The Clifford defect of a numerical semigroup

TL;DR

Addresses the Clifford defect for numerical semigroups by generalizing from Weierstrass semigroups and defining the σ-function to encode the defect for one-point codes. Develops general properties, including symmetry effects and behavior under maximal embedding dimension, and derives explicit defect formulas for several semigroup families (Pedersen–Sørensen, Klein, Hermitian quotients, norm-trace, Suzuki). Provides exact maximizers and closed-form σ-values that clarify the defect’s role in decoding error-correction and dimension bounds, with applications to decoding strategies and code performance. Concludes with open questions, notably the Clifford defect for semigroups with two generators, and suggests directions for future work.

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.

Figures (5)

• Figure 1: The Clifford defect of $S = \langle 15,16,17 \rangle$ is attained at $45$, shown in red.
• Figure 2: Klein semigroup for $m = 10$. The Clifford defect is attained at $46$, drawn in red.
• Figure 3: The Clifford defect for the semigroup $S = \langle 7,13 \rangle$ is attained at $39$, shown in red.
• Figure 4: The Clifford defect of the Pedersen-Sørensen semigroup for $q_0 = t = 5$ is attained at $g = 1550$, shown in red.
• Figure 5: The Clifford defect of the Suzuki semigroup for $q_0 = 8$ is attained at $g = 952$, drawn in red.

Theorems & Definitions (73)

• Remark 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Remark 2.4
• Proposition 2.5: fidalgo2024distributed
• Lemma 2.6
• proof
• Proposition 2.7
• proof