTG47UWNOJEVD3FV6CG2KJI

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

Yaoran Yang, Yutong Zhang

Cayley parametrizations and corrected projective-order congruences resolve seven open problems by giving exact existence criteria for constacyclic ovoid codes and constructions of negacyclic MDS codes that support complete 5-designs.

arxiv:2605.17371 v1 · 2026-05-17 · cs.IT · math.IT

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Claims

C1strongest claim

This paper gives a unified algebraic solution to seven open problems of Wang, Tang and Ding on cyclic, negacyclic and constacyclic codes supporting designs, including the exact existence criterion for constacyclic ovoid codes and constructions of consecutive-root negacyclic MDS codes yielding complete simple 5-designs.

C2weakest assumption

The 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, and the corrected projective-order congruence a=(q+1)c with c≡b mod (q-1) holds and applies to determine the orders for the ovoid code existence proof.

C3one line summary

Algebraically resolves seven open problems on cyclic and negacyclic codes supporting designs via Cayley parametrizations, quotient transports, and corrected congruences, yielding existence criteria for ovoid codes and MDS constructions for 5-designs.

References

12 extracted · 12 resolved · 0 Pith anchors

[1] Infinite families of cyclic and negacyclic codes supporting 3-designs, 2023

[2] E. F. Assmus, Jr. and H. F. Mattson, Jr., “New 5-designs,”Journal of Combinatorial Theory, vol. 6, no. 2, pp. 122–151, Mar. 1969 1969

[3] C. Ding and C. Tang,Designs From Linear Codes, 2nd ed. Singapore: World Scientific, 2022 2022

[4] W. C. Huffman and V . Pless,Fundamentals of Error-Correcting Codes. Cambridge, U.K.: Cambridge University Press, 2003 2003

[5] F. J. MacWilliams and N. J. A. Sloane,The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977 1977

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:55.046133Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c39d799b9fa59ae492a3d96be11b4a4a0a66414511a3935089e654a103c7fe65

Aliases

arxiv: 2605.17371 · arxiv_version: 2605.17371v1 · doi: 10.48550/arxiv.2605.17371 · pith_short_12: YOOXTG47UWNO · pith_short_16: YOOXTG47UWNOJEVD · pith_short_8: YOOXTG47

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/YOOXTG47UWNOJEVD3FV6CG2KJI \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: c39d799b9fa59ae492a3d96be11b4a4a0a66414511a3935089e654a103c7fe65

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "0b11ca84c421dece5f83ca319442937cf732f72608e550cd5fbfcb9674d8467b",
    "cross_cats_sorted": [
      "math.IT"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "primary_cat": "cs.IT",
    "submitted_at": "2026-05-17T10:27:58Z",
    "title_canon_sha256": "4d825e2fd60fcb1be16d7d46ff3a64b4e2e3115ee1384a6a98168e5c2eeaf067"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17371",
    "kind": "arxiv",
    "version": 1
  }
}