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PreprintCC BY 4.0arXiv

Cyclic Codes of Length 7_p^s over F_p^m + uF_p^m : Characterization, Duals, and Applications to Quantum and LCD Codes

  • Payel Chandra
  • Kalyan Hansda
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Abstract

Let R2 = 𝔽pm + u𝔽pm (u2 = 0), where p is an odd prime and m, s ∈ β„•. For p ≑ 3, 5 pmod 7 with gcd(m, 6) = 1, the cyclotomic polynomial Ξ¦7(x) is irreducible over 𝔽pm. This yields a direct sum decomposition C = C1 βŠ• C2 for any cyclic code C of length 7ps over R2, where C1 has length ps and C2 is a $7$-cyclotomic code of length 6ps. We classify $7$-cyclotomic codes into four disjoint generator-based types and calculate exact cardinalities using residue and torsion subcodes. Furthermore, explicit generators for the Euclidean dual codes CβŠ₯ are determined. As operational applications of these classified codes, we construct new families of quantum stabilizer codes via the CSS framework and establish parameter criteria for linear codes with complementary duals (LCD codes).

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Cite

BibTeXRIS
BibTeX
@misc{chandra2026cyclic,
  title = {{Cyclic Codes of Length 7_p^s over F_p^m + uF_p^m : Characterization, Duals, and Applications to Quantum and LCD Codes}},
  author = {Payel Chandra and Kalyan Hansda},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07181},
  eprint = {2609.07181},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07181},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Cyclic Codes of Length 7_p^s over F_p^m + uF_p^m : Characterization, Duals, and Applications to Quantum and LCD Codes
AU  - Payel Chandra
AU  - Kalyan Hansda
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07181
UR  - https://arxiv.org/abs/2609.07181
AB  - Let R_2 = 𝔽_p^m + u𝔽_p^m (u^2 = 0), where p is an odd prime and m, s ∈ β„•. For p ≑ 3, 5 pmod 7 with gcd(m, 6) = 1, the cyclotomic polynomial Ξ¦_7(x) is irreducible over 𝔽_p^m. This yields a direct sum decomposition C = C_1 βŠ• C_2 for any cyclic code C of length 7p^s over R_2, where C_1 has length p^s and C_2 is a $7$-cyclotomic code of length 6p^s. We classify $7$-cyclotomic codes into four disjoint generator-based types and calculate exact cardinalities using residue and torsion subcodes. Furthermore, explicit generators for the Euclidean dual codes C^βŠ₯ are determined. As operational applications of these classified codes, we construct new families of quantum stabilizer codes via the CSS framework and establish parameter criteria for linear codes with complementary duals (LCD codes).
ER  -