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PreprintCC BY 4.0arXiv

Weight spectra of some families of GRM codes

  • Minjia Shi
  • Zhaokang Xing
  • Patrick Sole
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Abstract

Let RMq(r,m) denote the generalized Reed--Muller code of order r and length qm over the finite field 𝔽q. We completely determine the weight spectra of RM3(2m−i,m) for i=3,4, RM4(3m−i,m) for i=2,3, RM5(4m−4,m), and RM7(6m−4,m), in the ranges of m specified in the corresponding results. The proofs proceed by induction on m, using a sumset inclusion for GRM weight spectra together with results on low-weight codewords. The required base cases are established by computations in Magma and, for certain ternary cases, exact constraint solving with Z3.

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Cite

BibTeXRIS
BibTeX
@misc{shi2026weight,
  title = {{Weight spectra of some families of GRM codes}},
  author = {Minjia Shi and Zhaokang Xing and Patrick Sole},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13653},
  eprint = {2609.13653},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13653},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Weight spectra of some families of GRM codes
AU  - Minjia Shi
AU  - Zhaokang Xing
AU  - Patrick Sole
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.13653
UR  - https://arxiv.org/abs/2609.13653
AB  - Let RM_q(r,m) denote the generalized Reed--Muller code of order r and length q^m over the finite field 𝔽_q. We completely determine the weight spectra of RM_3(2m−i,m) for i=3,4, RM_4(3m−i,m) for i=2,3, RM_5(4m−4,m), and RM_7(6m−4,m), in the ranges of m specified in the corresponding results. The proofs proceed by induction on m, using a sumset inclusion for GRM weight spectra together with results on low-weight codewords. The required base cases are established by computations in Magma and, for certain ternary cases, exact constraint solving with Z3.
ER  -