On the Minimum Distances of Some Families of Goppa Codes and BCH Codes

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 $\delta=t+1$, 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 $d=\delta$, 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 $pt\mid m$.