Generalized BCH Codes and Twisted Goppa Codes Attaining Their Designed Distances

2026-07-20 • Information Theory

Information Theory
AI summaryⓘ

The authors studied a difficult problem in coding theory: figuring out the exact minimum distance of certain error-correcting codes called alternant codes. They focused on two types, generalized BCH codes and twisted Goppa codes, analyzing their parity-check matrices to understand their minimum distances. They found clear conditions under which these codes reach their expected minimum distances and showed that many generalized BCH codes actually do meet these distances, creating broad, infinite families instead of just single examples. For twisted Goppa codes, they identified when these codes have a minimum distance equal to a specific predictable value and described families that meet this condition. Overall, their work provides a better understanding and new examples of codes achieving their designed error-correcting capabilities.

alternant codeminimum distancegeneralized BCH codetwisted Goppa codeparity-check matrixerror-correcting codedesigned distancecoding theoryinfinite familiescode characterization
Authors
Yaqi Chen, Hao Chen, Cunsheng Ding, Huimin Lao, Chao Liu, Conghui Xie
Abstract
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 $Γ(L,g,η)$ with $\operatorname{deg} g=t$ satisfies $d(Γ(L,g,η))=t+1$, and derive structured classes and infinite families attaining this distance.
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