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Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments

  • Himadri Shekhar Chakraborty
  • Mehedi Hasan Tanver
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Abstract

We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.

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Cite

BibTeXRIS
BibTeX
@misc{chakraborty2026harmonic,
  title = {{Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments}},
  author = {Himadri Shekhar Chakraborty and Mehedi Hasan Tanver},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.15864},
  eprint = {2608.15864},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.15864},
}
RIS
TY  - UNPB
TI  - Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments
AU  - Himadri Shekhar Chakraborty
AU  - Mehedi Hasan Tanver
PY  - 2026
DA  - 2026/08/21
PB  - arXiv
DO  - 10.48550/arxiv.2608.15864
UR  - https://arxiv.org/abs/2608.15864
AB  - We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.
ER  -