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Auxiliary Codes and the Generalized Packing-Covering Conjecture

  • Isaac Barouch Essayag
  • Aryeh Lev Zabokritskiy
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Abstract

The generalized packing--covering conjecture asks whether, at every order, the packing radius of a linear code is at most its covering radius. We prove the conjecture for every linear code of redundancy at most fourteen over every finite field, extending the previously established range of redundancies at most seven. We also prove the bound on generalized Hamming weights dt(C)≤2Rt(C)+1 whenever the alphabet size q satisfies q≥ Rt(C). Both results use an auxiliary-code criterion that converts a covering property in the syndrome space into a weight bound. For binary primitive BCH codes, the packing radius is strictly smaller than the covering radius for every fixed error parameter and order, both at least two, once the extension degree is sufficiently large; this follows from existing covering bounds.

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Cite

BibTeXRIS
BibTeX
@misc{essayag2026auxiliary,
  title = {{Auxiliary Codes and the Generalized Packing-Covering Conjecture}},
  author = {Isaac Barouch Essayag and Aryeh Lev Zabokritskiy},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19098},
  eprint = {2609.19098},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19098},
}
RIS
TY  - UNPB
TI  - Auxiliary Codes and the Generalized Packing-Covering Conjecture
AU  - Isaac Barouch Essayag
AU  - Aryeh Lev Zabokritskiy
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.19098
UR  - https://arxiv.org/abs/2609.19098
AB  - The generalized packing--covering conjecture asks whether, at every order, the packing radius of a linear code is at most its covering radius. We prove the conjecture for every linear code of redundancy at most fourteen over every finite field, extending the previously established range of redundancies at most seven. We also prove the bound on generalized Hamming weights d_t(C)≤2R_t(C)+1 whenever the alphabet size q satisfies q≥ R_t(C). Both results use an auxiliary-code criterion that converts a covering property in the syndrome space into a weight bound. For binary primitive BCH codes, the packing radius is strictly smaller than the covering radius for every fixed error parameter and order, both at least two, once the extension degree is sufficiently large; this follows from existing covering bounds.
ER  -