Hyper-derivative algebraic geometry codes extend error correction possibilities
Hyper-derivative Algebraic Geometry Codes via Local Expansions
Information Theory
Summary
Error-correcting codes help send information reliably over noisy channels. This paper extends a special kind of code, called hyper-derivative Reed-Solomon codes, from a simple setting to more complex algebraic curves. The authors develop new formulas to understand how these codes behave and find conditions when the codes have special symmetry properties. They also establish limits on how well these codes can perform as they grow larger using advanced mathematical tools.
What this means in practice
- •For data storage engineers: Design codes with better error detection and correction using advanced algebraic geometry techniques for higher reliability in storage devices.
- •For communication system designers: Construct robust error-correcting codes with controllable symmetry properties to improve secure or efficient data transmission protocols.
Authors
Xiaofeng Liu, Hengfeng Liu, Jun Zhang, Fang-Wei Fu, Chunming Tang
Abstract
In this paper, we develop a systematic construction framework of hyper-derivative algebraic geometry codes via local expansions, extending hyper-derivative Reed-Solomon codes from the rational function field to general algebraic function fields. Using the residue theorem, we determine their Euclidean duals and illustrate that the duals naturally reverse. We further give criteria for reverse self orthogonality and reverse self duality in terms of two classes of bilinear forms. Finally, we provide an asymptotic bound on the rate and relative distance via function field towers.