Core Concepts
The author presents results on Fqn-linear MRD codes of dimension three, addressing exceptional types and connections with algebraic varieties over finite fields.
Abstract
The content discusses the classification of Fqn-linear MRD codes, their properties, and connections to algebraic geometry. It explores exceptional scattered polynomials and their relation to MRD codes.
The study focuses on exceptional Fqn-linear MRD codes of type ⟨xqt, x + δxq2t, G(x)⟩ in Ln,q with detailed mathematical analysis.
Key points include rank metrics, scattered sequences, algebraic varieties, intersection multiplicities, and the investigation of branches at singular points.
Theoretical results are derived through algebraic transformations and geometric interpretations in the context of coding theory.
Stats
For a rank-metric code C containing at least two elements: d(C) = min(A,B∈C,A̸=B) rank(A-B)
Singleton-like bound: |C| ≤ qn(m−d+1)
Criteria for determining maximum rank-metric codes (MRD)
Definition and properties of scattered polynomials and sequences
Theorem linking Moore polynomial sets to MRD codes
Propositions on intersection multiplicities at singular points in algebraic curves