A Deterministic Dimension Property of Twisted Goppa Codes

TL;DR

This study reveals a deterministic relationship between macro-parameters and the dimension of twisted Goppa codes through extensive computational analysis, enhancing understanding of their structural properties.

Contribution

It uncovers a unique deterministic regularity linking macro-parameters to the code dimension, based on large-scale computational analysis of over 50,000 cases.

Findings

Dimension k is uniquely determined by macro-parameters (q,m,t,b,u).

The dimension remains constant when macro-parameters are fixed.

Large-scale analysis confirms the regularity across many parameter sets.

Abstract

This paper presents a large-scale computational study on the dimensional properties of twisted Goppa codes. Through the systematic analysis of over 50,000 parameter sets, we uncover a remarkable deterministic regularity: the actual dimension k of a twisted Goppa code is uniquely determined by a set of macro-parameters (q,m,t,b,u). Specifically, when the order of the finite field q, the extension degree m, the degree t of the Goppa polynomial, the translation parameter b of the automorphism, and the order u of the transformation are fixed, the dimension k of the generated code remains constant.