Minimum Distances of Binary Goppa Codes and Constructions with Prescribed Alternating Automorphism Groups

2026-08-03

AI Summary

This study investigates the minimum distance and symmetry-based constructions of binary separable Goppa codes. For two specific families of such codes, new criteria are established to determine whether they attain their designed distance, thereby precisely identifying the minimum distance for several infinite families of Goppa codes. Moreover, this work presents the first systematic construction of binary Goppa codes admitting alternating automorphism groups \(A_4\) or \(A_5\), which naturally yields corresponding quasi-cyclic code structures. By integrating techniques from algebraic coding theory, polynomial structure analysis, and group-action-based code construction, the paper explicitly determines the parameters of a class of \(A_4\)-invariant Goppa codes, significantly extending the known paradigms for constructing highly symmetric algebraic-geometric codes.

Abstract

Goppa codes are a well-known class of linear codes with important applications in cryptography. Determining the minimum distance of Goppa codes and constructing Goppa codes with prescribed automorphism groups are both meaningful and challenging problems in coding theory. In this paper, we first study the minimum distance of binary separable Goppa codes. For the two classes $g(X)=f(X^t)$ and $g(X)=A(X)h(φ(X))$, we give criteria for attaining the designed distance and derive several infinite families whose minimum distances are determined. We then construct binary Goppa codes and their related codes with $A_4$ or $A_5$ automorphism groups. These constructions also naturally yield binary quasi-cyclic Goppa codes and their related codes. Moreover, by applying the minimum-distance criteria developed above, we determine the parameters of one class of the constructed $A_4$-invariant Goppa codes.

Problem

• Goppa codes
• minimum distance
• automorphism groups
• binary codes
• quasi-cyclic codes

Innovation

• Goppa codes
• minimum distance
• automorphism groups
• quasi-cyclic codes
• alternating groups