Minimum Distances of Binary Goppa Codes and Constructions with Prescribed Alternating Automorphism Groups
2026-08-03 • Information Theory
Information Theory
AI summaryⓘ
The authors study a type of error-correcting codes called binary separable Goppa codes, which are important in cryptography. They focus on understanding the minimum distance of these codes, which tells how well the codes can detect and correct errors. They provide conditions for two specific classes of these codes to reach their designed minimum distances and identify infinite families where this holds true. Additionally, the authors construct Goppa codes with special symmetries (automorphism groups) related to groups named A4 and A5, and use their distance criteria to find parameters of some codes with A4 symmetry.
Goppa codesminimum distancebinary codesseparable polynomialsautomorphism groupsA4 groupA5 groupquasi-cyclic codescoding theorycryptography
Authors
Tianni He, Kangquan Li, Longjiang Qu
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.