Tracing AG Codes: Toward Meeting the Gilbert-Varshamov Bound

TL;DR

This paper explores the use of traces of algebraic-geometry codes to approach the Gilbert-Varshamov bound over binary fields, leveraging algebraic structures and new theorems to analyze their potential and limitations.

Contribution

It introduces a novel analysis of trace-of-AG (TAG) codes using a new Hasse-Weil-type theorem, providing insights into their performance relative to classical bounds.

Findings

TAG codes can potentially surpass the GV bound after tracing from larger fields.

A new Hasse-Weil-type theorem is developed for analyzing TAG codes.

TAG codes are less effective than concatenation in high-distance regimes.

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.