On QC and GQC algebraic geometry codes

2026-02-04

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

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Applications of cryptography

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

• quasi-cyclic codes
• generalized quasi-cyclic codes
• algebraic geometry codes
• Kummer extensions
• flexible co-index

Innovation

• quasi-cyclic codes
• generalized quasi-cyclic codes
• algebraic geometry codes
• Kummer extensions
• automorphism groups