A Note on the Inheritance of the Isometry-Dual Property under Puncturing AG Codes

TL;DR

This paper investigates conditions under which the isometry-dual property of algebraic geometry codes is preserved after puncturing, linking it to the structure of the Weierstrass semigroup at a defining point.

Contribution

It establishes a necessary condition involving maximum sparse ideals of the Weierstrass semigroup for the isometry-dual property to be maintained after puncturing AG codes.

Findings

Identifies a necessary condition related to maximum sparse ideals.

Connects the preservation of the isometry-dual property to algebraic properties of the semigroup.

Provides insights into the structure of AG codes under puncturing.

Abstract

Consider a sequence of AG codes evaluating at a set of evaluation points $P_1,\dots,P_n$ the functions having only poles at a defining point $Q$, with the sequence of codes satisfying the isometry-dual condition (i.e. containing at the same time primal and their dual codes). We prove a necessary condition under which, after taking out a number of evaluation points (i.e. puncturing), the resulting AG codes can still satisfy the isometry-dual property. The condition has to do with the so-called maximum sparse ideals of the Weierstrass semigroup of $Q$.