The Non-asymptotic Landscape of Abelian Codes: A Framework for (F, G)-Goodness
摘要
While abelian group codes are classically known to be asymptotically bad—constrained by strict structural limitations such as the Square-Root and Logarithmic barriers—this traditional, infinite-length perspective obscures their rich mathematical behavior at finite block lengths. In this contribution, we move beyond the standard paradigm of asymptotic metrics to investigate the non-asymptotic goodness of abelian codes. We introduce the framework of (F, G)-goodness and a machinery that enables us to rigorously classify abelian codes into distinct performance “tiers”.