The Clifford defect of a numerical semigroup

TL;DR

This paper investigates the Clifford defect, a rational number linked to algebraic curve semigroups, revealing its role in decoding error correction and providing explicit formulas for specific cases.

Contribution

It introduces explicit formulas for the Clifford defect of numerical semigroups from algebraic curves, enhancing understanding of their decoding capabilities.

Findings

Explicit formulas for Clifford defect of certain semigroups

Clifford defect relates to error-correcting performance

Applications extend to decoding algorithms and algebraic geometry

Abstract

The Clifford defect is a rational number associated to the Weierstrass semigroup at a given point of an algebraic curve. It describes the error-correcting capability of the so-called Modified Algorithm for decoding the corresponding one-point codes defined at the point. This defect also finds applications in other contexts involving one-point codes. We study the Clifford defect of some numerical semigroups arising from curves and give explicit formulas for them.