AI 中文总结
本文研究二元可分Goppa码的最小距离判定问题，给出两类码达到设计距离的准则及若干确定最小距离的无限族，同时构造具有A₄或A₅自同构群的二元Goppa码及相关准循环码，并确定一类A₄-不变Goppa码的参数。
AI 中文摘要
Goppa码是一类著名的线性码，在密码学中具有重要应用。确定Goppa码的最小距离以及构造具有指定自同构群的Goppa码，都是编码理论中既有意义又具挑战性的问题。本文首先研究了二元可分Goppa码的最小距离。针对$g(X)=f(X^t)$和$g(X)=A(X)h(ϕ(X))$这两类码，我们给出了达到设计距离的判定准则，并推导出若干个最小距离已确定的无限族。随后，我们构造了具有$A_4$或$A_5$自同构群的二元Goppa码及其相关码，这些构造还自然地产生了二元准循环Goppa码及其相关码。此外，应用上述提出的最小距离判定准则，我们确定了一类所构造的$A_4$-不变Goppa码的参数。
英文摘要
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.
