A General Construction of Codes from Drinfeld Modules
Abstract
We construct additive rank-metric and sum-rank-metric codes from Drinfeld modules by restricting bounded-degree morphisms to prime-to-characteristic torsion. For supersingular Drinfeld modules of rank r in characteristic p of degree d, the stabilization formula for morphism spaces yields rank-metric codes of 𝔽q-dimension mrt−c and minimum distance r−t+1, where c=r(r−1)(d−1)/2. Simultaneous restriction to ℓ distinct degree-m torsion modules gives additive sum-rank codes of the same dimension and minimum distance at least ℓ r−t+1. Their normalized Singleton defects tend to zero, while in characteristic (T) the module ϕT=τr makes the defect vanish and produces an explicit MSRD family. We identify this family with a skew Chinese remainder theorem code supported on central skew polynomials and prove that its poly-skew weight is exactly m times its sum-rank weight. This gives a specialized Singleton-type bound and a polynomial-time unique decoder up to the full sum-rank unique-decoding radius. We also derive a Welch-Berlekamp-type filter equation for the general supersingular sum-rank construction; it becomes an effective decoder whenever bases of the relevant morphism spaces and the restriction maps are computable.
Related
- Skew cyclic codes through 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{giannoni2026general,
title = {{A General Construction of Codes from Drinfeld Modules}},
author = {Alessandro Giannoni and Giacomo Micheli and Mihran Papikian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01484},
eprint = {2609.01484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01484},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A General Construction of Codes from Drinfeld Modules
AU - Alessandro Giannoni
AU - Giacomo Micheli
AU - Mihran Papikian
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01484
UR - https://arxiv.org/abs/2609.01484
AB - We construct additive rank-metric and sum-rank-metric codes from Drinfeld modules by restricting bounded-degree morphisms to prime-to-characteristic torsion. For supersingular Drinfeld modules of rank r in characteristic p of degree d, the stabilization formula for morphism spaces yields rank-metric codes of 𝔽_q-dimension mrt−c and minimum distance r−t+1, where c=r(r−1)(d−1)/2. Simultaneous restriction to ℓ distinct degree-m torsion modules gives additive sum-rank codes of the same dimension and minimum distance at least ℓ r−t+1. Their normalized Singleton defects tend to zero, while in characteristic (T) the module ϕ_T=τ^r makes the defect vanish and produces an explicit MSRD family. We identify this family with a skew Chinese remainder theorem code supported on central skew polynomials and prove that its poly-skew weight is exactly m times its sum-rank weight. This gives a specialized Singleton-type bound and a polynomial-time unique decoder up to the full sum-rank unique-decoding radius. We also derive a Welch-Berlekamp-type filter equation for the general supersingular sum-rank construction; it becomes an effective decoder whenever bases of the relevant morphism spaces and the restriction maps are computable.
ER -