The weight enumerators of irreducible cyclic codes of length 2^ℓ

@misc{podesta2026weight,
  title = {{The weight enumerators of irreducible cyclic codes of length $2^\ell$}},
  author = {Ricardo A. Podestá and Denis E. Videla},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14107},
  eprint = {2610.14107},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14107},
}

TY  - UNPB
TI  - The weight enumerators of irreducible cyclic codes of length 2^ℓ
AU  - Ricardo A. Podestá
AU  - Denis E. Videla
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.14107
UR  - https://arxiv.org/abs/2610.14107
AB  - Using the spectral relation between irreducible cyclic codes (ICC) and generalized Paley (GP) graphs recently found by the authors, we study the weight distribution of irreducible cyclic codes of length 2^ℓ, ℓ ∈ ℕ. These codes were previously studied by Sharma, Bakshi and Raka, who divided them into two families (each having two cases). First we characterize the GP-graphs spectrally related to these families of ICCs: they are complete, semiprimitive, Hamming, or Cartesian products of semiprimitive GP-graphs. We then compute the spectrum of these graphs and hence, by the mentioned spectral relation we finally obtain neat explicit expressions for the weight enumerators of all the irreducible cyclic codes of length 2^ℓ with ℓ ∈ ℕ.
ER  -