Tracing AG Codes: Toward Meeting the Gilbert-Varshamov Bound
Gil Cohen, Dean Doron, Noam Goldgraber, Tomer Manket
TL;DR
This work investigates trace-based alphabet reduction for algebraic-geometry codes (TAG codes) as a route to meeting or exceeding the Gilbert–Varshamov bound over binary alphabets. A key technical achievement is a Hasse–Weil–type bound tailored to TAG codes, producing distance guarantees that depend on the extension degree Delta rather than the base curve genus, and extending to Artin–Schreier and Kummer extensions to bound exponential sums. The results yield concrete TAG-instantiations on Hermitian, Norm–Trace, and Hermitian-tower function fields, including epsilon-balanced codes with explicit rate–distance tradeoffs, while showing limitations in the high-distance regime where concatenation with Hadamard can outperform TAG. The paper thus clarifies both the potential and the boundaries of trace-based AG-code constructions for achieving GV-type performance in the binary setting, and it provides a toolkit (bounds on N_L, exponential sums) with broader applications in coding theory and finite-field arithmetic.
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.