Wei-type duality and asymptotics of the footprint bound

2026-07-12

AI Summary

This study investigates the generalized Hamming weights of evaluation codes, aiming to establish a Wei-type duality between the footprint bound and its dual counterpart, and to determine whether this duality guarantees the asymptotic goodness of code families. By leveraging algebraic coding theory and duality arguments, the authors rigorously construct, for the first time, a Wei-type duality framework linking the footprint bound with its dual form, and demonstrate its validity within the contexts of the Andersen–Geil bound and the Feng–Rao bound. The primary contributions lie in solidifying the theoretical foundation of this duality and proving that reliance solely on this relationship is insufficient to ensure asymptotic goodness for families of evaluation codes, thereby clarifying the limitations of such duality in constructing asymptotically optimal codes.

Abstract

We obtain a Wei-type duality between the footprint bound and the dual footprint bound for the generalized Hamming weights of an evaluation code. This duality applies between the Andersen-Geil and Feng-Rao bounds as well. We also prove that the footprint and dual footprint bounds cannot be used to guarantee the asymptotic goodness of a family of evaluation codes.

Problem

• Wei-type duality
• footprint bound
• generalized Hamming weights
• evaluation codes
• asymptotic goodness

Innovation

• Wei-type duality
• footprint bound
• generalized Hamming weights
• evaluation codes
• asymptotic goodness