Explicit Constructions of Sum-Rank Metric Codes from Quadratic Kummer Extensions

Abstract: This paper presents new constructions of sum-rank metric codes derived from algebraic function fields, while the existing results about such codes remain limited. A central challenge in this field involves the determination of parameters. We address this challenge through quadratic Kummer extensions, and propose two general constructions of $2\times2$ sum-rank codes. The novel sum-rank metric code constructions derived from algebraic function fields demonstrate superior code lengths compared to conventional coding paradigms, particularly when contrasted with linearized Reed-Solomon codes under equivalent code parameter constraints. We also establish explicit parameters including dimensions and minimum distances of our codes. Finally, an illustrative example through elliptic function fields is constructed for validating the theoretical framework.