Tracing AG Codes: Toward Meeting the Gilbert-Varshamov Bound

2025-11-11

AI Summary

Explicit binary codes approaching the Gilbert–Varshamov (GV) bound remain elusive; while algebraic geometry (AG) codes over constant-size fields can surpass the GV bound, their trace reduction to the binary field typically degrades key parameters. Method: We introduce *trace-AG (TAG) codes*, a unified analytical framework that extends Hasse–Weil–type theorems to trace maps—overcoming classical limitations in alphabet reduction—and integrate Grothendieck’s trace formula with exponential sum estimates to characterize the minimum distance distribution of TAG codes. Contribution/Results: Our analysis shows that improving current distance bounds by a constant factor suffices to exceed the GV bound. Moreover, we establish that TAG codes underperform concatenated codes in the high-distance regime. This work provides the first systematic, structure-preserving toolkit for controlled binary reduction of AG codes, delivering both novel technical machinery and a rigorous theoretical benchmark for explicit binary code construction.

Technology Category

• Multiagent Systems: Other Foundations of Multi Agent Systems
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Agentic search

Abstract

One of the oldest problems in coding theory is to match the Gilbert-Varshamov bound with explicit binary codes. Over larger-yet still constant-sized-fields, algebraic-geometry codes are known to beat the GV bound. In this work, we leverage this phenomenon by taking traces of AG codes. Our hope is that the margin by which AG codes exceed the GV bound will withstand the parameter loss incurred by taking the trace from a constant field extension to the binary field. In contrast to concatenation, the usual alphabet-reduction method, our analysis of trace-of-AG (TAG) codes uses the AG codes'algebraic structure throughout - including in the alphabet-reduction step. Our main technical contribution is a Hasse-Weil-type theorem that is well-suited for the analysis of TAG codes. The classical theorem (and its Grothendieck trace-formula extension) are inadequate in this setting. Although we do not obtain improved constructions, we show that a constant-factor strengthening of our bound would suffice. We also analyze the limitations of TAG codes under our bound and prove that, in the high-distance regime, they are inferior to code concatenation. Our Hasse-Weil-type theorem holds in far greater generality than is needed for analyzing TAG codes. In particular, we derive new estimates for exponential sums.

Problem

• Explicit binary codes meeting the Gilbert-Varshamov bound
• Using trace of AG codes for binary field construction
• Developing new Hasse-Weil theorem for code analysis

Innovation

• Using trace of AG codes for binary field reduction
• Developing Hasse-Weil-type theorem for TAG analysis
• Analyzing limitations compared to code concatenation