Skip to content
DraconiticResearch
Search Sources API
190,332 studies31 sourcesUpdated 16 min ago
PreprintCC BY 4.0HAL

A proof of the generalized packing-covering conjecture

  • Gianira N. Alfarano
  • Giuseppe Marino
  • Alessandro Neri
  • Rocco Trombetti
Open at sourcePDFDOICite

Abstract

The generalized packing--covering conjecture of Elimelech, Firer and Schwartz asserts that, for every linear code π’ž and every admissible order t, the t-th generalized Hamming weight dt(π’ž) and the t-th generalized covering radius Rt(π’ž) satisfy dt(π’ž)≀ 2Rt(π’ž)+2. We give a computer-assisted proof of the conjecture for every linear code over every finite field and every admissible order. Combining a parity-check reformulation of the conjecture, bounds on the length of putative counterexamples, and successive puncturing arguments, we settle all orders tβ‰₯ 32 and reduce the remaining orders to finitely many parameter tuples, which we exclude by an exact computer verification.

Related

  1. Auxiliary Codes and the Generalized Packing-Covering Conjecture
    Isaac Barouch Essayag, Aryeh Lev Zabokritskiy
    arXivPreprintarXiv licenceFull textGeneral mathematicsarXiv
  2. On the Generalized Packing and Covering Radii of Codes
    Wenjun Yu, Moshe Schwartz
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. Proof of the generalized Chui conjecture
    Egor Bezrukov
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. Proof of a Conjecture of Cui, Gu and Tang on 18-Colored Generalized Frobenius Partitions
    Hirakjyoti Das, Hemjyoti Nath, Abhishek Sarma
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  5. Proof of Nobori's generalized BΓΆttcher-Wenzel inequality conjecture
    Shuo Shi, Juan Zhang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{alfarano2026proof,
  title = {{A proof of the generalized packing-covering conjecture}},
  author = {Gianira N. Alfarano and Giuseppe Marino and Alessandro Neri and Rocco Trombetti},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.34910},
  eprint = {2609.34910},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05772903v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A proof of the generalized packing-covering conjecture
AU  - Gianira N. Alfarano
AU  - Giuseppe Marino
AU  - Alessandro Neri
AU  - Rocco Trombetti
PY  - 2026
DA  - 2026/09/28
PB  - HAL
DO  - 10.48550/arxiv.2609.34910
UR  - https://hal.science/hal-05772903v1
AB  - The generalized packing--covering conjecture of Elimelech, Firer and Schwartz asserts that, for every linear code π’ž and every admissible order t, the t-th generalized Hamming weight d_t(π’ž) and the t-th generalized covering radius R_t(π’ž) satisfy d_t(π’ž)≀ 2R_t(π’ž)+2. We give a computer-assisted proof of the conjecture for every linear code over every finite field and every admissible order. Combining a parity-check reformulation of the conjecture, bounds on the length of putative counterexamples, and successive puncturing arguments, we settle all orders tβ‰₯ 32 and reduce the remaining orders to finitely many parameter tuples, which we exclude by an exact computer verification.
ER  -