Generalized BCH Codes and Twisted Goppa Codes Attaining Their Designed Distances

Determining the true minimum distance of an alternant code remains a notoriously difficult problem in coding theory.

In this paper, we study the minimum distances of generalized BCH codes and twisted Goppa codes through their parity-check matrices.

We first give a necessary and sufficient condition for an alternant code to attain its designed distance and apply it to generalized BCH codes.

As applications, we prove that broad classes of generalized BCH codes have minimum distances equal to their designed distances.

These classes provide explicit infinite families rather than isolated examples.

We characterize when a twisted Goppa code $\Gamma(L,g,\eta)$ with $\operatorname{deg} g=t$ satisfies $d(\Gamma(L,g,\eta))=t+1$, and derive structured classes and infinite families attaining this distance.