Rank metric codes from Drinfeld modules
Giacomo Micheli, Mihran Papikian
TL;DR
The paper develops a bridge between Drinfeld modules and rank-metric semifield codes by constructing $\ ext{F}_q$-linear subspaces of $\\mathrm{End}(\\phi)$ and examining their action on torsion points $\\phi[\\mathfrak{p}]$ via Tate modules; this yields semifield codes when the subspace dimension equals $r\\deg(\\mathfrak{p})$ and nonzero endomorphisms act invertibly. It provides a Drinfeld-module reinterpretation of Sheekey's construction, offering a concise proof of a key result and extending the framework to infinite families of semifield codes through division-algebra techniques and the Chebotarev density theorem. The work integrates Anderson motives, norm criteria on characteristic polynomials, and local-global invertibility criteria to certify semifield-code properties, and it culminates with explicit examples and a detailed analysis of the nuclear parameters of the resulting codes. Overall, the paper broadens the toolkit for constructing semifield codes by leveraging the algebraic structure of Drinfeld modules and related division algebras, with potential to yield previously unknown semifields through suitable parameter choices. The results enhance conceptual understanding and provide a pathway to new families of rank-metric semifield codes with provable structure and parameters.
Abstract
We establish a connection between Drinfeld modules and rank metric codes, focusing on the case of semifield codes. Our framework constructs rank metric codes from linear subspaces of endomorphisms of a Drinfeld module, using tools such as characteristic polynomials on Tate modules and the Chebotarev density theorem. We show that Sheekey's construction [She20] fits naturally into this setting, yielding a short conceptual proof of one of his main results. We then give a new construction of infinite families of semifield codes arising from Drinfeld modules defined over finite fields.