Frobenius profiles of polynomial-based maximum rank distance codes studied
Frobenius Profiles of Polynomial-Multiple MRD Codes
Information Theory
Summary
This paper explores special patterns called Frobenius intersection profiles in certain error-correcting codes known as maximum rank distance (MRD) codes. These codes are built using polynomials over finite fields, and the authors show that all possible profiles (within certain numerical limits) can be achieved with their constructions. They provide formulas to describe the structure and give bounds on how large the underlying fields need to be. They also identify which codes belong to the well-known Gabidulin family and discover new examples outside of that family with specific profile properties.
What this means in practice
- •For coding theory engineers: Design MRD codes with specific Frobenius intersection profiles to optimize error correction in data transmission over large finite fields.
- •For cryptography developers: Implement rank-metric based cryptographic primitives that use polynomial-based MRD codes with guaranteed existence of target structural profiles over large extensions.
A theory result. No direct application yet.
Authors
Yangcheng Li, Pingzhi Yuan
Abstract
We study Frobenius intersection profiles of maximum rank distance (MRD) codes arising from ordinary polynomial multiples over finite fields. We prove that every numerically admissible profile is realized within this family over every sufficiently large extension, using a root-exponent pattern independent of the extension degree. A factor-orbit gap formula provides the underlying structural description, while effective extension-degree bounds guarantee realization. We also establish a lower bound valid for all polynomial generators and show that the upper and lower bounds differ by only $O_q(\log n)$ along an infinite family of parameters. Finally, we characterize the generalized Gabidulin members and obtain non-generalized-Gabidulin realizations of every profile with initial loss at least two over infinitely many extensions.