Generalized Skew Multivariate Goppa Codes

2026-05-07

AI Summary

This study investigates the parameter analysis of a novel class of algebraic geometry codes—generalized skew multivariate Goppa codes. For the first time, multivariate Ore polynomials are introduced into the construction of Goppa codes, establishing a structural connection with subfield subcodes of generalized skew Reed–Solomon codes. This relationship is elucidated through a novel parity-check matrix that reveals their underlying algebraic correspondence. Building on this framework, the work derives theoretical bounds on the dimension and minimum distance of these codes, thereby providing rigorous theoretical support for their application as efficient error-correcting codes and opening a new avenue for parameter analysis in algebraic coding theory.

Abstract

We introduce Generalized Skew Multivariate Goppa codes relying on the theory of multivariate Ore polynomials. These codes contain, as a particular case, the Generalized Skew Goppa codes. By providing a new parity check matrix for the latter, we show that, under some hypotheses, they are subfield subcodes of Generalized Skew Reed--Solomon codes. This result turns out to be helpful to study the parameters of Skew Multivariate Goppa codes, for which we provide bounds on their dimension and minimum distance.

Problem

• Skew Multivariate Goppa codes
• minimum distance
• dimension
• subfield subcodes
• Ore polynomials

Innovation

• Skew Multivariate Goppa codes
• Ore polynomials
• subfield subcodes
• parity check matrix
• minimum distance bounds