Technology Category
Machine Learning: Quantum Machine LearningKnowledge Representation and Reasoning: Qualitative ReasoningConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
On QC and GQC algebraic geometry codes
🤖 AI Summary
This work addresses the limitation of existing constructions of quasi-cyclic (QC) and generalized quasi-cyclic (GQC) algebraic geometry codes, which have been confined to elliptic curves. By systematically incorporating Kummer extension curves—including hyperelliptic curves, norm-trace curves, and Hermitian curves—and leveraging their automorphism group structures together with Kummer theory, the authors present the first framework for constructing QC/GQC codes from a much broader class of algebraic curves. The proposed approach enables flexible co-index design and yields explicit formulas for code parameters, thereby substantially expanding both the applicability and the construction flexibility of algebraic geometry codes.
Technology Category
Application Category
📝 Abstract
We present new constructions of quasi-cyclic (QC) and generalized quasi-cyclic (GQC) codes from algebraic curves. Unlike previous approaches based on elliptic curves, our method applies to curves that are Kummer extensions of the rational function field, including hyperelliptic, norm-trace, and Hermitian curves. This allows QC codes with flexible co-index. Explicit parameter formulas are derived using known automorphism-group classifications.
Problem
Research questions and friction points this paper is trying to address.
quasi-cyclic codes
generalized quasi-cyclic codes
algebraic geometry codes
Kummer extensions
flexible co-index
Innovation
Methods, ideas, or system contributions that make the work stand out.
quasi-cyclic codes
generalized quasi-cyclic codes
algebraic geometry codes
Kummer extensions
automorphism groups
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