Goppa codes form an important class of alternant codes with wide applications in algebraic coding theory and code-based cryptography. Determining the true minimum distance of a Goppa code is a difficult problem. In this paper, we provide a necessary and sufficient criterion for a Goppa code to attain its designed distance $δ=t+1$, where $t$ is the degree of the Goppa polynomial. As applications, we determine the minimum distances of several classes of $q$-ary Goppa codes. In particular, we prove the tightness of the improved lower bound for a class of wild Goppa codes, and extend the family with $G(x)=x^t+A$ from the binary case to arbitrary odd prime powers. We then specialize the criterion to the monomial case $G(x)=x^t$, which is equivalent to primitive BCH codes. This leads to several infinite families of primitive BCH codes with $d=δ$, including the binary codes $\mathbf{C}_{(2,2^m-1,9,1)}$ and $\mathbf{C}_{(2,2^m-1,15,1)}$, the family $\mathbf{C}_{(p,p^p-1,2p+2,1)}$ with an odd prime $p$ and the family $\mathbf{C}_{(q,q^m-1,r\frac{q^m-1}{q-1}+1,1)}$ with $r\mid q-1$. In particular, we prove that the primitive BCH code $\mathbf{C}_{(q,q^m-1,q^t+1,1)}$ has minimum distance $q^t+1$ under the condition $t\mid m$, improving the previously known condition $pt\mid m$.

翻译：Goppa码是一类重要的交替码，在代数编码理论和基于编码的密码学中具有广泛应用。确定Goppa码的真实最小距离是一个难题。本文给出了Goppa码达到其设计距离$δ=t+1$的充要判据，其中$t$是Goppa多项式的次数。作为应用，我们确定了若干类$q$元Goppa码的最小距离。特别地，我们证明了一类野Goppa码改进下界的紧致性，并将族$G(x)=x^t+A$从二元情形推广到任意奇素幂情形。随后我们将该判据特化为单项式情形$G(x)=x^t$，该情形等价于原始BCH码。由此导出若干具有$d=δ$的原始BCH码无穷族，包括二元码$\mathbf{C}_{(2,2^m-1,9,1)}$和$\mathbf{C}_{(2,2^m-1,15,1)}$、奇素数$p$下的族$\mathbf{C}_{(p,p^p-1,2p+2,1)}$以及$r\mid q-1$时的族$\mathbf{C}_{(q,q^m-1,r\frac{q^m-1}{q-1}+1,1)}$。特别地，我们证明了在条件$t\mid m$下原始BCH码$\mathbf{C}_{(q,q^m-1,q^t+1,1)}$的最小距离为$q^t+1$，改进了此前要求的$pt\mid m$条件。