PreprintCC BY 4.0arXiv
Skew cyclic codes through Drinfeld modules
Abstract
We construct skew cyclic codes in the rank metric from the torsion of the supersingular Drinfeld module ϕT=t+τn over 𝔽qr, where t is a root of a monic irreducible p0∈𝔽q[T] of degree r coprime to n. For every prime p≠p0 of degree d and every divisor m of n, the torsion ϕ[p] is a free module over the subring R0=κ[πm] generated by a power of the Frobenius π=τr, κ=𝔽q[T]/p, and the codes are the left ideals of the centralizer R≅Matm(R0) of R0. We prove a lattice anti-isomorphism between codes and R0-submodules of ϕ[p], and a rank-BCH bound whose designed distance is the true distance.
Related
- A General Construction of Codes from Drinfeld Modules
- The Pila-Zannier strategy for Drinfeld modules and Drinfeld modular curves
- Coset representatives corresponding to Yetter-Drinfeld modules of modular group and continued fraction
- Cyclicity of Drinfeld modules modulo primes in arithmetic progressions
- Rank-Two Drinfeld Modules over Elliptic Coordinate Rings
BibTeX
@misc{micheli2026skew,
title = {{Skew cyclic codes through Drinfeld modules}},
author = {Giacomo Micheli and Mihran Papikian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32906},
eprint = {2609.32906},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32906},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Skew cyclic codes through Drinfeld modules
AU - Giacomo Micheli
AU - Mihran Papikian
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32906
UR - https://arxiv.org/abs/2609.32906
AB - We construct skew cyclic codes in the rank metric from the torsion of the supersingular Drinfeld module ϕ_T=t+τ^n over 𝔽_q^r, where t is a root of a monic irreducible p_0∈𝔽_q[T] of degree r coprime to n. For every prime p≠p_0 of degree d and every divisor m of n, the torsion ϕ[p] is a free module over the subring R_0=κ[π^m] generated by a power of the Frobenius π=τ^r, κ=𝔽_q[T]/p, and the codes are the left ideals of the centralizer R≅Mat_m(R_0) of R_0. We prove a lattice anti-isomorphism between codes and R_0-submodules of ϕ[p], and a rank-BCH bound whose designed distance is the true distance.
ER -